How to find the intersection of level curves?

Question

For two functions $f,g:\mathbb{R}^2\rightarrow\mathbb{R}$, how would I show that the level curves of these two different functions intersect at right angles?

I can give the specific functions, but I would like to know in a more general way.

Answer 1

Given $z=f(x,y)$, $\nabla f= \frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}$, at a given $(x, y)$, is tangent to the level curve of $z= f(x, y)$ for that $(x, y)$. Similarly, $\nabla g$ is tangent to the level curve of $z= g(x, y)$. The two level curves intersect at a right angle where those two tangent vectors are perpendicular: where their dot product is $0$.

Answer 2

A general approach for this sort of problem would be to find a parameterization of both curves around each intersection point. Checking that they intersect at right angles is now reduced to computing that the inner product of the two velocities (ie. the derivatives of the parameterizations) at the intersection and seeing if it vanishes.