Let \(f(x,y) = xy\). Show that \(\nabla f\) is orthogonal to the level curves of \(f\).
Solution
\(\nabla f = (y, x)\). Along a level curve \(f(x,y) = c\) one may take a parametrization \(\gamma\) with \(f(\gamma(t)) = c\), so \(\frac{d}{dt}f(\gamma(t)) = 0\). By the chain rule this inner product is \(\langle \nabla f(\gamma(t)), \gamma'(t)\rangle\), hence \(\nabla f\) is orthogonal to every tangent of the level curve.
