Page 112 - Intro to Tensor Calculus

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I 44. In Cartesian coordinates the vorticity tensor at a point in a fluid medium is defined

1 ∂V j ∂V i
ω ij = −
2 ∂x i ∂x j
where V i are the velocity components of the fluid at the point. The vorticity vector at a point in a fluid
1 ijk
i
medium in Cartesian coordinates is defined by ω = e ω jk . Show that these tensors are dual tensors.
2
I 45. Write out the relation between each of the components of the dual tensors
1 ijkl
T ˆ ij = e T kl i, j, k, l =1, 2, 3, 4
2
and show that if ijkl is an even permutation of 1234, then T ˆ ij = T kl .

2
1
3
I 46. Consider the general affine transformation ¯x i = a ij x j where (x ,x ,x )= (x, y, z)with inverse
transformation x i = b ij ¯x j . Determine (a) the image of the plane Ax + By + Cz + D = 0 under this
transformation and (b) the image of a second degree conic section
2
2
Ax +2Bxy + Cy + Dx + Ey + F =0.
I 47. Using a multilinear form of degree M, derive the transformation law for a contravariant vector of
degree M.
∂g ij ∂g ij
I 48. Let g denote the determinant of g ij and show that = gg .
∂x k ∂x k
I 49. We have shown that for a rotation of xyz axes with respect to a set of fixed ¯x¯¯ axes, the derivative
z
y
~
of a vector A with respect to an observer on the barred axes is given by
~ ~
dA dA
~
= + ~ω × A.
dt dt
f r
Introduce the operators
~
dA
~
D f A = = derivative in fixed system
dt
f
~
dA
~
D r A = = derivative in rotating system
dt
r
~
~
(a) Show that D f A =(D r + ~ω×)A.
~
(b) Consider the special case that the vector A is the position vector ~r. Show that D f ~r =(D r + ~ω×)~r

r
produces V = V + ~ω × ~ where V represents the velocity of a particle relative to the fixed system
~
~
~

f r f

and V represents the velocity of a particle with respect to the rotating system of coordinates.
~

r

(c) Show that ~a = ~a + ~ω × (~ω × ~)where ~a represents the acceleration of a particle relative to the
r

f r f

fixed system and ~a represents the acceleration of a particle with respect to the rotating system.

r
(d) Show in the special case ~ω is a constant that

~
~a =2~ω × V + ~ω × (~ω × ~)
r

f
~
~
where V is the velocity of the particle relative to the rotating system. The term 2~ω × V is referred to
r
as the Coriolis acceleration and the term ~ω × (~ω × ~) is referred to as the centripetal acceleration.