SUNY Geneseo Department of Mathematics

The Fundamental Theorem for Line Integrals

Tuesday, November 22

Math 223 01
Fall 2022
Prof. Doug Baldwin

Anything You Want to Talk About?

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Potential Functions and the Fundamental Theorem

Based on “Curves and Regions,” “Fundamental Theorem for Line Integrals,” and “Conservative Vector Fields and Potential Functions” in section 5.3 of the textbook.

Key Ideas

The overall topic is conservative vector fields and their potential functions (the function whose gradient is the field).

The fundamental theorem of line integrals: if vector field F has potential function f then any circulation line integral from point P₁ to point P₂ equals f(P₂) - f(P₁). The reading also includes a proof of this theorem.

Consequences of the fundamental theorem:

Circulation integrals around closed curves (curves that start and end at the same point) in conservative fields are always 0.
The value of a circulation integral in a conservative field doesn’t depend on the exact curve followed, just on its endpoints.

A method for finding potential functions (see below for examples).

Examples

Is the vector field F(x,y) = ⟨ y+1, x ⟩ conservative? If so, find a potential function for it. Also find the integrals of F(x,y) • T(x,y) along two paths from (0,0) to (1,1). One path follows a straight line between the points, the other follows the parabola y = x².

F of X and Y is vector Y plus 1 comma X and 2 paths from point 0 0 to point 1 1

One possible start is to use the cross partials test to see if the field is obviously not conservative. But beware that the test can’t show that a field is conservative, since some non-conservative fields satisfy it. So in this case the test succeeded, but we didn’t learn much from that.

Derivative of P with respect to Y is 1 as is derivative of Q with respect to X

A more definite way to tell whether a field is conservative is to try to find a potential function for it. Start doing this by noting that the derivative of the potential function with respect to x must be the first component of the field. So integrate that component to get an approximate function for the field, but “approximate” because the “constant” of integration can really be any function of all variables except x; here we denoted that as-yet-unknown function g(y). You can learn more about g(y) by taking the derivative of the approximate potential function with respect to y, and setting it equal to the second component of the field. This may produce a function for g’(y), from which you can reconstruct g(y) by integrating. Repeat this process of integrating and checking derivatives until you run out of variables, or you find that the as-yet-unknown part of the potential function is 0.

Finding potential function by alternately integrating parts of field and differentiating potential function

Once we found a potential function, we knew that the field is conservative, and that we can evaluate both integrals in one fell swoop thanks to path independence and the fundamental theorem:

Integral of field along two paths is 2, using fundamental theorem

Is the vector field G(x,y) = ⟨ 2y, 3x ⟩ conservative?

We set out to find a potential function again, but this time ran into a problem when one of the “constant”-of-integration functions that was supposed to depend only on y had to also depend on x. This sort of problem indicates that in fact there’s no potential function, and the field isn’t conservative.

Trying to find potential function leads to an impossible function, showing that field is not conservative

Potential functions and the fundamental theorem in more dimensions.

No new reading.

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