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What are the components of the vectors $mathbfZ_i$ with respect to the covariant basis $mathbfZ_j$?

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1

$begingroup$

I am studying the book Introduction to tensor analysis and the calculus of moving surfaces, where the covariant basis is defined as the collection of vectors $mathbfZ_i$ obtained from a position vector $mathbfR(Z)$, by differentiation with respect to each of the coordinates $Z^i$:

$$mathbfZ_i = fracpartialmathbfR(Z)partial Z^i$$

At a subsequent exercise, the question is: "What are the components of the vectors $mathbfZ_i$ with respect to the covariant basis $mathbfZ_j$ ?". The author claims that the answer is a "single symbol" introduced in a previous chapter. The only "single symbol" previously introduced is the Kronecker delta $delta^i_j$. However, in the book the Kronecker delta is defined as:

$$delta^i_j = fracpartial Z^ipartial Z^j$$

so it involves the components $Z^i$ and $Z^j$ and not the collection of vectors $mathbfZ_i$ and $mathbfZ_j$.

How is the question related to this possible answer then?

multivariable-calculus tensors

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asked Mar 21 at 23:50

RaphaRapha

61

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add a comment | 

1

$begingroup$

I am studying the book Introduction to tensor analysis and the calculus of moving surfaces, where the covariant basis is defined as the collection of vectors $mathbfZ_i$ obtained from a position vector $mathbfR(Z)$, by differentiation with respect to each of the coordinates $Z^i$:

$$mathbfZ_i = fracpartialmathbfR(Z)partial Z^i$$

At a subsequent exercise, the question is: "What are the components of the vectors $mathbfZ_i$ with respect to the covariant basis $mathbfZ_j$ ?". The author claims that the answer is a "single symbol" introduced in a previous chapter. The only "single symbol" previously introduced is the Kronecker delta $delta^i_j$. However, in the book the Kronecker delta is defined as:

$$delta^i_j = fracpartial Z^ipartial Z^j$$

so it involves the components $Z^i$ and $Z^j$ and not the collection of vectors $mathbfZ_i$ and $mathbfZ_j$.

How is the question related to this possible answer then?

multivariable-calculus tensors

share|cite|improve this question

asked Mar 21 at 23:50

RaphaRapha

61

$endgroup$

add a comment | 

$begingroup$

I am studying the book Introduction to tensor analysis and the calculus of moving surfaces, where the covariant basis is defined as the collection of vectors $mathbfZ_i$ obtained from a position vector $mathbfR(Z)$, by differentiation with respect to each of the coordinates $Z^i$:

$$mathbfZ_i = fracpartialmathbfR(Z)partial Z^i$$

At a subsequent exercise, the question is: "What are the components of the vectors $mathbfZ_i$ with respect to the covariant basis $mathbfZ_j$ ?". The author claims that the answer is a "single symbol" introduced in a previous chapter. The only "single symbol" previously introduced is the Kronecker delta $delta^i_j$. However, in the book the Kronecker delta is defined as:

$$delta^i_j = fracpartial Z^ipartial Z^j$$

so it involves the components $Z^i$ and $Z^j$ and not the collection of vectors $mathbfZ_i$ and $mathbfZ_j$.

How is the question related to this possible answer then?

multivariable-calculus tensors

share|cite|improve this question

asked Mar 21 at 23:50

RaphaRapha

61

$endgroup$

share|cite|improve this question

asked Mar 21 at 23:50

RaphaRapha

61

share|cite|improve this question

asked Mar 21 at 23:50

RaphaRapha

61

asked Mar 21 at 23:50

RaphaRapha

61

asked Mar 21 at 23:50

RaphaRapha

61