Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 940: 121

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To determine the sign of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ from the graph of some contours of $z = f(x, y)$, we need to check the following: If the contour levels have increasing values when moving from the point $(x_0, y_0)$: $\rightarrow$ In the positive $x$ direction, then $\frac{\partial f}{\partial x}(x_0, y_0)$ is positive. $\rightarrow$ In the positive $y$ direction, then $\frac{\partial f}{\partial y}(x_0, y_0)$ is positive. But if the contour levels have decreasing values when moving from the point $(x_0, y_0)$: $\rightarrow$ In the positive $x$ direction, then $\frac{\partial f}{\partial x}(x_0, y_0)$ is negative. $\rightarrow$ In the positive $y$ direction, then $\frac{\partial f}{\partial y}(x_0, y_0)$ is negative.