Question

Circulation, Curl, and Stokes Theorem troubles

I have a problem and a solution attached below, but I am having trouble understanding it. The formula for curl that I am using is also attached below. My question is:

When I apply the formula for curl to the vector field, I get what they give in the solution. . What I dont understand is why it is being dotted with <2x, 2y, 1> . Where did that come from?

Answer 1

@dan815 , is it because on the second line it is dotted with dS, and somehow dS becomes <2x, -2y, 1> dA?

Answer 2

hello again

Answer 3

so what are you confused about

Answer 4

R(x,y)=<x,y,y^2-x^2> <--- you must get your normal vector from this
Rx=<1,0,-2x>
Ry=<0,1,2y>
Rx X Ry =<2x,-2y,1>

Answer 5

that is your normal vector not normalized tho

Answer 6

in this case we actually dont need to normalize, i assume you know why so moving on...

Answer 7

now you just do doubleintegral CurlF dot<2x,-2y,1> da
where da is the unit circle so lets bet to convert to polar now

Answer 8

best*

Answer 9

for questions like this the trickiest part is just parametrizing

Answer 10

Sorry, I am having connection issues, thanks for the reply, that helps a lot!

Answer 11

are you solid with this stuff

Answer 12

i can refer you to some great lectures if you dont know the theory about why we are cross to get the little surface elements and stuff

Answer 13

or why stokes theorem ends up giving you the line integral of the boundary curve

Answer 14

Thanks but thats ok. I have a final on this stuff so Im just working through last minute problems, trying to make sure I get it.

Answer 15

I just don't understand what happens with dA. Why we use the cross production of the partials of one of the surfaces and not the other... Ug, Im hopeless lol

Answer 16

Ug nevermind, I get it. Once you dot the curl with whatever Rx X Ry is called, then the 2nd surface comes into play as a bound. Finally think I understand.