allie

please help me with this question for multivariable calculus

I dont know how to do this problem please help

Each transformation is continuously differentiable (C1) on its domain -- R^2, or R^2∖{(0,0)} for the function involving log -- so Inverse Function Theorem implies it has a C1

inverse near every point satisfying some conditions. Match each transformation with the set of points in its domain that DO NOT satisfy the conditions.

 1. f(x,y)=(xe^(x^2+y^2),ye^(x^2+y^2))

 2. f(x,y)=(e^(x+3y),xy+y^2)

 3. f(x,y)=(x/(e^(x^2+y^2)),y/(e^(x^2+y^2)))

 4. f(x,y)=(xlog(x^2+y^2),ylog(x^2+y^2))

 5. f(x,y)=(x^3,y^3)

A. the line y=x

B. the line y=−x

C. the x and y axes

D. the empty set

E. the circle 2x^2+2y^2=1

F. the circle (ex)^2+(ey)^2=1 and the origin

G. the origin

H. the circle (ex)^2+(ey)^2=1

I really don't understand how to do this problem, can someone please help and explain it??

Let F(u,v)=(e^u−4v,u−2v,4u^2+v^2), and let S=F(R^2)⊂R^3 be the surface parametrized by F. The point p=(1,2,65) is a regular point of S. Find an equation for the tangent plane to S at p.

I really do not know how to do this question please help

I found the gradient of f (2x+2xy  2y+x^2) and then the critical points (rad2,-1), (-rad2,-1) and plugged them into the hessian and found they were both local maximums but I do not know where to go from here.