- #1
sikrut
- 49
- 1
Use Stokes's theorem to show that line integral of ##\vec{F}(\vec{r})## over an curve ##L##, given by ##\int_L \vec{F}(\vec{r}) d\vec{r}##, depends only on the start and endpoint of ##L##, but not on the trajectory of ##L## between those two points.
Hint: Consider two different curves, ##L## and ##M##, say, which share a common start and endpoint. Construct from them a closed curve ##C## to which Stokes's theorem can be applied.
$$\vec{F}(\vec{r}) \equiv [F_x(\vec{r}), F_y(\vec{r}), F_z(\vec{r})] = [2z^2, 3z^2, (4x+6y)z] \rightarrow where \rightarrow \vec{r} \equiv [x,y,z].$$
Hint: Consider two different curves, ##L## and ##M##, say, which share a common start and endpoint. Construct from them a closed curve ##C## to which Stokes's theorem can be applied.
$$\vec{F}(\vec{r}) \equiv [F_x(\vec{r}), F_y(\vec{r}), F_z(\vec{r})] = [2z^2, 3z^2, (4x+6y)z] \rightarrow where \rightarrow \vec{r} \equiv [x,y,z].$$