Recent content by minderbinder

Homework Statement Show that the set defined by the equations x + y + z + w = sin(xyzw) x - y + z + w^2 = cos(xyzw) - 1 can be described explicitly by equation of the form (z, w) = f(x, y) near the point (0,0,0,0); find the first partial derivatives of f(x,y) at the point (0,0)...

Thought about this some more, and I think the solution should be: (u, v) = f(x, y) = ( \frac{y}{x^2}, xy) I checked some coordinates and it appears to work. However, I got this solution through trial and error. Can someone point out to me a way to find the solution in a systematic way?

Homework Statement Find a one-to-one C1 mapping f from the first quadrant of the xy-plane to the first quadrant of the uv-plane such that the region where x^2 \leq y \leq 2x^2 and 1 \leq xy \leq 3 is mapped to a rectangle. Compute the Jacobian det Df and the inverse mapping f^{-1}. The...

I just looked up the errata for the textbook I was using. It looks like the author made a mistake and the first equation is actually supposed to be x^2 + y^2 + z^2 = 6. So with that, and re-reading the theorem, I think the answer should be as follows: The mix-partial derivatives matrix of...

Homework Statement Can the equation x^2 + y^2 + z^2 = 3, xy + tz = 2, xz + ty + e^t = 0 be solved for x, y, z as C^1 functions of t near (x, y, z, t) = (-1, -2, 1, 0)? Homework Equations The Attempt at a Solution The mixed-partial derivatives matrix I got was: [2x, 2y, 2z, 0]...