Inverse function of a function of two variables

[closet mathemetician]

If I have z=f(x,y), then how would I go about finding the inverse function?

More specifically, say I have a parametric function of the form

f(u,v) = (x(u,v), y(u,v),0)

which is a coordinate transformation. How do I find the inverse of this function?

All references I can find on inverse functions deal with single variable functions.

Say I choose various 2-variable functions as coordinate charts that I can map to a 2-dimensional manifold. I can pick various functions easily enough, but I'm having trouble figuring out how to get their inverses.

I'm trying to understand this whole idea of contraviant and covariant vectors and raising and lowering indicies and I want to play with some examples, but I'm running into the problem above.

 

[mathwonk]

for invertible functions from 2 dimensions to 2 dimensions, try polar coordinatyes, x = rcos(t), y = rsin(t).

then r^2 = x^2 + y^2, and t = arctan(y/x).

on appropriate domains.

 

[tacman]

Finding the inverse function of a function of two variables, also known as the inverse of a parametric function, can be a bit more complex compared to finding the inverse of a single variable function. In order to find the inverse function, we need to follow a few steps.

Step 1: Rewrite the function in terms of the independent variables
In this case, we need to express the function f(u,v) = (x(u,v), y(u,v), 0) in terms of the independent variables u and v. This may involve some algebraic manipulation or using known relationships between the variables.

Step 2: Solve for one of the independent variables
Choose one of the independent variables, let's say u, and solve for it in terms of the other variable v. This will give us an expression for u in terms of v, which we can then use to eliminate u from the original function.

Step 3: Rewrite the function with only one independent variable
Using the expression we found for u in the previous step, we can now rewrite the function as f(v) = (x(v), y(v), 0). This will give us a single variable function that we can easily find the inverse of.

Step 4: Find the inverse function
To find the inverse function, we simply need to swap the positions of the variables in our rewritten function. This will give us the inverse function as g(x,y) = (v(x,y), w(x,y), 0).

In summary, to find the inverse function of a function of two variables, we need to rewrite the function in terms of the independent variables, solve for one of the variables, rewrite the function with only one independent variable, and then swap the positions of the variables to find the inverse function. This process can be applied to any parametric function, including the example provided in the content.