# Curl of a function

1. Jan 14, 2007

### nolanp2

if the grad of the curl of a function is always zero does this mean the magnitude of the curl is constant? or am i way off here?

2. Jan 14, 2007

### cristo

Staff Emeritus

Last edited: Jan 14, 2007
3. Jan 14, 2007

### uart

Firstly note that it is NOT true that the grad of the curl of a function is always zero.

Actually nolan I think you are mixing up two seperate properties of "curl" there to come to this misconception. The properties that you are thinking of are probably these two :

1. The div of the curl of a function is always zero. (div is not the same thing as grad ok).

2. If a given function can be written as the grad of another function then that given function has a curl of zero.

Note that neither of thse two things (taken seperately or together) imply what you have written.

4. Jan 14, 2007

### uart

Ok looking at your question again I can see that it's probably the second property that I wrote that is the one you have gotten mixed up. It's the curl of a grad that is zero (a zero vector) not the way you put it, grad of a curl is not neccessarily zero

5. Jan 14, 2007

### mathwonk

that the curl of the grad is zero is essentially trivial. it follows from the trivial fact that the integral of GRAD is zero around a closed curve. this is because that integral is evaluated by subtracting the values of some function at the two endpoints, which are equal.

that in turn is true by the FTC, which holds by the trivial fact that the derivative of the area function is the height functon, which holds trivially because the area of a rectangle is the base times the height.

on the other hand, sticking all these trivialities together, maybe we have a pathologically amazing theorem!

more interesting is the investigation of forms with zero curl which are not gradients, like dtheta.

6. Jan 14, 2007

### uart

One other thing I should have noticed about the original post. "Grad of Curl" isn't even defined. Grad is normally applied to a scaler field, giving a vector result. So "div of curl" makes sense but "grad of curl" doesn't. So my guess is that the OP is getting confused between grad and div.