(a) Suppose that h is any (0,2)-tensor. Show that h(s), defined by 1 h(s)(³) = ½ (h(ÂÂ) + h(‚Ã)), is a symmetric tensor. (b) Show that h(A), defined by h(4) (ÂÂ) = ½ (h(ÂÃ) — h(ÂÃ)), is an antisymmetric tensor.
(a) Suppose that h is any (0,2)-tensor. Show that h(s), defined by 1 h(s)(³) = ½ (h(ÂÂ) + h(‚Ã)), is a symmetric tensor. (b) Show that h(A), defined by h(4) (ÂÂ) = ½ (h(ÂÃ) — h(ÂÃ)), is an antisymmetric tensor.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve all parts of this question
![(a) Suppose that \( \mathbf{h} \) is any \( (0,2) \)-tensor. Show that \( \mathbf{h}_{(S)} \), defined by
\[
\mathbf{h}_{(S)}(\vec{A},\vec{B}) = \frac{1}{2} \left( \mathbf{h}(\vec{A},\vec{B}) + \mathbf{h}(\vec{B},\vec{A}) \right),
\]
is a symmetric tensor.
(b) Show that \( \mathbf{h}_{(A)} \), defined by
\[
\mathbf{h}_{(A)}(\vec{A},\vec{B}) = \frac{1}{2} \left( \mathbf{h}(\vec{A},\vec{B}) - \mathbf{h}(\vec{B},\vec{A}) \right),
\]
is an antisymmetric tensor.
(c) The tensors defined in the previous two parts are called the symmetric and antisymmetric parts of \( \mathbf{h} \), respectively. Find the components of the symmetric and antisymmetric parts of \( \vec{p} \otimes \vec{q} \), a \( (0,2) \)-tensor with components
\[
(\vec{p} \otimes \vec{q})_{\alpha \beta} = p_\alpha q_\beta.
\]
(d) Show that if \( \mathbf{h} \) is an antisymmetric \( (0,2) \)-tensor, then
\[
\mathbf{h}(\vec{A},\vec{A}) = 0
\]
for any vector \( \vec{A} \).
(e) For a general \( (0,2) \)-tensor \( \mathbf{h} \), find the number of independent components that \( \mathbf{h}_{(S)} \) and \( \mathbf{h}_{(A)} \) have.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd226013f-cf22-43f3-9ed7-6e0311814322%2F392e1c8d-6a4b-4190-a181-6900c3c0d9a8%2Fdgmns5q_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Suppose that \( \mathbf{h} \) is any \( (0,2) \)-tensor. Show that \( \mathbf{h}_{(S)} \), defined by
\[
\mathbf{h}_{(S)}(\vec{A},\vec{B}) = \frac{1}{2} \left( \mathbf{h}(\vec{A},\vec{B}) + \mathbf{h}(\vec{B},\vec{A}) \right),
\]
is a symmetric tensor.
(b) Show that \( \mathbf{h}_{(A)} \), defined by
\[
\mathbf{h}_{(A)}(\vec{A},\vec{B}) = \frac{1}{2} \left( \mathbf{h}(\vec{A},\vec{B}) - \mathbf{h}(\vec{B},\vec{A}) \right),
\]
is an antisymmetric tensor.
(c) The tensors defined in the previous two parts are called the symmetric and antisymmetric parts of \( \mathbf{h} \), respectively. Find the components of the symmetric and antisymmetric parts of \( \vec{p} \otimes \vec{q} \), a \( (0,2) \)-tensor with components
\[
(\vec{p} \otimes \vec{q})_{\alpha \beta} = p_\alpha q_\beta.
\]
(d) Show that if \( \mathbf{h} \) is an antisymmetric \( (0,2) \)-tensor, then
\[
\mathbf{h}(\vec{A},\vec{A}) = 0
\]
for any vector \( \vec{A} \).
(e) For a general \( (0,2) \)-tensor \( \mathbf{h} \), find the number of independent components that \( \mathbf{h}_{(S)} \) and \( \mathbf{h}_{(A)} \) have.
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