Question #36742

1 Answer
Mar 6, 2017

Answer:

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#