complex variable z ,
+jv(x,y)%20%5C%20%5C%20%5C%20%20x,y%20%5Cin%20Real)
Jacobian Matrix
For analytic functions , they satisfy Cauchy-Riemann Relations
Both u and v are separately solutions of Laplace’s equation in two dimensions, i.e.
=%20%5Cfrac%7B%5Cpartial%20%7D%7B%5Cpartial%20x%7D%5Cleft(%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20y%7D%5Cright)=%5Cfrac%7B%5Cpartial%20%7D%7B%5Cpartial%20y%7D%5Cleft(%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20x%7D%5Cright)=-%5Cfrac%7B%5Cpartial%20%7D%7B%5Cpartial%20y%7D%5Cleft(%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%5Cright)%20%5CRightarrow%20%5Cfrac%7B%5Cpartial%5E2%20u%7D%7B%5Cpartial%20x%5E2%7D+%5Cfrac%7B%5Cpartial%5E2%20u%7D%7B%5Cpartial%20y%5E2%7D%20=0)
=%20-%5Cfrac%7B%5Cpartial%20%7D%7B%5Cpartial%20x%7D%5Cleft(%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%5Cright)=-%5Cfrac%7B%5Cpartial%20%7D%7B%5Cpartial%20y%7D%5Cleft(%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x%7D%5Cright)=-%5Cfrac%7B%5Cpartial%20%7D%7B%5Cpartial%20y%7D%5Cleft(%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20y%7D%5Cright)%20%5CRightarrow%20%5Cfrac%7B%5Cpartial%5E2%20v%7D%7B%5Cpartial%20x%5E2%7D+%5Cfrac%7B%5Cpartial%5E2%20v%7D%7B%5Cpartial%20y%5E2%7D%20=0)
The gradients of u and v are orthogonal
Cauchy-Riemann Relations
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