Question

stokes thoerem

Answer 1

http://gyazo.com/ed762ac7878864d189dd48cd75b84b4f

Answer 2

start by finding the curl

Answer 3

(2yx^2 - 1, -2xy^2, -1)

Answer 4

looks good, next write ndS in terms of dxdy, take the dot product and setup the double integral

Answer 5

i have problems finding the normal vector

Answer 6

\[\large \hat{n} dS = \langle -f_x, -f_y, 1\rangle dx dy \]

Answer 7

you can use that as well for stokes?

Answer 8

z = f(x,y) = x^2+y^2

Answer 9

why not, its just a formula for normal vector and area element of ANY surface - works for all surfaces

Answer 10

ohh ok, for the double integral, so i switch it to polar coords?

Answer 11

do i*

Answer 12

yeah sure, the shadow will be just a circle i guess ?

Answer 13

yeah, 0 < r < 2 ; 0 < theta < 2pi?

Answer 14

looks good ^ whats the integrand ?

Answer 15

-4cos^3(theta) - 2cos(theta) - 4cos(theta)(sin(theta))^3 - 1?

Answer 16

whoa that looks complicated

Answer 17

yeah...did i do something wrong?

Answer 18

let me check...

Answer 19

ok

Answer 20

I am getting below :

\[\large \iint_S -4x^3y +2x + 4x^3y -1 ~ dydx\]

Answer 21

it evaluates to 4pi :
http://www.wolframalpha.com/input/?i=+%5Cint+%5Climits_%7B-2%7D%5E%7B2%7D+%5Cint+%5Climits_%7B-%5Csqrt%7B4-x%5E2%7D%7D%5E%7B%5Csqrt%7B4-x%5E2%7D%7D+%284x%5E3y+-2x+-+4xy%5E3+%2B1%29+dydx

Answer 22

I think switching to polar is not as bad as it looks... evaluating integrals will be easy..... go ahead, switch to polar and evaluate :)

Answer 23

okie thanks:D

Answer 24

the question is asking about two methods... any idea what are they ?

Answer 25

the other one green theorem?

Answer 26

because green theorem = stokes theorem righ?

Answer 27

green theorem = 2D
stokes theorem = 3D

both are same...

Answer 28

stoke's theorem becomes green's theorem in 2D

Answer 29

I think the second method is to calculate the line integral directly without using stokes theorem

Answer 30

\[\oint \overrightarrow{F} \bullet d\vec{r}\]

Answer 31

@dan815

Answer 32

@dan815 any idea on second method... the question is asking to calculate that work or whatever using two different methods...

Answer 33

Lower half of the ellisoid so we can find the ellipsde that is the intersection in the xy plane

Answer 34

we're still doing part a....

Answer 35

Oh okay, what part you on, i glanced over your work, it looks like you already evaluated it

Answer 36

they want it done using two different methods

Answer 37

ahh gotcha then do line integral :)

Answer 38

above was an attempt using stoke's theorem...
we need to work it using another method it seems..

Answer 39

oh whats the path/curve ?

Answer 40

ohh this isnt a surface question sorry

Answer 41

it is a closed surface right?

Answer 42

yeah top is a plane z=4 and bottom is paraboloid..

Answer 43

okay so im guessing the boundary is that circle on the z=4 plane

Answer 44

we can apply stokes on just that plane instead

Answer 45

i need to run for lunch... y'all have fun :)