5.9 Green's Theorem

Green's Theorem

Introduction

Circulation

Green's Theorem

What happens when vector fields are not smooth?

We will consider the vector field $\vF = \langle 2y,3x^2 y\rangle$, which is defined on the entire $xy$-plane. Suppose that we want to calculate the circulation of $\vF$ around the circle $C$ of radius $2$, centered at $(0,0)$, and oriented counterclockwise.

Verify that $\vF$ is not path-independent by calculating the circulation of $\vF$ around the circle $C$ (Use Theorem 5.5.2). The SageMath cell below is set up to assist you with this, but you will need to supply a parametrization of $C$ on line 4.

Work through this exercise and explain your reasoning step by step.

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

The curve $C_2$ is the triangle with vertices $(0,0)$, $(3,0)$, and $(3,3)$ (oriented counterclockwise) and the vector field is $\vG = \langle y^2, 3xy\rangle$.

Find the circulation density of $\vF$ (i.e., the integrand of the double integral in Theorem 5.9.1).

Suppose that $C$ is the unit circle centered at the origin. Without doing any calculations, what can you say about $\oint_C \vF\cdot d \vr$? What does this tell you about if $\vF$ is path-independent?

What would you get if you integrated the circulation density of $\vF$ over the region bounded by $C$?

Do the previous two parts contradict Theorem 5.9.1? Explain your reasoning.

Is the vector field $\vG = \displaystyle\frac{x}{x^2+y^2}\vi +\frac{y}{x^2+y^2}\vj$, which is shown in , path-independent? Why or why not?

Suppose that $C$ is the unit circle centered at the origin. Find $\oint_C \vG\cdot d \vr$. Can you do this using Theorem 5.9.1?

The types of regions in $\R^2$ to which Theorem 5.9.1 applies are formally called simply connected regions. To be precise, a simply connected region $D$ in $\R^2$ is a set so that there is a path between every pair of points in $D$ that stays inside $D$ and any simple closed curve in $D$ can be shrunk to a point while remaining inside $D$. We can think of a simply connected region as being a region that does not have any “holes”. There are many instances where we can find a way to apply Theorem 5.9.1 multiple times to work with regions that are not simply connected, however.

This exercise presents another occasion where Theorem 5.9.1 can be used to convert a double integral into a line integral. - Recall from Section 4.5 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 $D$. - Find the centroid of the triangle with vertices $(0,0)$, $(a,0)$, and $(0,b)$ for real numbers $a,b>0$.