Labware - MA35 Multivariable Calculus - Two Variable Calculus
 MA35 Labs 2 » Two Variable Calculus Contents2.4 Integration 2.5 Vector Analysis 2.5.2 Path Integrals 2.5.5 Conservative Vector Fields 2.5.6 Green's Theorem Search

Conservative Vector Fields

Text

A two-dimensional vector field F = (p(x,y),q(x,y)) is conservative if there exists a function f(x,y) such that F = ∇f.

Note that the circulation of ∇f(x,y) along a curve C equals ab fx(x(t),y(t))x'(t) + fy(x(t),y(t))y'(t) dt = ∫abd/dt(f(x(t),y(t)) dt = f(x(t),y(t))|ab = f(x(b),y(b)) - f(x(a),y(a)). In particular, if F(x,y) is a conservative vector field then the circulation of F along a closed curve is 0.

If f exists, then it is called the potential function of F.

If a two-dimensional vector field F(p,q) is conservative, then py = qx.

Suppose that F is conservative and so there does exist a function f such that F = ∇f. Then p = fx and q = fy. Now consider the mixed partials fxy = py and fyx = qx. By the equality of mixed partials, py = fxy = fyx = qx.

If a two-dimensional vector field F(p,q) is conservative, then its curl is identically zero.

Demos

 Conservative Vector Fields This demo allows you to test whether or not a field is conservative using two piecewise paths, each consisting of three line segments. Use the hotspots in the 2D window to move the endpoints of these line segments. In the 3D window, the wireframes display F⋅ds, and the green curves show the integral of F⋅ds over the two piecewise paths (circulation). You have the option of displaying the graph of some function f(x, y). If (p(x, y), q(x, y)) is the gradient of some function f(x, y) and the red and blue paths are started at the same point, you should be able to find a function graph on which the green curves lie. Two other conditions that will exist for conservative fields: If either the red or the blue curve is a closed curve, the circulation around that curve should be 0 (i.e. the green curve will end up at height 0 at the tip of the arrow, and the area of the wireframe above the x-y plane will equal the area of the wireframe below the x-y plane). If the red and blue curves have the same starting and ending points, then the circulation for each should be equal.

Exercises

• 1. For each of the following, use the demo to determine whether or not the vector field F is conservative. If it is conservative, find the potential function of F.
• F = (x, y)
• F = ( 1, y)
• F = ( 0, -9.8)
• F = (cos(x), sin(y))
• F = (-y, x)
• F = (x/(x2+y2), y/(x2+y2))
• 2. Why won't the green curves necessarily lie on the graph of f(x, y) (the function whose gradient is F) if the blue and red paths have different starting points? (Assume F is conservative.)