- #1

greg_rack

Gold Member

- 363

- 79

- Homework Statement
- Use Stokes' theorem to evaluate ##\iint_{\textbf{S}}^{}curl \textbf{F}\cdot d\textbf{S}##, with ##\textbf{F}=<xyz, xy, x^2yz>## and S consisting of the top and the four sides(no bottom) of the cube with vertices ##(\pm 1,\pm 1,\pm 1)##

- Relevant Equations
- Computable form of a line integral and Stokes' theorem

From Stokes we know that ##\iint_{\textbf{S}}^{}curl \textbf{F}\cdot d\textbf{S}=\int_{C}^{}\textbf{F}\cdot d\textbf{r}##.

Now, we can calculate the surface integral of the curl of F by calculating the line integral of F over the curve C.

The latter ends up being 0(I calculated it parametrizing all the segments of the cube, that was more than tedious).

I was wondering whether this result could've been determined/deduced even prior to performing all the calculations, maybe due to symmetry of the surface and/or to its "interaction" with this particular vector field; with my very poor abstraction skills, I couldn't find anything... I thought probably rewriting ##\int_{C}^{}\textbf{F}\cdot d\textbf{r}## in the scalar form ##\int_{C}^{}xyz\ dx + xy \ dy+ x^2yz \ dz ## could have helped, but I still don't see the trick, if any.

Now, we can calculate the surface integral of the curl of F by calculating the line integral of F over the curve C.

The latter ends up being 0(I calculated it parametrizing all the segments of the cube, that was more than tedious).

I was wondering whether this result could've been determined/deduced even prior to performing all the calculations, maybe due to symmetry of the surface and/or to its "interaction" with this particular vector field; with my very poor abstraction skills, I couldn't find anything... I thought probably rewriting ##\int_{C}^{}\textbf{F}\cdot d\textbf{r}## in the scalar form ##\int_{C}^{}xyz\ dx + xy \ dy+ x^2yz \ dz ## could have helped, but I still don't see the trick, if any.