aclotm

Let F(x,y,z) = (3x+z^3, 4x+3y, z) be a vector field. Can F be a gradient field (i.e. F= -(nabla)V for some V(x,y,z).

So I know that something is a gradient field if the curl (i.e. (nabla)x(nabla f) = 0, but I don't know how to decide whether it is for some V?

f(x,y) = x + y subject to x^2 + y^2 + z^2 = 1, y + z = 1

I'm having some trouble solving for the five equations with five unknowns. I'm stuck right now with x = y-z (using lambda = 1/(2x)) and y + z = 1 (given). I'm not sure how to get a third equation...help!

f:A->B and g:B->C, g o f is a bijection. Prove:

1. f:A->B is an injection, and

2. g:B->C is a surjection

Any ideas? I understand proving if the composition is injective/surjective given the maps, but not the other way around

Using Lagrange multipliers find the extrema:

1) f(x,y)=x^2-xy+y^2 subject to the constraint of x^2-y^2=1

2) f(x,y,z)=x^2-2y+2z^2 subject to the constraint of x^2+y^2+z^2=1