line integrals in goldstein

Line Integrals as Used in Goldstein PhL 7.29.08 Goldstein writes integrals of the form pdq done using a contour in the q-p plane, which I can think of as p(q)dq . I am unfamiliar with this kind of integral, so let's look at it more closely. Change variables I = y(x)dx // contour is in the x-y plane now Imagine this is around a closed contour of some arbitrary smooth shape (possibly convex). I want to claim that this integral gives the area inside the contour (it is not the arc length!), and all the examples I have played with seem to suggest that fact is true. Here is a way to prove this using the much more powerful and general Stokes theorem which I write as: ∫S curl F dA = C Fdl (1) Here we use a CCW contour and we have dA naively in the RHR direction. Consider the following very simple vector function F: F(x,y,z) = y so Fx = y and Fy = Fz = 0 Then on the RHS of Stokes we have Fdl = y dx Now (doing LHS) for our contour take a planar loop lying in the x,y plane, and for our surface S, take the plane region inside this contour. We then have dA = dA and curl F dA = (curl F)z dA = (∂xFy – ∂yFx) dA = – dA Then our LHS in (1) above is ∫S [– dA] = – Area of Loop Thus, my conclusion is this: y dx = Area of loop // if contour is done Clockwise! This agrees with any simple example you make, such as a first quadrant square. The loop referred to here is a loop in the x-y plane. Example: suppose the loop is a circle of radius a. We know then that x2+ y2 = a2 so y(x) = ± , where we use the + sign on the upper half of the contour and the – sign on the lower. Or, we could parameterize the loop with an angle θ and say y = a sinθ and x = acosθ and this would work over the entire loop, no need to select signs. Then dx = -asinθdθ and we have for our CCW loop y dx = a sinθ [ -asinθdθ] = -a2 sin2θ dθ = -4a2 sin2θ dθ = -4a2 (π/4) = -πa2 and we do in fact obtain the area of a circle. The subject of line integrals in general is discussed in my yellow Calculus II book and many other places.