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Multivariable Calculus: Partial Derivative

Let x = (1/2)(u^2-v^2), y=uv, and f=f(x,y)

[QUESTION 1] use the chain rule to derive the change of variables formula in matrix form:

(fu,fv)=A*(fx,fy) {actually it is vertical , so fu is at the top and fv is at the bottom. Same for fx and fy: fx is under fy; sorry for the notation I cant do it another way}

[QUESTION 2] invert the matrix A to obtain a formula expressing fx and fy in terms of fu and fv.

[QUESTION 5] If f(u,v)=ln(u^2+v^2), express fx and fy as functions of u and v.

[QUESTION 6] given a function g=g(x,y), express guu and gvv in terms of the partial derivatives of g with respect to x and y. (start from the expression of gu in a), and differentiate it with respect to u. We have to be careful with the notations because (gx)u not equals to (gu)x.)

[QUESTION 7] Assume that g solves the Laplace equation:
gxx+gyy=0, show that guu + gvv = 0

{{Sorry for the presentation I don't know why but I can't put free space. when I write guu it means the second partial derivative according to u.}}

Solution Summary

The solution determines the partial derivative in the given problem. #7 of the given questions is not solved.