# When the gradient of a vector field is symmetric?

1. Oct 4, 2011

### boyboy400

1. The problem statement, all variables and given/known data

"A gradient of a vector field is symmetric if and only if this vector field is a gradient of a function"
Pure Strain Deformations of Surfaces
Marek L. Szwabowicz
J Elasticity (2008) 92:255–275
DOI 10.1007/s10659-008-9161-5

f=5x^3+3xy-15y^3
So the gradient of this function is a vector field, right? Now the grad of this grad is a tensor which is symmetric and according to Marek it's always like that.
Can you guys think of any reason or proof for it?

2. Relevant equations

3. The attempt at a solution
Maybe it has something to do with double differentiation...but I can't figure out why...

Last edited: Oct 4, 2011
2. Oct 4, 2011

### HallsofIvy

It's based on the "mixed partial derivative property":
As long as the derivatives are continuous,
$$\frac{\partial f}{\partial x\partial y}= \frac{\partial f}{\partial y\partial x}$$