Angle between intersecting regular surfaces ($\theta =\arccos \frac{|\nabla f\cdot \nabla g|}{|\nabla f||\nabla g|}$)

For smooth level surfaces $f=0$, $g=0$ meeting at a point where both gradients are nonzero, their acute angle is the angle between two planes tangent to the surfaces there. Each gradient is a normal vector to its tangent plane, proving the displayed formula. Vanishing gradients require further analysis because the level-set equation then need not specify a unique tangent plane.