Definition of Curl. Can anyone derive the gradient operator?

[Curl]

"Definition" of Curl. Can anyone derive the gradient operator?

Can anyone prove why this equality is true?
http://en.wikipedia.org/wiki/Curl_(mathematics)#Definition

Wikipedia says it is defined, however that's BS since the gradient operator was already defined so this needs to be proven. Either you take this for a definition and prove that the little "inverted triangle" is a derivative operator, or you prove the equality and don't call it a definition.

I can't tell how to go about proving that differentiating a vector field with a weird determinant is EQUAL to the loop integral of F*dr divided by the area enclosed (as the are goes to zero).

Its probably not hard, the cross product comes out of the "moment" of the field about a point, however I don't quite see how the derivative comes in.

 

[Char. Limit]

Well, this is a slight abuse of notation, but here's another way that the operator is defined.

\nabla = \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k}

Obviously, this can be generalized to higher dimensions, and as you can see, this only makes sense when cross-producted with a vector...

at least, I think that's how it goes.

 

[lurflurf]

Why is definition in quotes? When was the gradient operator was already defined? What does gradient operator have to do with curl? That definition of curl is senseable and standard, though other definitions are possible. Just because we write
grad(something)=∇(something)
and
curl(something)=∇×(something)
does not mean that
curl(something)=∇×(something)=∇(×something)=grad(×something)
or
curl(something)=∇×(something)=∇(×)(something)=grad(×)(something)
are valid as ×something and grad(×) are not meaningful.
∇× should be thought of as a symbol for curl, not a gradient of anything.

 

[Curl]

Okay then prove that grad X Field is equal to the limiting value of the loop integral F*dr divided by area enclosed by the path, as the area approaches zero.

When we compute curl we differentiate the vector field using a determinant. How do we know that it gives the same answer as doing a tiny loop integral around the area of interest and dividing by the are enclosed by the loop?

EDIT: Okay I finally found my book and its proven using stokes' theorem.

 

[billiards]

I think you're confused in the meaning of the "upside down" triangle.

The upside down triangle is the del operator. http://en.wikipedia.org/wiki/Del

It is invoked in the definition of grad (and div and curl). I think you are thinking that the upside down triangle is grad?

 

[arildno]

Okay then prove that grad X Field is equal to the limiting value of the loop integral F*dr divided by area enclosed by the path, as the area approaches zero.

When we compute curl we differentiate the vector field using a determinant. How do we know that it gives the same answer as doing a tiny loop integral around the area of interest and dividing by the are enclosed by the loop?

EDIT: Okay I finally found my book and its proven using stokes' theorem.

It is certainly important to be able to connect together different definitions, by means of proof!

What you however should focus on, with the wiki-definition, is that they define something they call "curl" of a fluid as the net circulation contained in a unit area, as that area shrinks to zero.

Thus, the curl is a measure of the local rotation rate of the fluid.

That this can also be calculated directly by a swift differentiation operation, rather than by a tricky limiting operation upon an integral, is indeed, one of the many wonders of Stokes' theorem!