ES: 624: Nonlinear Elasticity

Homework - 2, Due Jan 27th

Note: You should use indicial notation in your proofs. Expanding out terms will not fetch you anypoints. Any tensor A can be represented with a chosen basis as A = Aij(eiej). So when we talkabout inner product between a 2nd order tensor and a vector we say A u = Aij(ei ej) bkek.On expanding this you get Aijbk(ej ek)ei = Aijbkjkei = Aijbjei. This is the same expression youobtained earlier in class. You may use this information while working on the following problems.

1) Evaluate (r ~)r2) Show that if A B = 0 is true for every tensor B, then A = 0.3) For a given vector field u(x) = x/|x|3 compute harmonic of u i.e the Laplacian.4) If F is an invertible tensor, show that FTF is symmetric and positive definite.

5) Given vectors u and v R3 show that det(u v) = 0.6) Problem assigned in class.

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