# Why elementary work is not an exact differential?

1. Dec 14, 2011

### phydev

Why elementary work is defined as δW=Fdr?
My ques. is not on the definition; it is on why it cannot be dW=Fdr?

2. Dec 14, 2011

### tommyli

It is only an exact differential int he case of a conservative force. Precisely, if there is a potential we write

dU = -Fdr

which implies that F = -dU/dr.

3. Dec 14, 2011

### phydev

I know that in case of conservative/potential field δW=-dU.

Reference: Fundamental Laws of Mechanics, IE irodov
from equation 3.1 to 3.49
wherever needed he used δA for elementary work, in general!

4. Dec 14, 2011

### D H

Staff Emeritus
Work is a line integral:

$$W=\oint \mathbf F \cdot d\mathbf r$$

It is path independent in the case of a conservative force, but in general the integral depends on the path. In other words, $\mathbf F \cdot d\mathbf r$ is not an exact differential by definition (an exact differential is path independent).

5. Dec 14, 2011

### phydev

6. Dec 14, 2011

### tommyli

Whenever you write df it implies that the differential operator is applied to the function f, so if you write work as an exact differentail dW implicitly you are saying force can be written as the derivative of some function of spatial coordinates.

7. Dec 15, 2011

### phydev

Yeah! right!
Now, what does it further imply?
Cannot force be derivative of a function of spacial coordinates?

I think I have got it, but request you to elaborate so that I may confirm.

Thanks!

8. Dec 15, 2011

### Hobold

If work was an exact differential, for any two points a and b, you could write that the work to go from one point is F(b) - F(a), where F' is work. But this is most certainly not true, as this is saying work is a function of state, i.e. if you have a point, you'd have a work associated to it. This is false, as work is something you use to go from one state to another. It's pretty much like heat. Heat is also not a function of state and depends on the path.

9. Dec 15, 2011

### tommyli

If force is a derivative of some function of spatial coordinates, this function is called the potential energy, and the force is conservative. Non-conservative forces are certainly not derivatives of any function.

10. Dec 17, 2011

### phydev

well... thanks,
I concluded the same!!