When the gradient of a vector field is symmetric?

In summary, the conversation discusses the concept of symmetry in a gradient of a vector field. It is mentioned that a vector field is symmetric if and only if it is a gradient of a function. The paper referenced by Marek L. Szwabowicz further explains this concept and provides a proof for it. It is suggested that the concept may be related to double differentiation and the mixed partial derivative property.
  • #1
boyboy400
9
0

Homework Statement



"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?

Homework Equations


The Attempt at a Solution


Maybe it has something to do with double differentiation...but I can't figure out why...
 
Last edited:
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  • #2
It's based on the "mixed partial derivative property":
As long as the derivatives are continuous,
[tex]\frac{\partial f}{\partial x\partial y}= \frac{\partial f}{\partial y\partial x}[/tex]
 

FAQ: When the gradient of a vector field is symmetric?

What is a vector field?

A vector field is a mathematical concept that assigns a vector to each point in a given space. The vectors are typically represented by arrows, with the length and direction of the arrow indicating the magnitude and direction of the vector at that point.

What does it mean for a gradient to be symmetric?

A gradient is considered symmetric if the order in which the partial derivatives are taken does not affect the result. In other words, if the partial derivatives are taken in any order and the resulting vector field is the same, then the gradient is considered symmetric.

Why is it important for a vector field's gradient to be symmetric?

Having a symmetric gradient allows for simpler calculations and makes it easier to solve certain problems in physics and engineering. It also helps to ensure that the resulting vector field is unique and well-behaved.

How can you determine if a vector field's gradient is symmetric?

The symmetry of a gradient can be determined by checking if the mixed partial derivatives are equal. If they are equal, then the gradient is symmetric. Additionally, the symmetry can also be verified by checking if the Jacobian matrix of the vector field is symmetric.

Are all vector fields' gradients symmetric?

No, not all vector fields have a symmetric gradient. The symmetry depends on the specific properties and equations of the vector field. However, symmetric gradients are commonly found in physical systems that exhibit certain symmetries, such as rotational or translational symmetry.

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