Green's Theorem Question

reap-a-rat

Hi all ! Was hoping to get some clarification on Green's theorem. I understand the concept and all of that. All of the questions I'm doing are along the lines of "verify Green's theorem in the plane x,y for (line integral whatever), and the region is usually bound by 2 lines and a curve.

I'm just wondering, say if it was bound by y = -x^2 + 4; x = 0 and y = 0, when I'm doing the line integrals for each of these, what are the lower and upper limits, or the points you integrate between, whatever they are called? How do I know like?

I've been drawing the graph and finding the points of intersection, but I'm just never quite sure where I should be integrating between for each function!

Its the only thing setting me back in this particular type of question, as I'm fairly confident evaluating the other side of the theorem equation! I would appreciate any help at all, please!!!:)

RolandIRL

You go in an anticlockwise direction when putting in the limits.



you're given two functions f and g
so the line integral of f dx + g dy over C is the sum of all the individual line integrals for the different parts

C1: choose y = 0. dy = 0. x goes from 0 to 2. so 0 is lower limit and 2 is upper, and you're integrating f dx since g dy = 0

C2: y = -x^2 + 4...do the same but choose whether to write dx in terms of dy or vice verse, and whatever you're integrating with respect to will determine your limits. say integrating wrt x. limits will be 2 and 0 but lower one will be 2 and upper will be 0...see picture. but choose dx or dy such that it's easier to write one in terms of the other eg for the above dy = -2x dx...it's much easier to integrate wrt x as integrating wrt y would involve square roots and inverses.

C3: x = 0. dx = 0....go through the procedure again.

sum C1, C2, C3 and that should equal the area integral gotten from the other side of green's theorem. if you want, i could go over some of this with you before monday give me a PM if you need any help

reap-a-rat

You're a legend, Whiteman ! I've been really hit and miss with them, sometimes I do it right, sometimes I do it wrong! Thank you so much, you really cleared it up!!

I want to do really well in maths phsics, and I'm confident with my other two questions, so confident I've been helping all the others in my class with them ! Just Green's is being a little awkward! Thanks for your help though, I'll practice them now today and if I've any problems I'll PM you, seeing as you're good enough for honours !

RolandIRL

no problem. give me a PM tonight if you need any help. i probably won't be online tomorrow til after the exam