Page 188 - Intro to Tensor Calculus

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r
I 16. Let r =(~ ·~) 1/2 = p x + y + z ) and calculate (a) ∇ (r) (b) ∇ (1/r)(c) ∇ (r ) (d) ∇ (1/r )
r
I 17. Given the tensor equations D ij = 1 (v i,j + v j,i ), i, j =1, 2, 3. Let v(1),v(2),v(3) denote the
2
physical components of v 1 ,v 2 ,v 3 and let D(ij) denote the physical components associated with D ij . Assume
2
3
1
2
the coordinate system (x ,x ,x ) is orthogonal with metric coefficients g (i)(i) = h ,i =1, 2, 3and g ij =0
i
for i 6= j.
(a) Find expressions for the physical components D(11),D(22) and D(33) in terms of the physical compo-
1 ∂V (i) X V (j) ∂h i
nents v(i),i =1, 2, 3. Answer: D(ii)= + no sum on i.
h i ∂x i h i h j ∂x j
j6=i
(b) Find expressions for the physical components D(12),D(13) and D(23) in terms of the physical compo-
1 h i ∂ V (i) h j ∂ V (j)

nents v(i),i =1, 2, 3. Answer: D(ij)= +
2 h j ∂x j h i h i ∂x i h j
I 18. Write out the tensor equations in problem 17 in Cartesian coordinates.
I 19. Write out the tensor equations in problem 17 in cylindrical coordinates.
I 20. Write out the tensor equations in problem 17 in spherical coordinates.
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I 21. Express the vector equation (λ +2µ)∇Φ − 2µ∇× ~ω + F = 0intensorform.
I 22. Write out the equations in problem 21 for a generalized orthogonal coordinate system in terms of
physical components.
I 23. Write out the equations in problem 22 for cylindrical coordinates.
I 24. Write out the equations in problem 22 for spherical coordinates.
I 25. Use equation (2.1.4) to represent the divergence in parabolic cylindrical coordinates (ξ, η, z).

I 26. Use equation (2.1.4) to represent the divergence in parabolic coordinates (ξ, η, φ).

I 27. Use equation (2.1.4) to represent the divergence in elliptic cylindrical coordinates (ξ, η, z).

Change the given equations from a vector notation to a tensor notation.

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I 28. B = ~v ∇· A +(∇· ~v) A
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~
~
d dA dB dC
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~
~
~
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~
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~
I 29. [A · (B × C)] = · (B × C)+ A · ( × C)+ A · (B × )
dt dt dt dt
d~v ∂~v
I 30. = +(~v ·∇)~v
dt ∂t
~
1 ∂H
~
I 31. = −curl E
c ∂t
~
dB
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~
~
I 32. − (B ·∇)~v + B(∇· ~v)= 0
dt