Jacobian Determinant/ mult. variable implicit differentiation

[Isaac Wiebe]

Homework Statement

Let F: x^2 + y^2 - z^2 + 2xy - 1 = 0 and G: x^3 + y^3 - 5y - 4 = 0. Calculate dz/dx. Note: This is NOT the partial derivative ∂z/∂x.

I do not need help in taking the derivative of many polynomials. What I need help in is setting up a Jacobian determinant to evaluate this.

Homework Equations

The Jacobian determinant, bit tough to explain in a short period of space. Exceptionally helpful with many equations with as many variables as equations, in which the partial derivative can be evaluated by the Jacobian Determinant.

The Attempt at a Solution

Okay, so dz/dx = [∂(F, G) /∂(z, y)] / ∂(F, G) / ∂(x, y) * (-1), but this is incorrect as this is ∂z/∂y. Essentially I need to know, what is a function of what, and how can I evaluate this? I do not care necessarily for the answer, simply for the setup.

 

[ehild]

The ultimate independent variable is x. G relates x and y so y is only function of x:y=y(x).
F is a relation among x, y, z. z depends both on y and x: z=z(x,y(x)). ehild

 

[lurflurf]

Here is the motivation
F(x,y,z)=0
G(x,y,z)=0
0=dF=F_x dx+F_y dy+F_z dz
0=dG=G_x dx+G_y dy+G_z dz
first we find
0=(G_y dF - F_y dG)/dx
which will have the Jacobians you seek
then think it through in general

What happens is you get F,G,x,y in one Jacobian and F,G,y,z in the other and F,G,y effectively cancel (chain rule) leaving dz/dx