Question

Question #36742

Answer 1

See below

Explanation:

The work done in a conservative force field is guaranteed to be zero only over a closed path.

By definition, a vector field is conservative if and only if there exists a potential function `V(mathbf x)` such that `mathbf F = - nabla V(mathbf r)`.

For a force field, we have the definition of work as:

`delta W = mathbf F cdot delta mathbf r`

`implies W = int_(Gamma(mathbf p, mathbf q)) mathbf F cdot d mathbf r`

....where `Gamma` is a curve from `mathbf p` to `mathbf q` in the field.

The Gradient Theorem , aka the Fudamental Theorem for Line Integrals, states that:

`int_(Gamma(mathbf p, mathbf q)) nabla V(mathbf r) cdot d mathbf r = V(mathbf q) - V(mathbf p)`

This is where the idea of path independence comes from.

Now, if `mathbf p = mathbf q`, then this becomes:

`oint nabla V(mathbf r) cdot d mathbf r = 0`