Greens

joemama69

1. The problem statement, all variables and given/known data

verify greens for the integral (xy^2 i - x^2y j) dot dr over c, where c = y = x^2 from (-1,1) to (1,1). Evaluate both sides independantly to achieve the same answer

2. Relevant equations



3. The attempt at a solution

so i took the partials and got

-2xy - 2yx = -4xy, shouldnt it equal o.

dx

Why should it equal zero? What exactly is the question asking you to check?

HallsofIvy

I'm confused by this. Green's theorem relates the integral around a closed path to the integral of the expression you give over the region bounded by the closed path.

"c = y = x^2 from (-1,1) to (1,1)" is not a closed path and so does not bound a region. Did you intend to include the line from (1, 1) back to (-1, 1)?

In any case that expression you give would be 0 only if the integral around the closed path were 0.

joemama69

ya it does say "from -1,1 to 1,1 and the line segment from 1,1 to -1,1. sorry i didnt realize that was important

HallsofIvy

Then I strongly suggest you revies Green's theorem!

benorin

Forgot [tex]d\vec{r}[/tex].

Use [tex]\oint_{C}\left( xy^2\hat{i} -x^2y\hat{j}\right)\cdot d\vec{r} = \oint_{C}\left( xy^2\hat{i} -x^2y\hat{j}\right)\cdot \left( dx\hat{i} +2x\, dy\hat{j}\right) = \oint_{C}xy^2\, dx -2x^3y\, dy[/tex]

joemama69

oook check this

[tex]\oint[/tex]xy²dx - 2x³dy

P = xy² Q = 2x³y

[tex]\int[/tex][tex]\int[/tex]6x²y - 2xy dy dx

-1 < x < 1, x²<y<1

[tex]\int[/tex]-3x⁶ + x³ + 3x² - x dx

= 8/7

benorin