Labware - MA35 Multivariable Calculus - Two Variable Calculus



Green's Theorem (Page: 1 | 2 )


Let P(x,y) and Q(x,y) be differentiable functions of x and y. Let D be a region in the plane and let C be the boundary of D. Green's theorem states that C P(x,y) dx + Q(x,y) dy = ∫∫D (- Py(x, y) +Qx(x, y)) dxdy. where D is the region in the plane bounded by the oriented curve C



  • 1. Let p(x, y) equal some function of x and let q(x, y) equal some function of y. What is the value of the integral shown at the bottom of the control window? Why does this occur?
  • 2. For either or both of these demos, let p(x, y) = -y and q(x, y) = 0 and observe the properties of the diagrams shown in the different windows. Try changing the size and location of the region. Then do the same for p(x, y) = 0 and q(x, y) = x. What do you notice in comparing the results of these two trials? What geometric property of the region D is being calculated?
  • 3. In the first demo, Green's theorem for rectangles, try moving the point that starts out as the top-right vertex of the rectangle so that it becomes the top-left vertex. How does the integral change? Why does this change occur?

    Now see what happens when the point is the bottom-left vertex or the bottom-right vertex. Explain why this occurs.