265L23 - LESSON 23 Green’s Theorem READ Section 18.1...

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Unformatted text preview: LESSON 23 Green’s Theorem READ: Section 18.1 NOTES: Line integrals of the form R C-→ F · d-→ r can be easily evaluated using the fundamental theorem of line integrals if-→ F is the gradient of some scalar function, that is, when-→ F is conservative. But if-→ F is not conservative, there is no short cut for line integrals of the form Z C-→ F · d-→ r = Z b a ( P ( x ( t ) ,y ( t ))-→ ı + Q ( x ( t ) ,y ( t ))-→ ) · ( x ( t )-→ ı + y ( t )-→ ) dt = Z b a ( P ( x ( t ) ,y ( t )) x ( t ) + Q ( x ( t ) ,y ( t )) y ( t )) dt = Z C P ( x,y ) dx + Q ( x,y ) dy. You’ll just have to find a parameterization of C , and plug and chug as the saying goes. Well, actually, there is one other special case of such a line integral of a function of two variables that can be evaluated in an alternate way that is sometimes easier than the back-to-basics method. The formula is given in Green’s Theorem. We will need a few definitions before stating Green’s Theorem. Green’s Theorem is concerned with a line integral over a simple closed curve C . A curve is closed provided the initial point of the curve matches the terminal point of the curve. The curve is simple provided it does not cross itself, except at the endpoints. A circle is the fundamental example of a simple closed curve. A simple closed curve has an inside (which will be a finite bounded area) and an outside (which is an infinite area). The other idea we need is that an orientation can be assigned to a simple closed curve. Imagine walking along a simple closed curve incan be assigned to a simple closed curve....
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This note was uploaded on 02/24/2011 for the course MATH 265 taught by Professor Jerrymetzger during the Winter '11 term at North Dakota.

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265L23 - LESSON 23 Green’s Theorem READ Section 18.1...

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