(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
Transcribed Image Text:(a) Suppose that h is any (0,2)-tensor. Show that h(s), defined by
h(s) (ÂÂ) = ½ (h(Â, B) + h(B‚Ã)),
2
is a symmetric tensor.
(b) Show that h(A), defined by
h(4) (Ã, B) = – (h(Ã, B) – h(B, Ã)),
is an antisymmetric tensor.
(c) The tensors defined in the previous two parts are called the symmetric and antisym-
metric parts of h, respectively. Find the components of the symmetric and antisym-
metric parts of pã, a (0,2)-tensor with components
(p 9) aß = Pα¶ß.
(d) Show that if h is an antisymmetric (0, 2)-tensor, then
h(A,A) = 0
for
any
vector Ā.
(e) For a general (0,2)-tensor h, find the number of independent components that h(s)
and h(A) have.
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