Can Green's theorem be used in a plane other than the xy-plane?

Question

In the following 2D case, Green's theorem solves the following problem:

$$\vec{F}=\langle{xy+\ln{(\sin{e^{x})},x^2+e^{y^2}}}\rangle$$

$$\oint_C\vec{F}\cdot{d\vec{r}}=\iint_Dx\space{dA}$$

where C is the unit circle $x^2+y^2=1$, and D is the unit disk $x^2+y^2\le{1}$.

But what if instead the problem were:

$$\vec{F}=\langle{\frac{\sqrt{2}}{2}(xy+\ln{(\sin{e^x}}),x^2+e^{y^2},-\frac{\sqrt{2}}{2}(xy+\ln{(\sin{e^x}})}\rangle$$

$$\oint_C\vec{F}\cdot{d\vec{r}}$$

And $C$ is instead the unit circle $\vec{r}(t)=\langle{\frac{\sqrt{2}}{2}\cos{t},\sin{t},-\frac{\sqrt{2}}{2}\cos{t}}\rangle$.

This is the same problem as the first one, but rotated 45 degrees about the y-axis. I know I could solve the second problem by simply rotating the path and the vector field back 45 degrees. But is there any way to apply Green's theorem to the second problem directly?

Are you trying to generalize Green's theorem to higher dimensions? — Mhenni Benghorbal, May 22 '13 at 13:30
@MhenniBenghorbal not exactly, because I'm still trying to find a line integral, and the path and vector field still lie within a plane. It's just a different plane than the xy-plane. — rurouniwallace, May 22 '13 at 13:36

score 2 · Accepted Answer · answered May 22 '13 at 14:15

2

Green's theorem is just a special case of Stokes' theorem (which is just a special case of the generalized Stokes' theorem).

$$\oint_{\partial M} d\ell \cdot F = \int_M dA \cdot (\nabla \times F)$$

answered May 22 '13 at 14:15

Muphrid

19,902

Can Green's theorem be used in a plane other than the xy-plane?

1 Answers1