2.6 Vector and Tensor Calculus
41
Finally, we emphasize a key feature of tensor analysis. Equation (2.48) 2 shows
that if the components of a tensor with respect to one basis vanish, then its
components with respect to any other basis vanish. Thus, an equation written in
terms of tensor components, like an equation written in direct notation, holds in any
coordinate system within a given frame of reference.
2.6 Vector and Tensor Calculus
2.6.1 Differentiation with Respect to Spatial Coordinates
Differentiating vectors and tensors can be a tricky business. However, the process
is relatively straightforward if these mathematical objects are expressed in terms of
base vectors in a particular coordinate system. Then, so long as the base vectors are
included in the differentiations, the rules are essentially the same as those for scalar
expressions.
Consider, for example, the vector a = a i e i and the tensor T = T ij e i e j , where
the e i are unit base vectors for a curvilinear coordinate system with coordinates x i .
Defining ( ), i ≡ ∂( )/∂x i , we obtain
a, i = (a j e j ), i = a j , i e j + a j e j , i
T, i =
T jk e j e k
, i = T jk , i e j e k + T jk e j , i e k + T jk e j e k , i ,
(2.55)
where indices are renamed to stay within the rules of the summation convention.
These equations, which are valid for any orthogonal coordinate system, require
derivatives of the base vectors.
Example 2.8 Consider the vector
u = u x (x, y) e x + u y (x, y) e y = u r (r, θ ) e r (θ ) + u θ (r, θ ) e θ (θ ),
which is expressed in terms of both Cartesian and cylindrical components, with
coordinate dependencies included for clarity. Differentiate u with respect to the
Cartesian coordinates x and y and the cylindrical coordinates r and θ .
Solution
Differentiating with respect to the Cartesian coordinates yields
u, x = u x , x e x + u y , x e y
u, y = u x , y e x + u y , y e y
41
Finally, we emphasize a key feature of tensor analysis. Equation (2.48) 2 shows
that if the components of a tensor with respect to one basis vanish, then its
components with respect to any other basis vanish. Thus, an equation written in
terms of tensor components, like an equation written in direct notation, holds in any
coordinate system within a given frame of reference.
2.6 Vector and Tensor Calculus
2.6.1 Differentiation with Respect to Spatial Coordinates
Differentiating vectors and tensors can be a tricky business. However, the process
is relatively straightforward if these mathematical objects are expressed in terms of
base vectors in a particular coordinate system. Then, so long as the base vectors are
included in the differentiations, the rules are essentially the same as those for scalar
expressions.
Consider, for example, the vector a = a i e i and the tensor T = T ij e i e j , where
the e i are unit base vectors for a curvilinear coordinate system with coordinates x i .
Defining ( ), i ≡ ∂( )/∂x i , we obtain
a, i = (a j e j ), i = a j , i e j + a j e j , i
T, i =
T jk e j e k
, i = T jk , i e j e k + T jk e j , i e k + T jk e j e k , i ,
(2.55)
where indices are renamed to stay within the rules of the summation convention.
These equations, which are valid for any orthogonal coordinate system, require
derivatives of the base vectors.
Example 2.8 Consider the vector
u = u x (x, y) e x + u y (x, y) e y = u r (r, θ ) e r (θ ) + u θ (r, θ ) e θ (θ ),
which is expressed in terms of both Cartesian and cylindrical components, with
coordinate dependencies included for clarity. Differentiate u with respect to the
Cartesian coordinates x and y and the cylindrical coordinates r and θ .
Solution
Differentiating with respect to the Cartesian coordinates yields
u, x = u x , x e x + u y , x e y
u, y = u x , y e x + u y , y e y