## Problem on using the chain rule to show gradients are perpendicular to level curves

Let $\mathbf{r}(t) = x(t) \ \mathbf{i} + y(t) \ \mathbf{j}$ be a path along a level curve of $f(x,y)$.
1. Let $F(t) = f(x(t), y(t))$. Argue that $dF/dt = 0$.

2. Use the chain rule to express $dF/dt$ as a dot product of two vectors.

3. Show that the gradient of $f$ is perpendicular to the level curves of $f$.
• ## Solution

#### Part (a)

Recall that
Because $\mathbf{r}(t)$ is a level curve of $F$, $F(t)$ is constant. Hence, $dF/dt= 0$.

Recall that

#### Part (c)

Combining (a) and (b), we have shown that for all paths along a level curve $$0 = \nabla f \cdot \frac{d \mathbf{r}} {dt}$$
Recall that
Thus, $\nabla f$ is perpendicular to $\frac{d \mathbf{r}}{dt}$.
Because $\mathbf{r}(t)$ traces out the level curve, $\frac{d \mathbf{r}}{dt}$ is a tangent vector to the level curve. Hence we have shown that $\nabla f$ is perpendicular to the tangent vector of the level curve. Thus, it is perpendicular to the level curve.