Introduction to Green’s Theorem

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Path Dependence/Independence

Picking up the discussion from yesterday.

Show that integrating F(x, y) = ⟨ -2y, 2x ⟩along two different paths from (-1,0) to (1,0) produces different values. One path is the straight line along the x axis, the other is a semicircle of radius 1.

What does this result say about whether this field is conservative?

Because integrals of this field are not path independent, you definitely know that the field is not conservative.

To what extent can you use path independence to show that a vector field is conservative?

Theoretically, a field with an open and connected domain (which is an easy criterion to satisfy) is conservative if its integrals are path independent. This sounds like a nice test, definitely more powerful than the cross-partials test.

But the practical problem with it is that you have to show that all integrals of the field are path independent, not just find one pair that happen to be. Proving this is probably harder than other definitive tests we know, such as finding a potential function.

Green’s Theorem

Based on “Extending the Fundamental Theorem of Calculus” and “Circulation Form of Green’s Theorem” in section 5.4 of the textbook.

Key Ideas

The equation for Green’s Theorem (see the textbook, e.g., equations 5.4.1 and 5.4.2).

Green’s Theorem relates line integrals to double integrals. As such it’s an extension of the Fundamental Theorem to 2 dimensions — specifically, both say that you can know an integral over some 1- or 2-dimensional interval by knowing something about another function at the boundary of the interval:

Green’s Theorem only works for 2-dimensional vector fields.

There’s an interesting way of thinking about the Theorem as using a double integral to invert or “undo” partial derivatives.

Questions

How does the proof of Green’s Theorem work?

The “proof” in the book is more a suggestion about how the full proof works, since the book’s proof is limited to a rectangular region. We went through it line by line in class; the core idea is that line integrals along the top and bottom sides (also left and right) use the same bounds for integration, but in opposite orders; one of these integrals is thus negated in order to put the bounds in the same order. Integrands for the line integral can then be collected into a difference of function values at opposite ends of an interval, which by the Fundamental Theorem is a definite integral of derivatives of those functions. This insight turns the line integral into a double integral, and the final form of the equation then just requires rearranging that double integral slightly.

Are there examples? Definitely, and we’ll look at some Friday.

Next

“Examples day” for the circulation form of Green’s Theorem.

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