The Gradient

Return to Course Outline

Previous Lecture

Anything You Want to Talk About?

(No.)

Gradients

Based on “Gradient” and “Gradients and Level Curves” in section 3.6, plus yesterday’s introduction to gradients.

Key Ideas

The equation for a gradient (see yesterday’s notes, the examples below, or the textbook for the exact equation).

Directional derivatives can be calculated as a dot product of the gradient with the unit direction vector.

Properties:

• If the gradient is 0, then all directional derivatives are 0
• The maximum directional derivative is in the direction of the gradient, and its value is the magnitude of the gradient
• The minimum directional derivative is in the negative of the direction of the gradient; its value is the negative of the magnitude of the gradient
• Gradients are perpendicular to level curves.

The definition of “level curve” is also important.

Examples

Geneseo Widget Works makes fine hand-crafted widgets by giving c craftspeople p pounds of refined widget minerals. The cost, k, to make one widget under these conditions is p/10 + (c + p) / c. If Geneseo Widget Works currently has 5 craftspeople and 10 pounds of minerals, in what direction should they “move” in the people-minerals space in order to reduce cost fastest?

The fastest reduction will be in the negative of the direction the gradient points. So find the gradient at point (10, 5), and negate:

So Geneseo Widget Works should decrease its supply of minerals and increase its number of craftspeople, at rates that lose 3/10 of a pound of minerals for every 2/5 of a craftsperson hired.

Explain why gradients (or at least non-zero ones) are always perpendicular to level curves.

Think about a function and its level curves. Each level curve is a curve in the xy plane (i.e., a curve through the inputs to the function). Each level curve corresponds to a contour line on the function’s graph, i.e., a line along the surface of the function with constant value. Since the level curve is a curve, we can describe it parametrically, and we can describe the corresponding contour line as the function applied to values on the curve:

Since the function has a constant value along the contour line, its derivative along that line must be 0. And we can work out that derivative using the chain rule:

But now the derivative found by the chain rule looks like (and is) a dot product between the gradient of the function and the tangent to the level curve. Since this dot product must be 0, it means that the gradient and the (tangent to) the level curve are perpendicular:

Next

Extreme value problems in multiple variables.

Please read “Critical Points” and “Second Derivative Test” in section 3.7 of the textbook.

Next Lecture