Green's Theorem

For each of the curves described below, find the circulation of the given vector field around the curve. Do this both by calculating the line integral directly as well as by calculating the double integral from Green's Theorem.

1. The curve $C_1$ is the circle of radius $3$ centered at the point $(2,1)$ (oriented counterclockwise) and the vector field is $\vF = \langle y^2, 5x+2xy\rangle$.

2. The curve $C_2$ is the triangle with vertices $(0,0)$, $(3,0)$, and $(3,3)$ (or…

Sometimes, the value in Green's Theorem is in converting a double integral into a line integral. Recall that the area of a region $R$ in the plane can be found by calculating $\iint_R 1\, dA$.

1. Find a smooth vector field $\vF$ such that the circulation density of $\vF$ is $1$ everywhere.

2. Write out both integrals in Green's Theorem for the vector field you selected in part.

3. For positive real numbers $a,b$, the ellipse $x^2/a^2 + y^2/b^2 = 1$ can be param…

This exercise presents another occasion where Green's Theorem can be used to convert a double integral into a line integral.

1. Recall from Applications of Double Integrals that the centroid of a lamina $D$ of area $A$ is given by . Find vector fields $\vF$ and $\vG$ so that , where $C$ is the boundary of the lamina $D$.

2. Find the centroid of the triangle with vertices $(0,0)$, $(a,0)$, and $(0,b)$ for real numbers $a,b>0$.