Hey so here's the problem:



For part a I just took the partial derivatives of each part and ended up with dz/du=(1,-1) and dz/dv=(-1,1) so when I add both I just get 0. I'm not sure if I'm supposed to be using some sort of chain rule where z=f(x,y) and x = u-v and y = v-u.

For part b I have no idea how to start it

Also just a general question but say you find the Jacobian matrix for D(f(g(x,y)) (f composed of g of x,y). What does each entry of the matrix mean or represent? If g maps from R^2 -> R^3 and f maps from R^3 -> R^3.

4 (a)
yes, to be formally correct you need to use the chain rule.

as you said, let x = u-v and y = v-u

∂z/∂u = ∂f/∂x ⋅ ∂x/∂u + ∂f/∂y ⋅ ∂y/∂u = ∂f/∂x ⋅ (1) + ∂f/∂y ⋅ (-1)
∂z/∂v = ∂f/∂x ⋅ ∂x/∂v + ∂f/∂y ⋅ ∂y/∂v = ∂f/∂x ⋅ (-1) + ∂f/∂y ⋅ (1)

∂z/∂u + ∂z/∂v = ∂f/∂x ⋅ (1) + ∂f/∂y ⋅ (-1) + ∂f/∂x ⋅ (-1) + ∂f/∂y ⋅ (1)
=0

(b)
let u = y/x, v = z/x

x∂w/∂x = x [x^3 (∂F/∂u⋅∂u/∂x + ∂F/∂v⋅∂v/∂x) + 3x^2 * F(u,v)] = x [x^3 (∂F/∂u⋅(-y/x^2) + ∂F/∂v⋅(-z/x^2)) + 3x^2 * F(u,v)] = -yx^2 * ∂F/∂u - -zx^2 * ∂F/∂v + 3x^3 * F(u,v)
when calculating ∂w/∂x you need to use the product rule and chain rule

y∂w/∂y = yx^3 [∂F/∂u⋅∂u/∂y + ∂F/∂v⋅∂v/∂y] = yx^3 [∂F/∂u⋅(1/x) + ∂F/∂v⋅(0)] = yx^2 * ∂F/∂u

z∂w/∂z = zx^3 [∂F/∂u⋅∂u/∂z + ∂F/∂v⋅∂v/∂z] = zx^3 [∂F/∂u⋅(0) + ∂F/∂v⋅(1/x)] = zx^2 * ∂F/∂v

x∂w/∂x + y∂w/∂y + z∂w/∂z = -yx^2 * ∂F/∂u - -zx^2 * ∂F/∂v + 3x^3 * F(u,v) + yx^2 * ∂F/∂u + zx^2 * ∂F/∂v

= 3x^3 * F(u,v)
but we know that w = x^3 * F(u,v) therfore = 3w

.
.
jacobian is the matrix a_ij of partial derivatives of the f_j with respect to x_i

This post was edited by cdexswzaq on Oct 2 2014 02:24pm

Wow thats very helpful thank you so much