Tangent Planes to Level Surfaces

A level surface, S, consists of all points where a three-variable function equals a constant value.

To understand the geometry at a specific point P on this surface, imagine a curve r(t) that passes through P.

To show that r(t) lies entirely on the level surface, the parametric components of the curve are substituted into the surface equation.

Since k is a constant, its derivative on time must be zero. The Chain Rule breaks this derivative into two distinct parts.

The first part includes partial derivatives showing elevation changes and forms the gradient vector. The second part includes derivatives of the curve’s path, representing the direction of travel and forming the tangent vector.

This computes the dot product of the gradient and a tangent vector. Because it is zero, the gradient is perpendicular to all tangent vectors, and so perpendicular to the tangent plane at P. So, the gradient serves as the plane’s normal vector.

Imagine a hiker on a hill. Walking at a constant altitude means moving perpendicular to the gradient. The gradient vector points directly uphill, marking the steepest ascent. The tangent plane is like a flat platform resting against the hillside at the hiker's position.