Green's Theorem and polar coordinates

In summary, the conversation discusses the use of Green's Theorem to solve a specific integral over a closed curve, using polar coordinates to determine the limits of integration. The final result, after converting to polar coordinates, is 0, which may seem suspicious but is in fact a valid answer.
  • #1
sailsinthesun
24
0

Homework Statement


Using Green's Theorem, (Integral over C) -y^2 dx + x^2 dy=____________
with C: x=cos t y=sin t (t from 0-->2pi)

Homework Equations


(Integral over C) Pdx + Qdy=(Double integral over D) ((partial of Q w.r.t. x)-(partial of P w.r.t. y))dxdy

The Attempt at a Solution



I'm stuck from here. I remember the professor said to use polar coordinates which makes sense to get the limits on D, but how do I convert the integral (-y^2 dx + x^2 dy) to polar?

In my method I go from the original integral over C to (double integral over D) 2x+2y dxdy. I convert this to polar to get limits on D and I get (integral from 0 to 2pi)(integral from 0 to 1) 2rcos(theta)+2rsin(theta)*r*dr*d(theta). Once you calculate all of this you get 0 which I don't believe is correct. Any help?
 
Last edited:
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  • #2
Why don't you think 0 is correct? It's quite a nice number, one of my favorites, actually.:smile:
 
  • #3
People sometimes want answers in round numbers. You can't get any rounder than zero.:smile:
 

1. What is Green's Theorem and how does it relate to polar coordinates?

Green's Theorem is a fundamental theorem in vector calculus that relates the line integral around a simple closed curve in the plane to the double integral over the region enclosed by the curve. It can be applied to both Cartesian and polar coordinates, providing a way to compute area and line integrals in polar coordinates.

2. How is the formula for Green's Theorem different in polar coordinates compared to Cartesian coordinates?

In Cartesian coordinates, the formula for Green's Theorem is ∬(∂Q/∂x - ∂P/∂y)dA, where P and Q are functions of x and y. In polar coordinates, the formula becomes ∬(1/r)(∂(rQ)/∂θ - ∂(rP)/∂r)dA, where P and Q are now functions of r and θ.

3. What are some applications of Green's Theorem in polar coordinates?

Green's Theorem in polar coordinates is commonly used in physics and engineering to calculate work done by a force field, as well as to find the center of mass and moment of inertia of a region. It also has applications in fluid dynamics and electromagnetism.

4. Can Green's Theorem be extended to higher dimensions?

Yes, Green's Theorem can be extended to higher dimensions through the use of the generalized Stokes' Theorem. This theorem relates surface integrals to line integrals, and can be applied to regions in three-dimensional space using cylindrical or spherical coordinates.

5. How can I use Green's Theorem to evaluate a line integral in polar coordinates?

To evaluate a line integral in polar coordinates using Green's Theorem, first express the integrand as a function of r and θ. Then, use the formula ∫C P(x,y)dx + Q(x,y)dy = ∬(∂Q/∂x - ∂P/∂y)dA, where P and Q are the components of the integrand in polar form. Finally, evaluate the resulting double integral over the region enclosed by the curve.

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