In a similar fashion one can differentiate the equations (12.51) with respect to
the variable yand obtain
∂F
∂y +∂F
∂u∂u
∂y +∂F
∂v∂v
∂y =0
∂G
∂y +∂G
∂u∂u
∂y +∂G
∂v∂v
∂y =0(12 .54)which produces two equations in the two unknowns ∂u∂y and ∂v∂y .Solving this system
of equations using Cramers rule produces the solutions
∂u
∂y =−∂(F,G )
∂(y,v)
∂(F,G )
∂(u,v)and
∂v
∂y =−∂(F,G )
∂(u,y)
∂(F,G )
∂(u,v)(12 .55)This solution is valid provided the Jacobian determinant
∂(F, G )∂(u, v ) is different from
zero.
Example 12-3. (Conversion of the Laplace equation)
Transform the Laplace equation
∇^2 U=∂(^2) U
∂x^2
+∂
(^2) U
∂y^2
= 0 (12 .56)
from rectangular (x, y )coordinates to (r, θ )polar coordinates.
Solution 1
Figure 12-5.
Rectangular (x, y )and polar (r, θ)coordinatesx=rcos θ
y=rsin θ