MA242.003
•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 3rd 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 3rd 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.
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
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
Let C be a positively oriented, piecewise-smooth simple
closed curved in the plane,
piecewise-smooth means composed of a finite
number of smooth sub-curves
13.4: Green’s Theorem
Let C be a positively oriented, piecewise-smooth simple
closed curved in the plane,
simple closed means starts and ends at the same
point, with no other self-intersections
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:
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
Now we must work on the right-hand-side of the formula
Now we must work on the right-hand-side of the formula
Now we must work on the right-hand-side of the formula
Now we must work on the right-hand-side of the formula
Now we must work on the right-hand-side of the formula
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
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 = <P,Q,R>
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 = <P,Q,R>
Stokes’ theorem is the 3 dimensional version of Greens’
theorem.
First we need the following definition:
First we need the following definition:
First we need the following definition:
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..
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:
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)
(continuation of example)