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 that

 
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).

Suppose that this structure (composite) is sequentially submerged into several different homogeneous fields v_i,    i= 1..N.
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 that

Ink (E) = curl ( curl E)^T=0

Here 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 = -R

Show 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) < 0

2. Find all such quadratic forms.