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)...
Homework Statement
Consider the mapping f: R^3 \rightarrow R^2 defined by
f (x_1, x_2, x_3) = (x_1 e^{x_2}, x_2 e^{x_3}).
i) Show that the mapping f(x) is differentiable and find its derivative at the origin.
ii) Show that the equation f(x) = a for any point a \in R^2 determines a...
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]...