curl

**arbolis** · April 16th 2010, 10:45 AM

If $\vec F=\vec \nabla \varphi$ , show that $\vec \nabla \times \vec F=\vec 0$ . $\vec F$ is a vector field and $\varphi$ is a scalar field.
I know it's an obvious question but I can't remember how to formally prove it.
I know it's true if $\frac{\partial ^2 \varphi}{\partial y \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial y} ,\frac{\partial ^2 \varphi}{\partial x \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial x}$ and $\frac{\partial ^2 \varphi}{\partial x \partial y} =\frac{\partial ^2 \varphi }{\partial y \partial x}$ in Cartesian coordinates, I've showed it for $\varphi (x,y,z)=f(x)g(y)h(z)$ but not for a general form of $\varphi$ . I think I'm over-complicating this.
Could you give me a tip?

**Failure** · April 16th 2010, 11:10 AM

Originally Posted by arbolis

If $\vec F=\vec \nabla \varphi$ , show that $\vec \nabla \times \vec F=\vec 0$ . $\vec F$ is a vector field and $\varphi$ is a scalar field.
I know it's an obvious question but I can't remember how to formally prove it.
I know it's true if $\frac{\partial ^2 \varphi}{\partial y \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial y} ,\frac{\partial ^2 \varphi}{\partial x \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial x}$ and $\frac{\partial ^2 \varphi}{\partial x \partial y} =\frac{\partial ^2 \varphi }{\partial y \partial x}$ in Cartesian coordinates, I've showed it for $\varphi (x,y,z)=f(x)g(y)h(z)$ but not for a general form of $\varphi$ . I think I'm over-complicating this.
Could you give me a tip?

It follows from Stokes Theorem, for example. But maybe that's not the kind of tip that you have been expecting...

**arbolis** · April 16th 2010, 12:56 PM

Originally Posted by Failure

It follows from Stokes Theorem, for example. But maybe that's not the kind of tip that you have been expecting...

I'm not expecting a particular kind of help so I appreciate yours.
I do not know how to relates Stokes' theorem with this problem.
I know that $\int _{\gamma} \vec F \cdot d\vec r = \int _{S} ( \vec \nabla \times \vec F ) \cdot d\vec S$ .
In the example given, I do not know where $\vec F$ and $\varphi$ are defined.
By intuition I could take $\gamma$ as a circle with radius R tending to infinite and S being the upper half sphere of radius R but I'm not sure it makes sense nor how do I use $\varphi$ . Am I, at least, on the right direction? If so I'll think more about it.

Edit: Now I'm thinking about using Clairaut's theorem. But I'd have to show that $\varphi$ is continuous and it is not given...

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