Advanced Engineering Mathematics, 7th Edition

9.6 Tangent Planes and Normal Lines

INTRODUCTION

The notion of the gradient of a function of two or more variables was introduced in the preceding section as an aid in computing directional derivatives. In this section we give a geometric interpretation of the gradient vector.

Geometric Interpretation of the Gradient—Functions of Two Variables

Suppose f(x, y) = c is the level curve of the differentiable function z = f(x, y) that passes through a specified point P(x₀, y₀); that is, f(x₀, y₀) = c. If this level curve is parameterized by the differentiable functions

x = g(t), y = h(t) such that x₀ = g(t₀), y₀ = h(t₀),

then the derivative of