SUNY Geneseo Department of Mathematics

Conservative Vector Fields

Friday, April 28

Math 223
Spring 2023
Prof. Doug Baldwin

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Circulation Example

Continuing the brewing example, what is the circulation of water around the bag of grain?

Water in kettle passing through circular bag, with equations for water flow and bag

Circulation is just the original “field in direction of tangent to curve” line integral, but around a closed loop. Set it up and integrate accordingly…

Circulation integral is integral from 0 to 2 Pi of V of R of T dot R prime of T

Conservative (Gradient) Vector Fields

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

Key Ideas

A “gradient field” is a vector field that’s the gradient of some function.

“Conservative field” is just a synonym for gradient field.

A gradient field’s “potential function” is a function whose gradient is the field.

A gradient field can have several potential functions, but only differing from each other by a constant (very much like antiderivatives differ from each other only by constants).

The cross partial property tells if fields might be conservative, or if they definitely aren’t. But it can’t tell whether a field definitely is conservative.

The cross partial properties for 2- and 3-dimensional vector fields:

Derivatives of field components with respect to variables from other positions in gradient are equal

Questions

A proof of the cross partial property? It comes from Clairaut’s Theorem, since the relevant derivatives of component functions in the vector field are all mixed second derivatives of the potential function. For example, for 2 dimensions…

Derivative of P with respect to Y is mixed 2nd derivative of potential function, thus derivative of Q with respect to X

And similarly for 3…

Derivatives of components in 3 dimensions are also mixed second derivatives of potential function

Examples

Does F(x, y) = ⟨ y, -2x ⟩satisfy the cross partial property?

No, the derivatives differ:

Derivative of P with respect to Y is 1 but derivative of Q with respect to X is negative 2

So we can be sure that F isn’t conservative.

But on the other hand G(x, y) = ⟨ -y, -x ⟩ does satisfy it:

Derivative of P with respect to Y is negative 1 and so is derivative of Q with respect to X

So G might be conservative, but we don’t know for sure.

Problem Set

Problem set 13 (the last one!) is available.

It’s about conservative vector fields and their potential functions, with a little bit more about line integrals.

Work on it most of next week, grade it during the last days of classes.

Next

How do you really know that a field that satisfies the cross partials property is conservative?

The one way we have to be certain is showing that it has a potential function.

Please read “Conservative Vector Fields and Potential Functions” in section 5.3 of the textbook.

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