<Text-field style="Heading 1" layout="Heading 1">Introdu\303\247\303\243o</Text-field>

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C\303\241lculo Tensorial com Maple

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O subpacote Tensor do pacote DifferentialGeometry suplanta o hoje obsoleto pacote tensor no Maple. O pacote tensor fez essencial o uso do pacote hoje obsoleto linalg, ent\303\243o embora folhas de worksheets que usam esses pacotes obsoletos ainda funcionarem, \303\251 imperativa a mudan\303\247a para usar os novos formalismos no pacote Tensor.

Esse artigo \303\251 um "guia de sobreviv\303\252ncia" para implementar o c\303\241lculo tensorial no novo pacote Tensor. Ele explica as constru\303\247\303\265es nesse pacote da perspectiva da nota\303\247\303\243o cl\303\241ssica (i.e., indicial) de tensor.

O pr\303\263prio pacote DifferentialGeometry cont\303\251m 34 comandos mas tamb\303\251m seis subpacotes, eles mesmo contribuindo para um total de 184 comandos. Esses subpacotes, e uma medida de seus "tamanhos" est\303\243o listados na tabela 1.

Subpacote

N\303\272mero de Comandos

Tensor

Tools

JetCalculus

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEjMjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGLw==

GroupActions

Library

LieAlgebras

Tabela 1 Subpacotes no pacote DifferentialGeometry

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A complexidade de dominar o pacote DifferentialGeometry aumenta mais quando subpacotes tais como GroupActions possuem eles mesmo subpacotes (MovingFrames). Um tutorial compreens\303\255vel no pacote completo DifferentialGeometry exigiria mais que um livri texto, mas claramente, essa n\303\243o \303\251 nossa ambi\303\247\303\243o aqui. Felizmente, o pr\303\263prio pacote cont\303\251m dois conjuntos de tutoriais muito \303\272teis, uma cole\303\247\303\243o compreens\303\255vel de Lessons (Li\303\247\303\265es), e um conjunto de Tutorials (Tutoriais). As folhas de trabalho Lessons fornecem uma abordagem sistem\303\241tica para o aprendizado dos comandos nos subpacotes DifferentialGeometry, Tensor, LieAlgebras e JetCalculus. Cada li\303\247\303\243o tamb\303\251m cont\303\251m um conjunto de exerc\303\255cios que possuem graus de dificuldade que v\303\243o de simples computa\303\247\303\265es a problemas de programa\303\247\303\243o. Solu\303\247\303\265es s\303\243o dadas. Os tutoriais apresentam aplica\303\247\303\265es especializadas do pacote DifferentialGeometry.

Nossa meta mais modesta para esse artigo \303\251 mostrar como inscrever tensores covariantes e contravariantes, computar suas derivadas covariantes, obter as equa\303\247\303\265es de transporte paralelo e geod\303\251sica, e computar o tensor b\303\241sico de relatividade geral.

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<Text-field style="Heading 1" layout="Heading 1">Inicializa\303\247\303\265es</Text-field>

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

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

<Text-field style="Heading 1" layout="Heading 1">Declarando a Constru\303\247\303\243o em DifferentialGeometry</Text-field>

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Antes que um vetor ou tensor seja inscrito, um frame (estrutura) precisa ser declarado declarando sua vari\303\241veis e dando a ele um nome. Por exemplo, para declarar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 como o espa\303\247o Cartesiano com vari\303\241veis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, execute

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

SS9mcmFtZX5uYW1lOn5SMkc2Ig==

O comportamento padr\303\243o para DifferentialGeometry \303\251 tal que desse ponto em diante, o prompt poderia ser modificado para exibir o nome da estrutura enquanto a estrutura estivesse ativa. Desse modo, sem que tenhamos suspendido esse padr\303\243o com o comando Preferences na se\303\247\303\243o Inicializa\303\247\303\265es, o prompt seguiria o comando DGsetup poderia ser o mostrado na Tabela 2.

Tabela 2 A modifica\303\247\303\243o padr\303\243o do prompt

N\303\263s escolhemos suspender esse comportamento padr\303\243o por tr\303\252s raz\303\265es. Primeiro, a edi\303\247\303\243o interativa de uma folha de trabalho pode criar uma exibi\303\247\303\243o confusa. Se em uma se\303\247\303\243o onde um nome de estrutura aparece em todos os prompts, uma nova estrutura \303\251 definida, os prompts subsequentes que mostram o nome anterior n\303\243o mudam para mostrar o novo nome. Somente um "novo" prompt ir\303\241 exibir o novo nome da estrutura. O resultado \303\251 uma folha de trabalho com prompts mostrando diferentes nomes de estruturas onde eles podem n\303\243o ser relevantes. Tais prompts extraviados devem ser deletados manualmente se eles n\303\243o est\303\243o fornecendo informa\303\247\303\265es corretas.

Segundo, esses prompts modificados s\303\243o persistentes - eles n\303\243o podem ser removidos por nenhum comando do Maple. Eles devem ser removidos por elimina\303\247\303\243o. (O comando interface do Maple que modifica prompts n\303\243o cascata a mudan\303\247a atrav\303\251s de prompts existentes. Ele somente modifica prompts novos.)

Finalmente, se os comandos s\303\243o executados em Document Blocks (Blocos de Documento), e n\303\243o em prompts, n\303\243o haver\303\241 prompt vis\303\255vel para modificar. Desse modo, o prompt modificado \303\251 uma folha de trabalho paradigma que n\303\243o carrega todo o uso do Maple. Por todas essas raz\303\265es, n\303\243o teremos nomes de estruturas vis\303\255veis em nossos prompts.

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<Text-field style="Heading 1" layout="Heading 1">Os Paradigmas Covariante/Contravariante</Text-field>

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Qualquer um que teve mesmo um mergulho superficial nas \303\241guas do c\303\241lculo tensorial sabe que existem duas palavras, covariante e contravariante, que precisam ser enfrentadas. N\303\243o seremos capazes de inscrever um tensor no pacote Tensor sem fazer a distin\303\247\303\243o entre esses dois termos. Usando a convens\303\243o do somat\303\263rio de Einstein (\303\255ndices repetidos, um elevado e um baixo, s\303\243o somados), a Tabela 3 define vetores contravariantes e covariantes.

Tipo de Vetor

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiVkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvRjZRJ25vcm1hbEYn

Tipo de Base

Lei de Transforma\303\247\303\243o

Contravariante

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JJW1zdWJHRiQ2JS1GLzYnUSJlRicvJSVib2xkR0Y0L0YzUSZmYWxzZUYnL0Y2USVib2xkRicvJStmb250d2VpZ2h0R0ZLRjgvJS9zdWJzY3JpcHRzaGlmdEdGPy9GNlEnbm9ybWFsRic=

Tangentee \303\240 curvas coordenadas

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

Covariante

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSVtc3VwR0YkNiUtRi82J1EiZUYnLyUlYm9sZEdGNC9GM1EmZmFsc2VGJy9GNlElYm9sZEYnLyUrZm9udHdlaWdodEdGS0Y4LyUxc3VwZXJzY3JpcHRzaGlmdEdGPy9GNlEnbm9ybWFsRic=

Gradientes (Normais \303\240 superf\303\255cies coordenadas)

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

Tabela 3 Vetores contravariante e covariante

Na coluna mais \303\240 direita da Tabela 3 se usa a nota\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn para componentes expressos no sistema coordenado LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, mas LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImtGJ0YyRjUtSSNtb0dGJDYtUSInRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnRjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQQ== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImtGJ0YyRjUtSSNtb0dGJDYtUSInRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnRjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkE= para componentes no sistema coordenado LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Livros tamb\303\251m denotam o novo sistema coordenado pelo uso de uma barra sobre a componente, ou uma letra mai\303\272scula \303\240 esquerda da vari\303\241vel.

Os componentes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn s\303\243o os componentes contravariante e covariante, respectivamente, do vetor V. Os vetores base LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRJ25vcm1hbEYn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GOVEnbm9ybWFsRic= s\303\243o rec\303\255procos, tal que 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. Desse modo, uma base ortonormal \303\251 auto-rec\303\255proca. Isso porque a distin\303\247\303\243o entre bases contravariante e covariante n\303\243o \303\251 importante em espa\303\247os Cartesianos.

Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy \303\251 o mapeamento de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJy9GNlEnbm9ybWFsRic= via fun\303\247\303\265es da forma LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EidUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYvUSI9RidGL0Y1RjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZORistSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYnUSJ4RidGL0YyRjVGOC9GNlEnbm9ybWFsRidGWUZZ, ent\303\243o os vetores gradientes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GOVEnbm9ybWFsRic= s\303\243o as linhas da matriz Jacobiana 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, onde o \303\255ndice superior LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 interpretado como um \303\255ndice de kinha, e o \303\255ndice inferior LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, como um \303\255ndice de coluna.

Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYkUSIxRidGNS9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNQ== \303\251 o mapeamento de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= via fun\303\247\303\265es da forma LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYvUSI9RidGL0Y1RjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZORistSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYnUSJ1RidGL0YyRjVGOC9GNlEnbm9ybWFsRidGWUZZ, ent\303\243o os vetores tangentes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRJ25vcm1hbEYn s\303\243o as colunas da matriz Jacobiana 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.

JSFH

Para facilitar a implementa\303\247\303\243o da lei de transforma\303\247\303\243o do contravariante, escrevendo os componentes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= como um vetor coluna v significa que somas com a matriz Jacobiana s\303\243o feitas ao longo de uma linha e atravessando as colunas. Portanto, a matriz produto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYnUSJKRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPy1GIzYlLUY0NiVRJyYjOTM0O0YnRjovRj5RJ25vcm1hbEYnL0Y7RjkvRj5RJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGR0ZHLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUY0NidRInZGJ0Y3RjpGPUZARkc= implementa a transforma\303\247\303\243o contravariante. Escrevendo o componente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn como o vetor linha w significa que somas com a matriz Jacobiana s\303\243o descendo uma coluna mas atravessando as linhas. Portanto, a matriz produto 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 implementa a transforma\303\247\303\243o covariante. Essa distin\303\247\303\243o inerente entre bases tangentes e bases normais indeuz a distin\303\247\303\243o entre contravariante e covariante. Usando vetores linha e coluna para expressar essa diferen\303\247a \303\251 um dispositivo visual conveniente em c\303\241lculo tensorial cl\303\241ssico.

<Text-field style="Heading 1" layout="Heading 1">O Tensor como um Objeto Multilinear</Text-field>

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Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 um espa\303\247o vetorial, digamos, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 com bases 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, ent\303\243o um tensor classe-dois \303\251 um objeto multilinear LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RLCZiaWdvdGltZXM7RicvJSVzaXplR1EiOEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZPRitGPA==, o produto direto de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= com ele mesmo, possuindo objetos base indexados duplamente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYnUSJlRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPy1GIzYlLUY0NiVRI2lqRicvRjtGOS9GPlEnaXRhbGljRidGR0ZILyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRj5RJ25vcm1hbEYnRk0vJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRk0=. O tensor \303\251 atualmente o objeto

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linear em ambos \303\255ndices. \303\211 claro, os LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn s\303\243o os componentes contravariantes do tensor; e assim como para vetores, haveriam os componentes covariantes equivalentes, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==. Existem tamb\303\251m tensores mistos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImpGJ0YyRjVGMkY1LUYjNiUtRi82JVEiaUYnRjJGNUYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkQvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnL0Y2USdub3JtYWxGJw== que transformam contravari\303\242ncia em um \303\255ndice e covari\303\242ncia em outro.

Na pr\303\241tica, um manipula somente os componentes do tensor, e quase nunca exibe explicitamente os objetos base. Entretanto, no pacote DifferentialGeometry, vetores e tensores exigem um uso expl\303\255cito dos objetos base.

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<Text-field style="Heading 1" layout="Heading 1">Bases e Seus Duais</Text-field>

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A base 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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 poderia ser inscrita como

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

NyQtSSRfREdHNiI2IzckNyVRJ3ZlY3RvckYlSSNSMkdGJTciNyM3JDcjIiIiRi8tRiQ2IzckRig3IzckNyMiIiNGLw==

ou poderia ser extra\303\255da do Maple com o comando DGinfo do pacote Tools.

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(Para digitar o sublinhado no modo matem\303\241tica (2D), pressione o caractere de escape (\134) primeiro. Alternativamente, entre tal express\303\243o em modo texto (linear, 1D) e converta para modo matem\303\241tica via Menu Contexto.)

A base rec\303\255proca (ou dual) 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 \303\251 ent\303\243o

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Na realidade, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn \303\251 considerado uma forma diferencial, mais de acordo com a concep\303\247\303\243o moderna da geometria diferencial.

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<Text-field style="Heading 1" layout="Heading 1">Representando Vetores como Objetos de DifferentialGeometry </Text-field>

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No pacote DifferentialGeometry, o vetor contravariante cujos componentes s\303\243o 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 \303\251 dado por

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkjUjJHRiQ3IjckNyQ3IyIiIi1JIkFHRiQ2JEkieEclKnByb3RlY3RlZEdJInlHRjM3JDcjIiIjLUkiQkdGJEYx

ou por

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEmREd6aXBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYqLUZXNiYtRiM2KUYrLUYjNictRiw2JVEiQUYnRjZGOUY8LUZXNiQtRiM2Jy1GLDYlUSJ4RidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ5RidGNkY5LyUrZXhlY3V0YWJsZUdGREZARkBGXnBGQEZlby1GIzYnLUYsNiVRIkJGJ0Y2RjlGPEZeb0ZecEZARitGXnBGQEZALyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmVvLUYsNiVRI0IxRidGNkY5RmVvLUkjbXNHRiQ2I1ElcGx1c0YnRitGXnBGQEZARl5wRkBGK0ZecEZARitGXnBGQA==

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkjUjJHRiQ3IjckNyQ3IyIiIi1JIkFHRiQ2JEkieEclKnByb3RlY3RlZEdJInlHRjM3JDcjIiIjLUkiQkdGJEYx

Os comandos evalDG e DGzip s\303\243o os dois caminhos mais simples de criar um objeto cuja estrutura de informa\303\247\303\243o \303\251 intr\303\255nseca ao pacote DifferentialGeometry. Quando se usa o comando evalDG, o asterisco \303\251 o operador multiplicativo expl\303\255cito. Se esse vetor tiver sido inscrito em modo matem\303\241tica, o eco do asterisco seria um ponto centralizado. A alternativa ao asterisco seria o espa\303\247o.

O vetor covariante cujas componentes s\303\243o 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 \303\251 isncrito como

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

LUkkX0RHRzYiNiM3JDclUSVmb3JtRiRJI1IyR0YkIiIiNyQ3JDcjRiotSSJhR0YkNiRJInhHJSpwcm90ZWN0ZWRHSSJ5R0YyNyQ3IyIiIy1JImJHRiRGMA==

ou como

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

LUkkX0RHRzYiNiM3JDclUSVmb3JtRiRJI1IyR0YkIiIiNyQ3JDcjRiotSSJhR0YkNiRJInhHJSpwcm90ZWN0ZWRHSSJ5R0YyNyQ3IyIiIy1JImJHRiRGMA==

O que n\303\243o parece ser um modo simples de representar um vetor como uma coluna ou vetor linha. O subpacote Tools fornece o comando DGinfo com o qual as componentes de um vetor podem ser extra\303\255dos. Seu uso \303\251 ilustrado por

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

NyQ3JDcjIiIiLUkiQUc2IjYkSSJ4RyUqcHJvdGVjdGVkR0kieUdGKzckNyMiIiMtSSJCR0YoRik=

ou por

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

NyQtSSJBRzYiNiRJInhHJSpwcm90ZWN0ZWRHSSJ5R0YoLUkiQkdGJUYm

<Text-field style="Heading 1" layout="Heading 1">Representando Tensores como Objetos de DifferentialGeometry </Text-field>

JSFH

Um tensor contravariante classe dois poderia ser isncrito como

JSFH

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkjUjJHRiQ3JDckUShjb25fYmFzRiRGLDciNyY3JDckIiIiRjFJJFQxMUdGJDckNyRGMSIiI0kkVDEyR0YkNyQ3JEY1RjFJJFQyMUdGJDckNyRGNUY1SSRUMjJHRiQ=

O construto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEkRF94RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIiZGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRInRGJ0YvRjJGNUYrRjk= corresponde ao elemento de base di\303\241ticaca LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1JI21uR0YkNiRRIjFGJy9GOVEnbm9ybWFsRicvRjZGNC9GOVEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRkM=, etc.

Um tensor covariante classe dois poderia ser inscrito como

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkjUjJHRiQ3JDckUShjb3ZfYmFzRiRGLDciNyY3JDckIiIiRjFJJHQxMUdGJDckNyRGMSIiI0kkdDEyR0YkNyQ3JEY1RjFJJHQyMUdGJDckNyRGNUY1SSR0MjJHRiQ=

O construto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEiJkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVEidEYnRi9GMkY1RitGOQ== corresponde ao elemento de base di\303\241tica LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1JI21uR0YkNiRRIjFGJy9GOVEnbm9ybWFsRicvRjZGNC9GOVEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGQw==, etc.

Os componentes de um tensor podem ser recuperados com o comando DGinfo

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

NyY3JDckIiIiRiVJJFQxMUc2IjckNyRGJSIiI0kkVDEyR0YnNyQ3JEYqRiVJJFQyMUdGJzckNyRGKkYqSSRUMjJHRic=

ou com o comando convert.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1GLDYlUShjb252ZXJ0RidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRJ1QyX2NvbkYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRKERHQXJyYXlGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZccEZARitGXHBGQEYrRlxwRkBGK0ZccEZA

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

Os componentes do tensor classe dois s\303\243o frequentemente representados como as entradas de uma matriz. Se uma matriz \303\251 usada para tal representa\303\247\303\243o, \303\251 poss\303\255vel converter a matriz para um tensor de DifferentialGeometry.

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkjUjJHRiQ3JDckUShjb25fYmFzRiRGLDciNyY3JDckIiIiRjFJJFQxMUdGJDckNyRGMSIiI0kkVDEyR0YkNyQ3JEY1RjFJJFQyMUdGJDckNyRGNUY1SSRUMjJHRiQ=

<Text-field style="Heading 1" layout="Heading 1">Contra\303\247\303\243o de \303\215ndices</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Dados dois tensores de dimens\303\265es confort\303\241veis, digamos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lrRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2prRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==, formando a soma dos produtos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lrRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSVtc3ViR0YkNiUtRi82JVEiVkYnRjJGNS1GIzYlLUYvNiVRI2prRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R0Y/L0Y2USdub3JtYWxGJw== \303\251 a opera\303\247\303\243o contra\303\247\303\243o dos \303\255ndices. Recorde que a conven\303\247\303\243o do somat\303\263rio de Einstein indica uma soma de mesmo \303\255ndice quando ele aparece uma vez levantado e uma vez abaixado. Essa opera\303\247\303\243o, implementada no pacote Tensor com o comando ContractIndices, \303\251 ilustrada para os dois tensores

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da se\303\247\303\243o anterior. Para simplificar a informa\303\247\303\243o de entrada, renomeamos a primeira como T e a segunda, como V.

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As quatro poss\303\255veis contra\303\247\303\265es que resultam em um tensor classe dois s\303\243o dadas na Tabela 4, onde seus componentes s\303\243o exibidos como elementos de matrizes.

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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Tabela 4 Quatro tensores classe dois formados pela contra\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2ttRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==

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Uma inspe\303\247\303\243o cuidadosa dos quatro tensores na Tabela 4 (veja especialmente a coluna mais \303\240 direita) mostra que todos s\303\243o diferentes.

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<Text-field style="Heading 1" layout="Heading 1">O Tensor M\303\251trico</Text-field>

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A geometria de uma multiplicidade \303\251 capturada primeiro no tensor covariante m\303\251trico 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 ou sua contraparte contravariante or 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. N\303\243o existe "c\303\241lculo" em c\303\241lculo tensorial sem primeiro obter esse tensor essencial. Portanto, \303\251 imperativo que existem maneiras eficientes de obter esse tensor. V\303\241rias dessas t\303\251cnicas ser\303\243o ilustradas para o plano Cartesiano no qual coordenadas polares foram impostas.

<Text-field style="Heading 2" layout="Heading 2">M\303\251todo 1 - Obter <Equation executable="false" style="Heading 2" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZ0YnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEjaWpGJ0YyRjVGOEYyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YyL0Y5USdub3JtYWxGJw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZ0YnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEjaWpGJ0YyRjVGOEYyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YyL0Y5USdub3JtYWxGJw==</Equation> como uma Matriz e Converter para um Tensor</Text-field>

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Defina o mapa LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYkUSIxRidGNS9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNQ== com equa\303\247\303\265es da forma LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUEYrLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2J1EidUYnRi9GMkY1RjhGPkY+Rj4= via

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e o vetor raio (posi\303\247\303\243o) via

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Ent\303\243o, uma representa\303\247\303\243o dos vetores base 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 \303\251 dado por

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de modo que uma matriz cujas entradas s\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw== \303\251

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

Para converter isso no tensor m\303\251trico em coordenadas polares, precisamos definir a estrutura polar com

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

em cujo caso caso n\303\263s ent\303\243o temos

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyRRKGNvdl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpSSJyRyUqcHJvdGVjdGVkR0Y0RjE=

A forma covariante desse tensor m\303\251trico, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, pode ser obtido com

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyRRKGNvbl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpSSJyRyUqcHJvdGVjdGVkR0Y0ISIi

A matriz representando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn \303\251 a matriz inversa da matriz representando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==.

<Text-field style="Heading 2" layout="Heading 2">M\303\251todo 2 - Tranformar o Tensor M\303\251trico Cartesiano</Text-field>

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Come\303\247e com o tensor m\303\251trico contravariante no espa\303\247o Cartesiano LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1:

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Defina a transforma\303\247\303\243o de Cartesiano para coordenadas polares como

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LUkkX0RHRzYiNiM3JDcmUS90cmFuc2Zvcm1hdGlvbkYkNyQ3JEkiUEdGJCIiITckSSNSMkdGJEYsNyI3Iy1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjL0kkJWlkR0YkIipja2R4IjckNyQqJkkickdGNCIiIi1JJGNvc0dGMzYjSSZ0aGV0YUdGNEY+SSJ4R0Y0NyQqJkY9Rj4tSSRzaW5HRjNGQUY+SSJ5R0Y0

e use isso para converter a m\303\251trica Euclidiana para

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<Text-field style="Heading 1" layout="Heading 1">Levantando e Abaixando \303\215ndices</Text-field>

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O mesmo vetor V pode ser dado com respeito aos vetores base tangentes naturais LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRJ25vcm1hbEYn ou com respeito \303\240 base rec\303\255proca de vetores de base gradiente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GOVEnbm9ybWFsRic=. Assim, tem-se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2J1EiVkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1JJW1zdXBHRiQ2JS1GLDYlUSJ2RicvRjNGMS9GNlEnaXRhbGljRictRiM2JS1GLDYlUSJrRidGWUZaRllGWi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSVtc3ViR0YkNiUtRiw2J1EiZUYnRi9GMkY1RjhGZm4vJS9zdWJzY3JpcHRzaGlmdEdGXW9GOi1GX282JUZWRmZuRmRvLUZUNiVGYW9GZm5GW29GPg==. A conven\303\247\303\243o entre componentes contravariante e covariante de V \303\251 efetuada por contra\303\247\303\243o com o tensor m\303\251trico:

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<Text-field style="Heading 2" layout="Heading 2">Exemplo 1</Text-field>

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Dado o vetor contravariante

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Os componentes covariantes s\303\243o dados pelo comando RaiseLowerIndices no pacote Tensor. Desse modo, temos

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onde "g" denota o tensor m\303\251trico covariante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==. Alternativamente, o tensor covariante

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os componentes contravariantes s\303\243o

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyNRKGNvbl9iYXNGJDciNyQ3JDcjIiIiLUkiYUdGJDYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRjY3JDcjIiIjKiYpRjVGOiEiIi1JImJHRiRGNEYx

onde "LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=" denota o tensor m\303\251trico contravariante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Os Coeficientes de Conex\303\243o</Text-field>

JSFH

Uma vez que o tensor \303\251 conhecido, o modo como os vetores base mudam de ponto a ponto pode ser determinado. Acontece que a taxa de varia\303\247\303\243o dos vetores base podem ser expressos como combina\303\247\303\265es lineares desse mesmos vetores. Os coeficientes dessas combina\303\247\303\265es lineares s\303\243o chamados de coeficientes de conex\303\243o, ou de S\303\255mbolos de Christoffel. Dependendo da forma usada para esses s\303\255mbolos, eles s\303\243o chamados s\303\255mbolos de Christoffel de primeiro tipo ou de segundo tipo. Desde que exista um comando embutido que providencie os s\303\255mbolos de Christoffel de segundo tipo, n\303\263s os obteremos via

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

LUkkX0RHRzYiNiM3JDclUStjb25uZWN0aW9uRiRJIlBHRiQ3JDclUShjb3ZfYmFzRiRRKGNvbl9iYXNGJEYsNyI3JTckNyUiIiIiIiNGMyokSSJyRyUqcHJvdGVjdGVkRyEiIjckNyVGM0YyRjMsJEY1Rjc3JDclRjNGM0YyRjQ=

Livros cl\303\241ssicos em geometria diferencial usam a nota\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEnJiM5MTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRI2prRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LUYjNiUtRi82JVEiaUYnRj1GP0Y9Rj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkgvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnRjU=, 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, ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSNqa0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiaUYnRjRGN0Y+Rj4vJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRj4= para s\303\255mbolos de Christoffel de segundo tipo, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSNqa0YnRjRGN0Y+Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPg== or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSNqa0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiaUYnRjRGN0Y+Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPg== para s\303\255mbolos de Christoffel de primeiro tipo. Consequentemente, a interpreta\303\247\303\243o correta da sa\303\255da do comando Christoffel \303\251 capturada, \303\241rdua e trabalhosamente, na Tabela 5.

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

Tabela 5 S\303\255mbolos de Christoffel de segundo tipo para coordenadas polares

O mais pr\303\263ximo que o pacote Tensor chega para articular os s\303\255mbolos de Christoffel \303\251

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

NyU3JDclIiIiIiIjRiYqJEkickclKnByb3RlY3RlZEchIiI3JDclRiZGJUYmLCRGKEYqNyQ3JUYmRiZGJUYn

onde a lista LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNigtSSNtaUdGJDYlUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJqRidGNEY3RjotRjE2JVEia0YnRjRGN0Y+Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPg== mapeia para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEnJiM5MTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRI2lrRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LUYjNiUtRi82JVEiakYnRj1GP0Y9Rj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkgvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnRjU=. Desse modo, o \303\255ndice do meio na lista \303\251 aumentado uma vez, e o primeiro e o terceiro s\303\243o os menores.

<Text-field style="Heading 2" layout="Heading 2">Exemplo 2</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Para coordenadas polares, mostre que

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

Portanto, mostre que a derivada do vetro base pode ser expressa como uma combina\303\247\303\243o linear dos vetores base, e que os coeficientes dessas combina\303\247\303\265es lineares s\303\243o os s\303\255mbolos de Christoffel.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Para mostrar que derivadas de vetores base podem ser expressas como combina\303\247\303\265es lineares de vetores base, comece expressando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYnUSJpRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPC1JI21vR0YkNi1RIixGJy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYnUSJqRidGNEY3RjpGPUZDRkMvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRkM= em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiYtSSVtc3ViR0YkNiUtSSNtaUdGJDYnUSJlRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y+USdub3JtYWxGJy9GO0Y5L0Y+USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRkgvJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlRjMtRiM2JS1GRTYkUSIyRidGSEZKRktGTUZIRkgvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnLUY0NiNRIUYnRkg=. Isso \303\251 feito na Tabela 6 onde os s\303\255mbolos i e j s\303\243o introduzidos explicitamente e o confrontos de nomea\303\247\303\265es s\303\243o evitadas pela introdu\303\247\303\243o dos nomes alternados 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 para 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.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCQvSSJpRzYiLCQqJiwmKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmdGhldGFHRi0iIiImSSJlR0YlRi9GMUYxKihJInJHRi1GMSZGMzYjRjVGMS1JJGNvc0dGLEYvRjEhIiJGMUY1RjpGOi9JImpHRiUqJkY1RjosJiooRjVGMUYqRjFGNkYxRjEqJkYyRjFGOEYxRjFGMQ==

Tabela 6 Expressando 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 em termos de 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

A Tabela 7, que lista as derivadas dos vetores base, mostra que os s\303\255mbolos de Christoffel s\303\243o de fato os coeficientes na combina\303\247\303\243o linear dos vetores base que expressam as derivadas dos vetores base.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIh

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

LCQqJkkickclKnByb3RlY3RlZEciIiImSSJlRzYiNiNGJEYmISIi

Tabela 7 Diferencia\303\247\303\243o de vetores base em coordenadas polares

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A Tabela 8 fornece uma f\303\263rmula para computar os s\303\255mbolos de Christoffel de segundo tipo para os componentes do tensor m\303\251trico.

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Tabela 8 S\303\255mbolos de Christoffel em termos do tensor m\303\251trico

JSFH

Um atalho pr\303\241tico notacional \303\251 o uso de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbW9HRiQ2LVEnJnBhcnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjExMTExMTFlbUYnLUYjNiUtSSNtaUdGJDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnRlBGUy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjI= para o operador de diferencia\303\247\303\243o 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. A f\303\263rmula na Tabela 8 seria mais f\303\241cil de escrever (e de lembrar) se esse dispositivo notacional \303\251 usado.

<Text-field style="Heading 1" layout="Heading 1">A Derivada Covariante</Text-field>

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Em c\303\241lculo, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYnUSJ1RicvJSVib2xkR1EldHJ1ZUYnRjIvRjZRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkEvRjNGPy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYvNiVRImZGJ0ZERkVGNUY1LUYvNiNRIUYnRjU= \303\251 a derivada direcional da fun\303\247\303\243o escalar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tomada na dire\303\247\303\243o do vetor unit\303\241rio u. O vetor gradiente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEmJkRlbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8= surge naturalmente do c\303\241lculo da derivada

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que \303\251 como a derivada direcional \303\251 definida quando 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiUEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1JI21uR0YkNiRRIjBGJy9GOVEnbm9ybWFsRicvRjZGNC9GOVEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdGQi1JI21vR0YkNi1RIj1GJ0ZDLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGaG4tRi82JVEieEYnRkVGRi1GSzYtUSJ+RidGQ0ZORlBGUkZURlZGWEZaL0ZnblEmMC4wZW1GJy9Gam5GYm8tRi82J1EiaUYnRjJGNUY4RjstRks2LVEiK0YnRkNGTkZQRlJGVEZWRlhGWi9GZ25RLDAuMjIyMjIyMmVtRicvRmpuRltwLUYvNiVRInlGJ0ZFRkZGXm8tRi82J1EiakYnRjJGNUY4RjtGQw==, e 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. \303\211 claro, esse c\303\241lculo se extende para dimens\303\265es maiores. Mas mais importante, note como come\303\247ando como uma fun\303\247\303\243o escalar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, o vetor quantidade LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEmJkRlbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8= deve ser definido, e a derivada direcional desejada \303\251 um produto pontual desse vetor gradiente com o vetor dire\303\247\303\243o.

A derivada covariante surge da mesma maneira, isto \303\251, da defini\303\247\303\243o de uma derivada direcional de um vetor. O novo objeto que deve ser criado \303\251 a derivada covariante, um tensor classe dois, e a taxa de varia\303\247\303\243o atual do vetor em uma dada dire\303\247\303\243o \303\251 o "produto pontual" desse novo tensor com um vetor especificando a dire\303\247\303\243o. Em particular, a Tabela 9 d\303\241 as express\303\265es da derivada covariante de vetores covariantes e contravariantes, um tensor misto, e o tensor m\303\251trico.

Vetor Contravariante

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Vetor Covariante

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Tensor misto

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Tensor m\303\251trico

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Tabela 9 F\303\263rmulas para derivadas covariantes

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Em coordenadas polares, derivadas covariantes de vetores covariantes e contravariantes

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LUkkX0RHRzYiNiM3JDclUSVmb3JtRiRJIlBHRiQiIiI3JDckNyNGKi1JImFHRiQ2JEkickclKnByb3RlY3RlZEdJJnRoZXRhR0YyNyQ3IyIiIy1JImJHRiRGMA==

respectivamente, s\303\243o obtidas no pacote Tensor com o comando CovariantDerivative. Isso requer como segundo argumento os coeficientes de conex\303\243o, que podem ser dados tanto como s\303\255mbolos de Christoffel (de segundo tipo)

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ou como a conex\303\243o

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(No pacote Tensor, os s\303\255mbolos de Christoffel n\303\243o est\303\243o seoparados. O argumento para o comando Connection \303\251 a mesma soma de termos que o pacote Tensor usa para expressar os s\303\255mbolos de Christoffel.)

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A derivada covariante do vetor contravariante \303\251

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mais facilmente lida como a matriz

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A derivada covariante do vetor covariante \303\251

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mais facilmente lida como a matriz

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<Text-field style="Heading 1" layout="Heading 1">A Derivada Covariante Direcional</Text-field>

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A derivada covariante 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 \303\251 o tensor classe dois que surge quando a derivada direcional do vetor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= \303\251 definida. Como tal, ela \303\251 a generaliza\303\247\303\243o do vetor gradiente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEmJkRlbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8= que surge quando a derivada direcional da fun\303\247\303\243o escalar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 definida. Assim como a derivada direcional da fun\303\247\303\243o escalar requer um produto pontual com um adire\303\247\303\243o vetorial, o mesmo acontece com a derivada do vetor. Portanto, a derivada direcional do vetor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= na dire\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRImpGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjU= \303\251 dada por 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, a contra\303\247\303\243o da derivada covariante com o vetor dire\303\247\303\243o.

Esse c\303\241lculo \303\251 implementado no pacote Tensor pelo comando DirectionalCovariantDerivative.

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<Text-field style="Heading 2" layout="Heading 2">Exemplo 3</Text-field>

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A derivada do vetor covariante

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSdwdl9jb25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLUkiQUdGJDYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRjM3JDcjIiIjLUkiQkdGJEYx

na dire\303\247\303\243o do vetor

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiSSJ1R0YkNyQ3IyIiI0kidkdGJA==

\303\251 dada por

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyNRKGNvbl9iYXNGJDciNyQ3JDcjIiIiLCgqKEkickclKnByb3RlY3RlZEdGMUkidkdGJEYxLUkiQkdGJDYkRjRJJnRoZXRhR0Y1RjEhIiIqJkkidUdGJEYxLUklZGlmZkdGNTYkLUkiQUdGJEY5RjRGMUYxKiZGNkYxLUY/NiRGQUY6RjFGMTckNyMiIiMqJiwqKiZGPUYxRjdGMUYxKiZGNkYxRkFGMUYxKihGPUYxLUY/NiRGN0Y0RjFGNEYxRjEqKEY2RjEtRj82JEY3RjpGMUY0RjFGMUYxRjRGOw==

Se n\303\263s escrevermos a derivada covariante como a matriz

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LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKT9ZVCYp

e escrevermos a derivada covariante direcional como o vetor coluna

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LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpL1tUJik=

podemos ver mais facilmente que a soma em 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 \303\251 bem modelada pelo produto da matriz pelo vetor dire\303\247\303\243o.

<Text-field style="Heading 1" layout="Heading 1">Paralelismo e Geod\303\251sicas</Text-field>

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O vetor contravariante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= definido ao longo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, a curva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUEYrLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEicEYnL0YzRjEvRjZRJ2l0YWxpY0YnRj5GPkY+, \303\251 dito ser paralelamente ao longo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se

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

a derivada absoluta (ou intr\303\255seca) ao longo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= desaparece.

Defini\303\247\303\243o 1 Paralelismo ao longo de uma curva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

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O comando ParallelTransportEquations no pacote Tensor gera as equa\303\247\303\265es impl\303\255citas pela Defini\303\247\303\243o 1.

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<Text-field style="Heading 2" layout="Heading 2">Exemplo 4</Text-field>

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Em coordenadas polares LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUScmIzk1MjtGJy9GNUZCRj5GPkY+Rj4=, deixe a curva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ser dada por

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

NyQqJClJInBHNiIiIiMiIiIsJiomRidGKEYlRihGKCIiJEYo

O vetor contravariante

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLUkiQUdGJDYjSSJwR0YkNyQ3IyIiIy1JIkJHRiRGMQ==

\303\251 paralelo ao longo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se as equa\303\247\303\265es impl\303\255citas por

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLCYqKCIiI0YuLUkiQkdGJDYjSSJwR0YkRi4pRjVGMUYuISIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIkFHRiRGNEY1Ri43JDcjRjEqJiwoKihGMUYuRjJGLkY1Ri5GLiomRjFGLkY8Ri5GLiomLUY5NiRGMkY1Ri5GNkYuRi5GLkY2Rjc=

s\303\243o v\303\241lidas. Para acessar as componentes desse vetor e escrever as equa\303\247\303\265es individuais, use

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

LywmKigiIiMiIiItSSJCRzYiNiNJInBHRilGJilGK0YlRiYhIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiQUdGKUYqRitGJiIiIQ==

LywoKigiIiMiIiItSSJCRzYiNiNJInBHRilGJkYrISIiRiYqKEYlRiYtSSJBR0YpRipGJilGK0YlRixGJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGJ0YrRiYiIiE=

onde a sintaxe estra isola as duas equa\303\247\303\265es em uma \303\272nica ordem. O Maple fornece uma solu\303\247\303\243o para essas equa\303\247\303\265es e as condi\303\247\303\265es iniciais LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUYsNiVRIkJGJ0YvRjJGNUZARjpGPg==.

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

LCYqJiIiJCIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiI0YlSSJwR0YrRiVGJUYvISIiRiVGJSomRiRGJS1JJHNpbkdGKEYsRiVGJQ==

KiYsJiomIiIkIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJiomIiIjRiZJInBHRixGJkYmRjAhIiJGJkYyKiZGJUYmLUkkY29zR0YpRi1GJkYmRiYpRjFGMEYy

aqui novamente, n\303\263s fornecemos sintaxe extra para extrair e processar unicamente itens. Estamos procurando por alguma evid\303\252ncia de que ao longo da curva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= o vetor 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 permanece paralelo. Uma vez que o plano polar \303\251 coincidente com o plano Cartesiano, vamos olhar para suas evidentes coordenadas Cartesianas.

Come\303\247amos por escrever a curva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= na forma vetor raio (posi\303\247\303\243o):

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

Ent\303\243o, formamos o vetor 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 e o avaliamos ao longo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

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

A Figura 1 \303\251 um gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= junto com vetores que foram transportados paralelamente ao vetor inicial. Mesmo sem o gr\303\241fico \303\251 claro da express\303\243o alg\303\251brica para o campo ao longo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= que sua constante no sentido Euclidiano, e, portanto, paralelo.

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

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Figura 1 Campo paralelo ao longo da curva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

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JSFH

Defini\303\247\303\243o 2 Campo vetorial paralelo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Exemplo 5</Text-field>

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A derivada covariante do campo vetorial

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLUkiQUdGJDYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRjM3JDcjIiIjLUkiQkdGJEYx

\303\251, na forma de matriz,

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LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKSNcWmEp

O desaparecimento dessa derivada covariante requer a solu\303\247\303\243o de um conjunto de quatro equa\303\247\303\265es diferenciais parciais, cuja solu\303\247\303\243o geral \303\251

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Para qualquer dire\303\247\303\243o inicial do vetor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= como um vetor em coordenadas polares, o vetor equivalente em coordenadas Cartesianas \303\251

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LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpP3VXJik=

Portanto, temos um campo vetorial constante, paralelo no plano Euclidiano.

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Deixe LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ser o comprimento de arco ao longo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, a curva definida por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUEYrLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEic0YnL0YzRjEvRjZRJ2l0YWxpY0YnRj5GPkY+. Se o vetor unidade tangente ao longo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 paralelo ao longo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 uma geod\303\251sica. A condi\303\247\303\243o que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= precisa satisfazer \303\251

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

Defini\303\247\303\243o 3 Geod\303\251sica

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Exemplo 6</Text-field>

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As equa\303\247\303\265es para curvas geod\303\251sicas em coordenadas polares s\303\243o

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

LywmKiYpLUklZGlmZkclKnByb3RlY3RlZEc2JC1JJnRoZXRhR0YoNiNJInNHNiJGLSIiIyIiIi1JInJHRihGLEYwISIiLUYnNiQtRic2JEYxRi1GLUYwIiIh

LywmKioiIiMiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkickdGKTYjSSJzRzYiRi5GJi1GKDYkLUkmdGhldGFHRilGLUYuRiZGKyEiIkYmLUYoNiRGMEYuRiYiIiE=

Se n\303\263s usarmos o ponto sobrescrito para expressar diferencia\303\247\303\243o com respeito a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, podemos escrever essas equa\303\247\303\265es como

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

essas equa\303\247\303\265es s\303\243o consistentes com a Defini\303\247\303\243o 3, como vemos recordando os s\303\255mbolos de Christoffel na Tabela 5. Da Defini\303\247\303\243o 3 temos

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Na primeira equa\303\247\303\243o, 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 e os s\303\255mbolos restantes de Christoffel s\303\243o zero; no segundo, 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 e os outros s\303\255mbolos de Christoffel s\303\243o zero.

A solu\303\247\303\243o geral da geod\303\251sica em coordenadas polares \303\251

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnZHNvbHZlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEjREVGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjtGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGPUZA

NiQ3JDwjLy1JJnRoZXRhRyUqcHJvdGVjdGVkRzYjSSJzRzYiSSRfQzNHRis8Iy8tSSJyR0YoRiksJiomSSRfQzFHRisiIiJGKkY0RjRJJF9DMkdGK0Y0NyQ8Iy9GJiwmLUknYXJjdGFuRzYkRihJKF9zeXNsaWJHRis2IywkKigjRjQiIilGNCwmKigiIiNGNClGNUZFRjRGKkY0RjQqKEZFRjRGLEY0RkZGNEY0RjRGNSEiIkY0RjRJJF9DNEdGK0Y0PCMvRi8qJkYzRjQpLUklZGlmZkdGKDYkRiZGKiNGNEZFRkg=

A primeira curvatura \303\251 um segmento de reta radial; o segundo, um segmento de reta arbitr\303\241rio - ambos de acordo com nossas expectativas para geod\303\251sicas no plano.

<Text-field style="Heading 1" layout="Heading 1">Curvatura</Text-field>

JSFH

A f\303\263rmula comutativa para deriva\303\247\303\243o covariante \303\251

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

onde

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

e 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. O tensor misto classe quatro 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 \303\251 geralmente chamado de tensor curvatura de Riemann-Christoffel de segundo tipo. Outros nomes incluem tensor curvatura, tensor de Riemann, tensor Riemann-Christoffel, e tensor misto de Riemann-Christoffel. Al\303\251m disso, varia\303\247\303\265es na pr\303\263pria f\303\263rmula aparecem nos livros, especialmente quando a deriva\303\247\303\243o covariante repetida 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 \303\251 denotada por 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 or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUkjbW9HRiQ2LVEifkYnL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYmLUYjNiUtRi82JVEia0YnRjJGNUYyRjVGPi8lJW9wZW5HUSJ8Z3JGJy8lJmNsb3NlR0Zobi1GLzYlUSJoRidGMkY1RjJGNS1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0Zlby9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRidGPg==. Existe uma invers\303\243o inerente da ordem l\303\251xica com respeito \303\240 ordem do operador. Quando a There is an inherent reversal of the lexical order with respect to the operator order. Quando a inscri\303\247\303\243o invertida \303\251 traduzida para 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, um sinal de menos \303\251 introduzido \303\240 defini\303\247\303\243o. O leitor \303\251 a minus sign is introduced into the definition. O leitor \303\251 aconselhado a ser mais cuidadoso quando comparar o Maple aos livros.

JSFH

O tensor curvatura de Riemann-Christoffel de primeiro tipo \303\251 definido como 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 e \303\251 \303\240s vezes chamado de tensor curvatura covariante, e mesmo de tensor de Riemann. De novo deve se ter grande cuidado quando se ler e comparar diferentes livros.

O tensor curvatura de Riemann (veja como essas frases facilmente influenciam na escrita?) mede a curvatura de um espa\303\247o. Um espa\303\247o no qual esse tensor \303\251 zero \303\251 chamado plano, e o plano Cartesiano \303\251 plano n\303\243o importa qual sistema de coordenadas \303\251 imposto, como vemos a partir de

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyZRKGNvbl9iYXNGJFEoY292X2Jhc0YkRi1GLTciNyM3JDcmIiIiRjJGMkYyIiIh

o tensor curvatura para o plano sob coordenadas polares.

Uma superf\303\255cie masi interessante \303\251 aquela da esfera unit\303\241ria centrada na origem. O tensor m\303\251trico para essa superf\303\255cie \303\251 obtido pelos seguintes c\303\241lculos.

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LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvdl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJJHBoaUdGOkY0RjE=

Consequentemente, o tensor curvatura de Riemann-Christoffel de segundo tipo \303\251

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyZRKGNvbl9iYXNGJFEoY292X2Jhc0YkRi1GLTciNyY3JDcmIiIiIiIjRjJGMyokKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSRwaGlHRjlGM0YyNyQ3JkYyRjNGM0YyLCRGNCEiIjckNyZGM0YyRjJGM0ZANyQ3JkYzRjJGM0YyRjI=

ou melhor,

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

NyY3JDcmIiIiIiIjRiVGJiokKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRwaGlHRixGJkYlNyQ3JkYlRiZGJkYlLCRGJyEiIjckNyZGJkYlRiVGJkY0NyQ3JkYmRiVGJkYlRiU=

A partir disso, constru\303\255mos a representa\303\247\303\243o expl\303\255cita na Tabela 10.

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

Tabela 10 Componentes n\303\243o-zeros de 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 para a esfera

JSFH

O tensor curvatura de Riemann-Christoffel de primeiro tipo \303\251

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyZRKGNvdl9iYXNGJEYsRixGLDciNyY3JDcmIiIiIiIjRjFGMiokKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSRwaGlHRjhGMkYxNyQ3JkYxRjJGMkYxLCRGMyEiIjckNyZGMkYxRjFGMkY+NyQ3JkYyRjFGMkYxRjM=

ou melhor,

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEnREdpbmZvRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2KC1GLDYlUSpSQzFzcGhlcmVGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUkjbXNHRiQ2I1ExT2JqZWN0Q29tcG9uZW50c0YnRisvJStleGVjdXRhYmxlR0ZERkBGQEZib0ZARitGYm9GQEYrRmJvRkA=

NyY3JDcmIiIiIiIjRiVGJiokKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRwaGlHRixGJkYlNyQ3JkYlRiZGJkYlLCRGJyEiIjckNyZGJkYlRiVGJkYzNyQ3JkYmRiVGJkYlRic=

A partir disso, constru\303\255mos a Tabela 11.

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

Tabela 11 Componentes n\303\243o-zeros de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJWlqaGtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn para a esfera

JSFH

Note que para um espa\303\247o de dimens\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk= existe somente uma componente distinto necess\303\241rio para descrever o tensor completo, e essa componente \303\251 geralmente pega como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFElMTIxMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+. Em geral, existem 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 componentes distintas do tensor curvatura de Riemann de primeiro tipo, ent\303\243o para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5Rjk= existem 6, mas para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjRGJ0Y5Rjk= existem 20.

No pacote DifferentialGeometry, o tensor de Ricci \303\251 definido como 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. Obtido por essa defini\303\247\303\243o, o tensor \303\251

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvdl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJJHBoaUdGOkY0RjE=

Obtido com o comando RicciTensor, esse tensor \303\251

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvdl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJJHBoaUdGOkY0RjE=

Vemos que o tensor \303\251 o mesmo sob qualquer computa\303\247\303\243o.

O escalar Ricci, dado por

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

IiIj

\303\251 definido como 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, uma defini\303\247\303\243o que podemos testar via

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

IiIj

Para uma multiplicidade de duas dimens\303\265es a curvatura Gaussiana (ou total) \303\251 dado por 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, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 o determinante da matriz representando o tensor m\303\251trico LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==. Uma vez que para a esfera 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser computado no pacote Tensor via

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

KiYpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJHBoaUdGJyIiIyIiIi1JJF9ER0dGKTYjNyQ3JVEndGVuc29yRilJIlNHRik3JDciNyM3JFEkYmFzRilGLDcjNyRGNiokRiNGLSEiIg==

Finalmente, notamos que 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 \303\251 o tensor de Einstein. Para a esfera, esse tensor \303\251

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvbl9iYXNGJEYsNyI3IzckNyQiIiJGMSIiIQ==

e ent\303\243o ele desaparece, como podemos verificar por inspe\303\247\303\243o ap\303\263s obter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImpGJ0YyRjVGMkY1LUYjNiUtRi82JVEiaUYnRjJGNUYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkQvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnL0Y2USdub3JtYWxGJw== via

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvbl9iYXNGJFEoY292X2Jhc0YkNyI3JDckNyQiIiJGMkYyNyQ3JCIiI0Y1RjI=

J\303\241 que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk=, e a matriz a partir de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImpGJ0YyRjVGMkY1LUYjNiUtRi82JVEiaUYnRjJGNUYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkQvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnL0Y2USdub3JtYWxGJw== \303\251 a matriz identidade, 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 \303\251 a matriz zero.JSFH

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