The Gradient

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Student Representative

The math department recently changed its rules for department meetings to include 4 math student representatives. These representatives will be able to vote on lots of department business, and are far and away the most effective way for students to influence what the department does.

The department further decided that each of its student clubs (PRISM, AWM, SIAM, and Actuary) would recommend one of these students. PRISM needs someone to recommend.

You don’t have to be a PRISM member, and your job as a representative to department meetings would be to represent student views as a whole, not specifically PRISM’s. So any math major (of any form, e.g., BA, BS, adolescence ed., etc.) can volunteer for this. It looks good on a resume, and gives you a certain amount of influence in the department.

Meetings are every other Tuesday from 2:30 - 4:00.

If you’re interested, contact PRISM president Darien Connnolly at prism@geneseo.edu.

Gradients

From “Gradient” and “Gradients and Level Curves” in section 3.6 of the textbook.

Key Ideas

The equation that defines the gradient:

Relations between gradients and other things:

• The gradient of f points in the direction (in the space of inputs to f) of fastest increase in f, i.e., the direction that has the greatest directional derivative for f. The negative of the gradient points in the direction of fastest decrease.
• The gradient at a point is orthogonal to the level curve(s) through that point.

Examples

Find the gradient of f(x,y) = 4x²y - 5xy².

Following the definition of gradient, this just requires finding the partial derivatives of f and putting them in a vector:

How about ∇g(x,y,z) where g(x,y,z) = x sin(y+z) + z tan(x² - y²)?

This uses the same ideas as above, except that the gradient has 3 components because the function has 3 variables.

Applications

Suppose you were standing at point (π/6, π/4, √2/4) on a plot of the egg carton function z = sin x cos y. What direction should you move in to climb fastest up the surface? To descend fastest down it? To move while keeping your height constant?

This problem uses the fact that the gradient points in the direction of fastest increase in a function, while its negative points in the direction of fastest decrease, and it’s orthogonal to level curves. So find the gradient function, and then evaluate it at x = π/6, y = π/4 to get a vector from which to answer the parts of the question:

To find a vector in the direction of no change in height, we used the facts that level curves are curves along which a function’s value remains constant, and that the gradient is orthogonal, i.e., perpendicular, to the level curve through point (π/6, π/4, √2/4). One easy way to find a vector orthogonal to a known 2-dimensional one is to switch the known vector’s components and then negate one. Taking the dot product of the given and calculated vectors shows that they’re orthogonal:

What if you’re at the top or bottom of a surface, how can the gradient point in “the” direction of greatest increase or decrease when that’s all directions at the same time? When you’re at a local maximum or minimum, the gradient ends up being 0, which doesn’t point anywhere, but is at least a signal that something unusual is happening:

Next

Start looking at some applications of partial derivatives, particularly finding local extreme values of multivariable functions.

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

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