Unit 7: Vector Fields, Path Integrals & Green's Theorem

1. Vector Fields in Space

A vector field on $\mathbb{R}^3$ is a function $\vec{F}$ that assigns to each point $(x, y, z)$ a three-dimensional vector:

Physical examples include fluid velocity fields $\vec{v}(x, y, z)$, gravitational fields $\vec{F}_g = -\frac{G M m}{r^3}\vec{r}$, and electrostatic fields $\vec{E} = \frac{q}{4\pi\epsilon_0 r^3}\vec{r}$.

If there exists a scalar function $f$ such that $\vec{F} = \nabla f$, then $\vec{F}$ is called a conservative vector field, and $f$ is called a scalar potential for $\vec{F}$.

2. Divergence of a Vector Field

Definition: The divergence of a vector field $\vec{F} = \langle P, Q, R \rangle$ is the scalar field defined by the symbolic dot product with the del operator $\nabla = \langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \rangle$:

Physical Meaning:

The divergence measures the net rate of outward fluid flux per unit volume at a given point:

• $\text{div} \, \vec{F} > 0$: The point is a source (fluid is being produced or expanding).
• $\text{div} \, \vec{F} < 0$: The point is a sink (fluid is draining or compressing).
• $\text{div} \, \vec{F} = 0$: The field is incompressible or solenoidal (e.g., magnetic fields $\nabla \cdot \vec{B} = 0$).

3. Curl of a Vector Field

Definition: The curl of $\vec{F} = \langle P, Q, R \rangle$ is the vector field defined by the symbolic cross product with $\nabla$:

Physical Meaning:

The curl measures the tendency of particles to rotate about the axis parallel to $\text{curl} \, \vec{F}$ (vorticity). If a tiny paddle wheel is placed in a fluid flow, it rotates with angular velocity proportional to $|\text{curl} \, \vec{F}|$.

• If $\text{curl} \, \vec{F} = \vec{0}$ everywhere, the vector field is called irrotational.

4. Fundamental Vector Differential Identities

1. Identity 1 (Curl of a Gradient is Zero):

If $f$ has continuous second partial derivatives:

Significance: Every conservative vector field is irrotational.

2. Identity 2 (Divergence of a Curl is Zero):

If $\vec{F}$ has continuous second partial derivatives:

Significance: A vector field cannot be the curl of another field unless its divergence is identically zero.

3. The Laplacian:

1. Line Integrals of Scalar Fields

Let $C$ be a smooth space curve parameterized by $\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$ for $a \le t \le b$. The line integral of a scalar function $f$ along $C$ with respect to arc length is:

If $f(x, y, z) = \rho(x, y, z)$ is linear mass density, the integral gives the total mass of the wire.

2. Line Integrals of Vector Fields (Work and Circulation)

The line integral of a vector field $\vec{F} = \langle P, Q, R \rangle$ along an oriented curve $C$ is the integral of the tangential component of $\vec{F}$:

• Work: If $\vec{F}$ is a force field, $\int_C \vec{F} \cdot d\vec{r}$ represents the total work done by the field in moving a particle along curve $C$.
• Circulation: When $C$ is a closed curve, $\oint_C \vec{F} \cdot d\vec{r}$ is called the circulation of $\vec{F}$ around $C$.

3. The Fundamental Theorem for Line Integrals

Theorem: Let $C$ be a smooth curve parameterized by $\vec{r}(t)$ for $a \le t \le b$. If $f$ is a continuously differentiable scalar function, then:

Proof:

By the Chain Rule:

By the single-variable Fundamental Theorem of Calculus:

Fundamental Consequences:

1. Path Independence: The line integral of a conservative vector field depends only on the starting point $A = \vec{r}(a)$ and ending point $B = \vec{r}(b)$, and is completely independent of the path taken between them.

2. Closed Loops: For any closed loop $C$ ($\vec{r}(a) = \vec{r}(b)$):

George Green (1828) discovered a profound theorem connecting a line integral around the boundary of a plane region to a double integral over the interior of the region.

1. Statement of Green's Theorem

Theorem (Green's Theorem): Let $D$ be a positively oriented, piecewise-smooth, simply connected region in the $xy$-plane, bounded by a simple closed curve $C = \partial D$. If $P(x, y)$ and $Q(x, y)$ have continuous partial derivatives on an open region containing $D$, then:

Positive Orientation: $C$ is traversed counterclockwise, so that the region $D$ always remains on the left as a traveler walks along the boundary.

2. Proof for Simple Regions

Assume $D$ is simultaneously Type I ($a \le x \le b, g_1(x) \le y \le g_2(x)$) and Type II.

We prove $\oint_C P \, dx = -\iint_D \frac{\partial P}{\partial y} \, dA$:

Now evaluate $\oint_C P \, dx$. The boundary consists of four paths: bottom curve $C_1$, right vertical edge $C_2$, top curve $C_3$ (traversed right-to-left), and left edge $C_4$. Along vertical edges, $dx = 0$.

Thus:

An identical argument treating $D$ as a Type II region proves $\oint_C Q \, dy = \iint_D \frac{\partial Q}{\partial x} \, dA$. Adding the two equations yields Green's Theorem. $\blacksquare$

3. Planar Area via Boundary Line Integrals

If we choose functions $P, Q$ such that $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1$:

1. $P = 0, Q = x \implies \text{Area} = \oint_C x \, dy$
2. $P = -y, Q = 0 \implies \text{Area} = -\oint_C y \, dx$
3. $P = -\frac{y}{2}, Q = \frac{x}{2} \implies \mathbf{A(D) = \frac{1}{2}\oint_C (x \, dy - y \, dx)}$

This formula allows exact calculation of enclosed area solely by integrating along the outer boundary perimeter.