1. curl

If $\displaystyle \vec F=\vec \nabla \varphi$, show that $\displaystyle \vec \nabla \times \vec F=\vec 0$. $\displaystyle \vec F$ is a vector field and $\displaystyle \varphi$ 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 $\displaystyle \frac{\partial ^2 \varphi}{\partial y \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial y} ,\frac{\partial ^2 \varphi}{\partial x \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial x}$ and $\displaystyle \frac{\partial ^2 \varphi}{\partial x \partial y} =\frac{\partial ^2 \varphi }{\partial y \partial x}$ in Cartesian coordinates, I've showed it for $\displaystyle \varphi (x,y,z)=f(x)g(y)h(z)$ but not for a general form of $\displaystyle \varphi$. I think I'm over-complicating this.
Could you give me a tip?

2. Originally Posted by arbolis
If $\displaystyle \vec F=\vec \nabla \varphi$, show that $\displaystyle \vec \nabla \times \vec F=\vec 0$. $\displaystyle \vec F$ is a vector field and $\displaystyle \varphi$ 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 $\displaystyle \frac{\partial ^2 \varphi}{\partial y \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial y} ,\frac{\partial ^2 \varphi}{\partial x \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial x}$ and $\displaystyle \frac{\partial ^2 \varphi}{\partial x \partial y} =\frac{\partial ^2 \varphi }{\partial y \partial x}$ in Cartesian coordinates, I've showed it for $\displaystyle \varphi (x,y,z)=f(x)g(y)h(z)$ but not for a general form of $\displaystyle \varphi$. 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...

3. 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 $\displaystyle \int _{\gamma} \vec F \cdot d\vec r = \int _{S} ( \vec \nabla \times \vec F ) \cdot d\vec S$.
In the example given, I do not know where $\displaystyle \vec F$ and $\displaystyle \varphi$ are defined.
By intuition I could take $\displaystyle \gamma$ 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 $\displaystyle \varphi$. 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 $\displaystyle \varphi$ is continuous and it is not given...