Line Integrals (continued)

It is possible that the line integral of the function, F, is independent of the curve (path), C,

in the x-y plane. This situation occurs when the force, F,

F = P(x,y) i + Q(x,y) j

is conservative In such a case, the curl of F must be zero.

curl of F = ∂Q/∂x - ∂P/∂y = 0 if the force is conservative

So ∂P/∂y = ∂Q/∂x

And the (vector field) force, F , can be expressed in terms of the gradient

of a potential function (scalar function, φ )

φ(x,y) . i.e.

F = grad (φ)

Suppose you were to evaluate the line integral

∫ P(x, y) dx + ∫ Q(x, y) dy

C C

where the path (curve) C is somewhat complicated. Then you might first check

to see if the “force” (vector field) is conservative.

F = P(x,y) i + Q(x,y) j

i.e. Does ∂P/∂y = ∂Q/∂x ? If so, the force, F, is conservative.

Then the line integral, ∫ F · dr , is independent of its path and you can simplify the

C

calculation by selecting an easier path.

Click here for examples involving line integrals.

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