SUNY Geneseo Department of Mathematics

Gradient Vector Fields

Wednesday, November 16

Math 223 01
Fall 2022
Prof. Doug Baldwin

Return to Course Outline

Previous Lecture

Anything You Want to Talk About?

(No.)

Thanksgiving Pi Day

PRISM will be celebrating Thanksgiving Pi Day this Friday (November 18), from 4:30 - 6:00 in South 309.

(Pi Day is officially March 14, but since that’s almost always during spring break, PRISM celebrates it around Thanksgiving time as an excuse to share some pi(e) with math students and professors.)

Mathematica and Vector Fields

The VectorPoints option does work with both VectorPlot and VectorPlot3D, no idea why it wouldn’t work yesterday. It’s very helpful for decluttering a vector field plot.

Another helpful option is VectorScaling -> Automatic, which makes Mathematica scale the sizes of vectors in a plot according to their magnitudes.

You can download a notebook that demonstrates both of these options from Canvas.

Gradient Fields

Also known as conservative fields.

Based on “Gradient Fields (Conservative Fields)” in section 5.1 of the textbook.

Questions

Terminology? “gradient field” and “conservative field” are synonyms, both mean a vector field defined by something’s gradient.

Key Ideas

The definition of “gradient field” above.

If F(x,y,...) is the gradient of f(x,y,...) then f is the “potential function” of F.

The cross partials property: all gradient fields have it, but so do some non-gradient fields. For 2 dimensions it says that for vector field ⟨ P(x,y), Q(x,y) ⟩, ∂P/∂y = ∂Q/∂x. In 3 dimensions there are more comparisons, namely…

D P D Y equals D Q D X; D Q D Z equals D R D Y; D R D X equals D P D Z

Examples

Find the gradient field defined by ∇f(x,y) where f(x,y) = x2y - xy2.

The field is simply the gradient of f:

Field is gradient of F, namely vector 2 X Y minus Y squared comma X squared minus 2 X Y

Verify that it has the cross partials property. To do this, find the specified derivatives and check that they’re equal:

Derivative of P with respect to Y equals 2 X minus 2 Y which equals derivative of Q with respect to X

Curiously, the cross partials property doesn’t hold in the other direction, i.e., ∂P/∂x is not equal to ∂Q/∂y. This is because the cross partials property is basically Clairaut’s Theorem in action:

Derivatives of P with respect to Y and Q with respect to X are the mixed second derivatives of F

The derivatives of P with respect to y and Q with respect to x are mixed second derivatives of the potential function, and so are expected to be equal. The derivatives of P with respect to x and Q with respect to y, however, are second derivatives with respect to x twice and y twice, and so aren’t covered by Clairaut’s Theorem.

If I want an example of a conservative vector field, how do you suppose I generate it?

Make whatever function comes to mind of the right number of variables, and find its gradient:

The gradient of an arbitrary scalar function of 4 variables is a 4-dimensional gradient vector field

What does the cross partials property look like for 4 variables? Similar to 3 variables, except that there are a lot more mixed second derivatives of the potential function that can be equal to each other:

Some of the cross partials equalities in a 4-dimensional gradient vector field

Next

Line integrals of vector fields.

Please read “Vector Line Integrals” in section 5.2 of the textbook.

Next Lecture