Page 132 - Intro to Tensor Calculus

127

¯ i
I 42. Let Y ,i =1, 2, 3 denote the components of a field of parallel vectors along the curve C defined by
y
i
¯ i
i
the equations y =¯y (t), i =1, 2, 3 inaspace withmetrictensor ¯g ij , i, j =1, 2, 3. Assume that Y and d¯ i
dt
¯
are unit vectors such that at each point of the curve C we have
y
d¯ j
¯ g ij Y ¯ i =cos θ = Constant.
dt
¯
(i.e. The field of parallel vectors makes a constant angle θ with the tangent to each point of the curve C.)
i
¯ i
3
i
i
1
2
Show that if Y and ¯y (t) undergo a transformation x = x (¯y , ¯y , ¯y ), i =1, 2, 3 then the transformed
vector X m = Y ¯ i ∂x m
j makes a constant angle with the tangent vector to the transformed curve C given by
∂¯ y
3
i
i
1
2
x = x (¯y (t), ¯y (t), ¯y (t)).
-
∂x i
I 43. Let J denote the Jacobian determinant | j |. Differentiate J with respect to x m and show that
∂x
p
∂J α ∂x r
= J − J .
∂x m αp ∂x m rm
Hint: See Exercise 1.1, problem 27 and (1.4.7).
W
I 44. Assume that φ is a relative scalar of weight W so that φ = J φ. Differentiate this relation with
k
respect to x . Use the result from problem 43 to obtain the transformation law:
" #
m
∂φ α W ∂φ r ∂x
− W φ = J − W φ .
∂x k αk ∂x m mr ∂x k
The quantity inside the brackets is called the covariant derivative of a relative scalar of weight W. The
covariant derivative of a relative scalar of weight W is defined as
∂φ r
φ ,k = − W φ
∂x k kr
and this definition has an extra term involving the weight.
It can be shown that similar results hold for relative tensors of weight W. For example, the covariant
derivative of first and second order relative tensors of weight W have the forms

∂T i i m r i
i
T = + T − W T
,k k
∂x km kr
∂T j i i σ r
i
σ
i
T j,k = + T − T − W T j i
j
σ
∂x k kσ jk kr
When the weight term is zero these covariant derivatives reduce to the results given in our previous definitions.
i
I 45. Let dx i = v denote a generalized velocity and define the scalar function of kinetic energy T of a
dt
particle with mass m as
1 i j 1 i j
T = mg ij v v = mg ij ˙x ˙x .
2 2
Show that the intrinsic derivative of T is the same as an ordinary derivative of T. (i.e. Show that δT = dT .)
δT dt