What are the components of the vectors $mathbf{Z}_i$ with respect to the covariant basis $mathbf{Z}_j$? ...
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What are the components of the vectors $mathbf{Z}_i$ with respect to the covariant basis $mathbf{Z}_j$?
The 2019 Stack Overflow Developer Survey Results Are InPhysical components of a third-order tensorRelationsip between two definitions of the christoffel symbol?Div, grad, curl in curvilinear coordinatesFinding the basis one forms (covectors) corresponding to a particular formulation of basis vectorsThe Differential Geometry of a 2-D SurfaceCovariant derivative : calculation on basis vectorsExpressing contravariant basis vectors in terms of position vectorIndex notation of double contraction with second order tensor derivativeProving that the covariant derivative of a vector-valued tensor is a tensorIs the covariant basis covariant?
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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 $mathbf{Z}_i$ obtained from a position vector $mathbf{R}(Z)$, by differentiation with respect to each of the coordinates $Z^i$:
$$mathbf{Z}_i = frac{partialmathbf{R}(Z)}{partial Z^i}$$
At a subsequent exercise, the question is: "What are the components of the vectors $mathbf{Z}_i$ with respect to the covariant basis $mathbf{Z}_j$ ?". 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:
$${delta^i}_j = frac{partial Z^i}{partial Z^j}$$
so it involves the components $Z^i$ and $Z^j$ and not the collection of vectors $mathbf{Z}_i$ and $mathbf{Z}_j$.
How is the question related to this possible answer then?
multivariable-calculus tensors
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 $mathbf{Z}_i$ obtained from a position vector $mathbf{R}(Z)$, by differentiation with respect to each of the coordinates $Z^i$:
$$mathbf{Z}_i = frac{partialmathbf{R}(Z)}{partial Z^i}$$
At a subsequent exercise, the question is: "What are the components of the vectors $mathbf{Z}_i$ with respect to the covariant basis $mathbf{Z}_j$ ?". 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:
$${delta^i}_j = frac{partial Z^i}{partial Z^j}$$
so it involves the components $Z^i$ and $Z^j$ and not the collection of vectors $mathbf{Z}_i$ and $mathbf{Z}_j$.
How is the question related to this possible answer then?
multivariable-calculus tensors
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 $mathbf{Z}_i$ obtained from a position vector $mathbf{R}(Z)$, by differentiation with respect to each of the coordinates $Z^i$:
$$mathbf{Z}_i = frac{partialmathbf{R}(Z)}{partial Z^i}$$
At a subsequent exercise, the question is: "What are the components of the vectors $mathbf{Z}_i$ with respect to the covariant basis $mathbf{Z}_j$ ?". 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:
$${delta^i}_j = frac{partial Z^i}{partial Z^j}$$
so it involves the components $Z^i$ and $Z^j$ and not the collection of vectors $mathbf{Z}_i$ and $mathbf{Z}_j$.
How is the question related to this possible answer then?
multivariable-calculus tensors
asked Mar 21 at 23:50
RaphaRapha
61
$endgroup$
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 $mathbf{Z}_i$ obtained from a position vector $mathbf{R}(Z)$, by differentiation with respect to each of the coordinates $Z^i$:
$$mathbf{Z}_i = frac{partialmathbf{R}(Z)}{partial Z^i}$$
At a subsequent exercise, the question is: "What are the components of the vectors $mathbf{Z}_i$ with respect to the covariant basis $mathbf{Z}_j$ ?". 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:
$${delta^i}_j = frac{partial Z^i}{partial Z^j}$$
so it involves the components $Z^i$ and $Z^j$ and not the collection of vectors $mathbf{Z}_i$ and $mathbf{Z}_j$.
How is the question related to this possible answer then?
multivariable-calculus tensors
multivariable-calculus tensors
asked Mar 21 at 23:50
RaphaRapha
61
asked Mar 21 at 23:50
RaphaRapha
61
asked Mar 21 at 23:50
RaphaRapha
61
asked Mar 21 at 23:50
RaphaRapha
61
asked Mar 21 at 23:50
RaphaRapha
61
61
add a comment |
add a comment |
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