Fall 2002 M,W 12:30-2:00
1. Express vector Z through vectors A, B,
C, and constants a, b, c, if
A.Z=a, B . Z = b, C
. Z= c
(Here, (.) is the scalar product
2. Express all solutions R to the equation
R. A =m
through A and m.
3. Prove that
a) curl(curl A)= grad (div A)- div( grad A)
b) div( A x B ) = (B . curl A )- (A . curl B)
4. Let S and T be the square matrices
. Differentiate
d /d S ( Tr ( S^{-1} T)
First challenge project (due in 2 weeks, at September 28); will be followed by the oral presentation in class.
Consider a (generally anisotropic) structure made of a conducting material with the conductivity k and a perfect isolator. Denote by c the fraction of the conductor in the structure . It is known thatSuppose that this structure (composite) is sequentially submerged into several different homogeneous fields v_i, i= 1..N.
1. The conductivity tensor K of conducting structures of an arbitrary geometry satisfies the inequality(*) Tr[( K - k I) ^{-1} ]< c/(1-c) (1/k) S
where S is a nonnegative matrix with unit trace, S >=0, Tr S =1.
2. The bound (*) is exact: There exist structures that deliver equality in (*) for all permitted values of S. Tensor S depends on the structure and determines the degree of anisotropy (see for details Cherkaev, 2000).
The dissipated energy density w_i of each field is
w_i= v_i^T S v_i= Tr [S . (v_iv_i^T)]. Suppose also that you are free to choose the structure from the set (*)Find
1. The minimal value of the sum of the energies w_1 + ... w_n.2. Optimal value of structural matrix S and anisotropic conductivity K
1. Consider a tensor E
E= def u = (1/2) [grad u + (grad u)^T]
where u is a differentialble vector
Show thatInk (E) = curl ( curl E)^T=0Here Ink is a llinear second-rank differential operator (from inkompabilit"{a}t).Show that
div (Ink G) =0 forall tensors G.2. Consider two 1-periodic vectors
v_1= grad f1 and v_2= grad f2, and the antisymmetric tensor R: R^T = -RShow that
<v_1. R . v_2>=<v_1> . R . <v_2>
where < > is the average over a unit cube.
Consider a tensor V and assume that
(*) div V=0 .
1. Find a quadratic function phi(V)=V:Q:V of elements of V such that
(i) phi (V) >= 0 if (*) holds
(ii) There exist V such that phi(V) < 02. Find all such quadratic forms.