Suppose I have the following curvilinear coordinate (s, n) transform to Cartesian coordinates (x, y, z)

where f(s, n), g(s, n), and h(s, n) are inaccessible functions (i.e. they can’t be expressed explicitly, I have them as an interpolation).

Is there a way to plot a heatmap or contour of h(s, n) in the space of \left(x, y\right)?

I’ve been studying curvilinear coordinates and Jacobians and doing

```
s = range(0, S, length = 101)
n = range(0, N, length = 101)
heatmap(f.(s, n'), g.(s, n'), h.(s, n'))
```

doesn’t throw an error, but doesn’t produce anything (i.e. it plots an empty axes) (as a wild attempt just to see if it would work, and I guessed it wouldn’t work but it was worth a shot (that said, I’m aware of curvilinear coordinates having potential regions of surjectivity in their transformation to Cartesian space, I just wanted to see what Julia would do)).

Extra note: When I went down the path of coordinate transforms theory, I could obtain the inverse transform Jacobian matrix (taking the interpolated functions f and g as differentiable), but that’s still a function of (s, n) so it doesn’t make sense to integrate it to obtain (s, n) as a function of (x, y) because… well… you can’t (AFAIK). So extra question, is there a way to handle the transform and obtain F(s, n) as a function of (x, y)?