Section 7.6 Summary

The directional derivative. Definition 7.1 defines \(\left(D_{\mathbf u} f\right)_{P_0}\) as the limit in (7.1); it measures the rate of change of \(f\) at \(P_0\) in the direction of a unit vector \(\mathbf u\text{.}\) It is computed directly from this limit in Example 7.6, and Figure 7.23 shows how the plane and space pictures fit together.

The gradient vector. Using the chain rule we obtain Definition 7.8, which gives the shortcut \(\left(D_{\mathbf u} f\right)_{P_0} = \left(\nabla f\right)_{P_0}\cdot\mathbf u\) in (7.5). This is applied in Example 7.13.

Fastest increase, fastest decrease, and no change. Writing \(D_{\mathbf u} f = |\nabla f|\cos\theta\) in (7.11) shows that \(f\) increases most rapidly in the direction of \(\nabla f\text{,}\) decreases most rapidly in the direction of \(-\nabla f\text{,}\) and does not change in the directions orthogonal to \(\nabla f\text{.}\) These directions are found in Example 7.18.

The gradient is perpendicular to the level curves. Theorem 7.10 proves that \(\nabla f\) is perpendicular to the level curve through each point. Equivalently, the directions of no change are tangent to the level curve, while the directions of most rapid increase and decrease are perpendicular to it.