46
2 Vector and Tensor Analysis
Example 2.10 In Cartesian coordinates, let (x, y, z) = (x 1 , x 2 , x 3 ). Compute the
gradient of the scalar function φ(x i ), as well as the gradient, divergence, and curl of
the vector function a(x i ).
Solution
With ds i = dx i , Eq. (2.66) gives ∇ = e i ∂/∂x i . Substituting this into (2.67) and
recognizing that the e i are constant yield
∇φ =
e i
∂
∂x i
φ = e i φ, i
∇a =
e i
∂
∂x i
(a j e j ) = e i (a j e j ), i = a j , i e i e j
∇ · a = e i · (a j e j ), i = a j , i e i · e j = a j , i δ ij = a i , i
∇ × a = e i × (a j e j ), i = a j , i e i × e j = a j , i ij k e k .
Equation (2.4) 2 was used in the last equation.
Example 2.11 Determine the gradient and divergence of the 2D velocity field
v(r, θ ) = v r e r + v θ e θ in cylindrical polar coordinates.
Solution
For cylindrical coordinates, Eq. (2.62) gives ds 1 = dr, ds 2 = r dθ, and ds 3 = dz,
and Eq. (2.66) yields
∇ = e r
∂
∂r
+
e θ
r
∂
∂θ
+ e z
∂
∂z
.
(2.68)
In two dimensions, the z-term can be omitted, and the velocity gradient is given by
∇v =
e r
∂
∂r
+
e θ
r
∂
∂θ
(v r e r + v θ e θ )
= e r (v r , r e r + v r e r , r +v θ , r e θ + v θ e θ , r )
+ r
−1 e θ (v r , θ e r + v r e r , θ +v θ , θ e θ + v θ e θ , θ )
= v r , r e r e r + v θ , r e r e θ + r
−1 (v r , θ −v θ )e θ e r + r
−1 (v r + v θ , θ )e θ e θ ,
where Eqs. (2.3) have been used.