This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: LESSON 23 Greenâ€™s Theorem READ: Section 18.1 NOTES: Line integrals of the form R Câ†’ F Â· dâ†’ r can be easily evaluated using the fundamental theorem of line integrals ifâ†’ F is the gradient of some scalar function, that is, whenâ†’ F is conservative. But ifâ†’ F is not conservative, there is no short cut for line integrals of the form Z Câ†’ F Â· dâ†’ r = Z b a ( P ( x ( t ) ,y ( t ))â†’ Ä± + Q ( x ( t ) ,y ( t ))â†’ ) Â· ( x ( t )â†’ Ä± + y ( t )â†’ ) 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. Youâ€™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 backtobasics method. The formula is given in Greenâ€™s Theorem. We will need a few definitions before stating Greenâ€™s Theorem. Greenâ€™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....
View
Full
Document
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.
 Winter '11
 JerryMetzger
 Calculus, Integrals, Scalar

Click to edit the document details