Page 191 - Intro to Tensor Calculus

186

I 44. Verify for orthogonal coordinates the relation

h i 1 ∂ 1 ∂(h 2 h 3 A(1)) ∂(h 1 h 3 A(2)) ∂(h 1 h 2 A(3))
~
∇ ∇· A · ˆ (i) = + +
e
h (i) ∂x (i) h 1 h 2 h 3 ∂x 1 ∂x 2 ∂x 3
I 45. Verify the relation
3
h i X A(k) ∂B(i) X B(k) ∂h (i) ∂h k
~
~
(A ·∇)B · ˆ e (i) = + A(i) − A(k)
h (k) ∂x k h k h (i) ∂x k ∂x (i)
k=1 k6=i
I 46. The Gauss divergence theorem is written
∂F ∂F ∂F 1 2 3
ZZZ 1 2 3 ZZ
+ + dτ = n 1 F + n 2 F + n 3 F dσ
∂x ∂y ∂z
V S
i
i
where V is the volume within a simple closed surface S. Here it is assumed that F = F (x, y, z) are
continuous functions with continuous first order derivatives throughout V and n i are the direction cosines
of the outward normal to S, dτ is an element of volume and dσ is an element of surface area.
(a) Show that in a Cartesian coordinate system
∂F 1 ∂F 2 ∂F 3
i
F = + +
,i
∂x ∂y ∂z
ZZZ ZZ
i
i
and that the tensor form of this theorem is F dτ = F n i dσ.
,i
V S
(b) Write the vector form of this theorem.
(c) Show that if we define
∂u ∂v m
u r = , v r = and F r = g rm F = uv r
∂x r ∂x r
i
then F = g im F i,m = g im (uv i,m + u m v i )
,i
(d) Show that another form of the Gauss divergence theorem is
ZZZ ZZ ZZZ
m
g im u mv i dτ = uv m n dσ − ug im v i,m dτ
V S V
Write out the above equation in Cartesian coordinates.
 
1 1 2
I 47. Find the eigenvalues and eigenvectors associated with the matrix A =  1 2 1  .
2 1 1
Show that the eigenvectors are orthogonal.
 
1 2 1
I 48. Find the eigenvalues and eigenvectors associated with the matrix A =  2 1 0  .
1 0 1
Show that the eigenvectors are orthogonal.
 
1 1 0
I 49. Find the eigenvalues and eigenvectors associated with the matrix A =  1 1 1  .
0 1 1
Show that the eigenvectors are orthogonal.
I 50. The harmonic and biharmonic functions or potential functions occur in the mathematical modeling
2
of many physical problems. Any solution of Laplace’s equation ∇ Φ = 0 is called a harmonic function and
4
any solution of the biharmonic equation ∇ Φ = 0 is called a biharmonic function.
(a) Expand the Laplace equation in Cartesian, cylindrical and spherical coordinates.
(b) Expand the biharmonic equation in two dimensional Cartesian and polar coordinates.
4
2
2
2
4
Hint: Consider ∇ Φ= ∇ (∇ Φ). In Cartesian coordinates ∇ Φ=Φ ,ii and ∇ Φ=Φ ,iijj .