MA Day 67 April 22, 2013 Section 13.7: Stokes’s Theorem Section 13.4: Green’s Theorem

Remark about Final Exam: 1. Use your 4 tests (and their study guides) to prepare for the final exam.

Remark about Final Exam: 1. Use your 4 tests (and their study guides) to prepare for the final exam. 2.All problems on the final (except the LAST TWO) will be of the same TYPE as the problems on the 4 tests

Remark about Final Exam: 1. Use your 4 tests (and their study guides) to prepare for the final exam. 2.All problems on the final (except the LAST TWO) will be of the same TYPE as the problems on the 4 tests For example, you should expect a double integral problem on the final exam because double integrals were covered on the 3 rd test.

Remark about Final Exam: 1. Use your 4 tests (and their study guides) to prepare for the final exam. 2.All problems on the final (except the LAST TWO) will be of the same TYPE as the problems on the 4 tests For example, you should expect a double integral problem on the final exam because double integrals were covered on the 3 rd test. 3. There will be one problem each covering Stokes’ theorem (13.7) and the Divergence Theorem (13.8)

In sections 13.7 and 13.8 we will study two famous integral theorems of vector calculus.

The theorems may be thought of as the 2 and 3 dimensional versions of the following integral formulas

Fundamental Theorem of Calculus

The theorems may be thought of as the 2 and 3 dimensional versions of the following integral formulas Fundamental Theorem of Calculus Fundamental Theorem for Line Integrals

We will first discuss Green’s theorem before considering Stokes’ Theorem, since Green’s theorem is needed in the proof of Stokes’ Theorem.

13.4: Green’s Theorem Let C be a positively oriented, piecewise-smooth simple closed curved in the plane,

13.4: Green’s Theorem Let C be a positively oriented, piecewise-smooth simple closed curved in the plane, positively oriented means counter clockwise

13.4: Green’s Theorem piecewise-smooth means composed of a finite number of smooth sub-curves Let C be a positively oriented, piecewise-smooth simple closed curved in the plane,

13.4: Green’s Theorem simple closed means starts and ends at the same point, with no other self-intersections Let C be a positively oriented, piecewise-smooth simple closed curved in the plane,

13.4: Green’s Theorem Let C be a positively oriented, piecewise-smooth, simple closed curved in the plane, and let D be the region bounded by C. If P and Q have continuous partial derivatives on an open region containing D, then

13.4: Green’s Theorem Let C be a positively oriented, piecewise-smooth, simple closed curved in the plane, and let D be the region bounded by C. If P and Q have continuous partial derivatives on an open region containing D, then

Proof : First a definition and a lemma we will need in the proof.

Proof : First a definition and a lemma we will need in the proof.

Proof : First a definition and a lemma we will need in the proof. Lemma:

Proof of Green’s Theorem for a special type of region: Notice first that since P(x,y) and Q(x,y) are independent of each other, to prove Green’s theorem

Proof of Green’s Theorem for a special type of region: Notice first that since P(x,y) and Q(x,y) are independent of each other, to prove Green’s theorem It is sufficient to prove the following two formulas separately:

Proof of Green’s Theorem for a special type of region: Notice first that since P(x,y) and Q(x,y) are independent of each other, to prove Green’s theorem It is sufficient to prove the following two formulas separately: I’m going to prove the second formula

Proof of Green’s Theorem for a special type of region:

Let’s first work on the left-hand-side of the formula using

Now we must work on the right-hand-side of the formula

Left-hand-side:

Now we must work on the right-hand-side of the formula Left-hand-side: Right-hand-side: Hence we have proved formula *** ***

The proof of the other formula

Is essentially the same as the above and we leave this part as an exercise.

13.7: Stokes’ Theorem Notice that the integrand in the double integral in Green’s Theorem

13.7: Stokes’ Theorem Notice that the integrand in the double integral in Green’s Theorem Is the z-component of the curl of F =

13.7: Stokes’ Theorem Notice that the integrand in the double integral in Green’s Theorem Is the z-component of the curl of F = Stokes’ theorem is the 3 dimensional version of Greens’ theorem.

First we need the following definition:

We will have more to say about this after stating Stokes’ theorem.

See your textbook for the proof of this theorem for special types of surfaces.

This means that in order to determine “positive orientation” for a surface and its boundary curve, we may “deform the surface” WITHOUT tearing it to make the decision easier..

Example:

Remark about problem STATEMENTS:

1. If a problem tells you to “USE STOKE’S THEOREM to compute

Remark about problem STATEMENTS: 1. If a problem tells you to “USE STOKE’S THEOREM to compute then you should compute

Remark about problem STATEMENTS: 1. If a problem tells you to “USE STOKE’S THEOREM to compute then you should compute 2. If a problem tells you to “USE STOKE’S THEOREM to compute

Remark about problem STATEMENTS: 1. If a problem tells you to “USE STOKE’S THEOREM to compute then you should compute 2. If a problem tells you to “USE STOKE’S THEOREM to compute then you should compute

(continuation of example)