- #1
bugatti79
- 794
- 1
Homework Statement
Calculate the folowing directly and with greens theorem
Homework Equations
[itex] \int (x-y) dx + (x+y) dy[/itex]
C= x^2+y^2=4
The Attempt at a Solution
Directly
[itex]x= r cos \theta, y=r sin \theta, r^2=4, dx = -r sin \theta d \theta, dy= r cos \theta d \theta[/itex]
Substituting I get
[itex] \displaystyle \int_0^{2 \pi} (-r^2 sin \theta cos \theta +r^2 sin^2 \theta) d \theta+(r^2 cos^2 \theta +r^2 sin \theta cos \theta) d \theta[/itex]
[itex]=4 \int_0^{2 \pi} d \theta= 8 \pi[/itex]
Greens theorem
[itex] \displaystyle \int \int_R (G_x -F_y)dA= \int_0^{2 \pi}\int_0^2 2 r dr d \theta = 2 \pi[/itex]...? I can't spot the error!