Cauchy-Riemann equations

Calculus Level 4

Let u ( x , y ) u(x,y) and v ( x , y ) v(x,y) be continuous functions that satisfy the following equations:

u x = v y u y = v x u_x = v_y \\ u_y = -v_x

Suppose the first and second partial derivatives of u u and v v are continuous in both x x and y y .

Find u x x + u y y u_{xx}+u_{yy} .


The answer is 0.

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1 solution

u x x + u y y u_{xx} + u_{yy} = v y x v x y = 0 v_{yx} - v_{xy} = 0 because second derivatives of v v are continuous.

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