Unit 6 - Notes

MTH166 6 min read

Unit 6: Vector calculus II

1. Line Integrals

1.1 Definition

A line integral generalizes the concept of a definite integral to a curve in a plane or in space. It integrates a function (scalar or vector) along a curve $C$ .

Vector Line Integral (Work Integral):
If $\mathbf{F}(x, y, z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ is a vector field defined on a curve $C$ given by the position vector $\mathbf{r}(t)$ , the line integral of $\mathbf{F}$ along $C$ is:

$\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt$

Where:

$t$ varies from $a$ to $b$ .
$d\mathbf{r} = dx\mathbf{i} + dy\mathbf{j} + dz\mathbf{k}$ .
Physical Interpretation: This integral represents the Work Done by the force field $\mathbf{F}$ moving a particle along curve $C$ .

1.2 Evaluation Methods

Parametric Form: Convert $x, y, z$ into functions of a single parameter $t$ (e.g., $x(t), y(t), z(t)$ ), substitute into the integral, and integrate with respect to $t$ .
Explicit Form: If $y = f(x)$ , substitute $y$ and $dy = f'(x)dx$ to integrate with respect to $x$ .

1.3 Path Independence and Conservative Fields

A vector field $\mathbf{F}$ is conservative if the line integral depends only on the endpoints of the path, not the path itself.

Condition: $\mathbf{F} = \nabla \phi$ (F is the gradient of a scalar potential $\phi$ ).
Test for Conservatism: $\text{curl } \mathbf{F} = \nabla \times \mathbf{F} = \mathbf{0}$ .
Fundamental Theorem for Line Integrals:
$\int_A^B \nabla \phi \cdot d\mathbf{r} = \phi(B) - \phi(A)$

2. Green's Theorem

Green's Theorem relates a line integral along a simple closed curve $C$ to a double integral over the plane region $R$ bounded by $C$ .

2.1 Statement

Let $C$ be a positively oriented (counter-clockwise), piecewise smooth, simple closed curve in the plane, and let $D$ be the region bounded by $C$ . If $P(x, y)$ and $Q(x, y)$ have continuous partial derivatives on an open region containing $D$ , then:

$\oint_C (P \, dx + Q \, dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$

2.2 Applications

Simplifying Integration: Converting difficult line integrals into easier area integrals (or vice-versa).
Area Calculation: The area of region $D$ can be found using line integrals:
$\text{Area}(D) = \frac{1}{2} \oint_C (x \, dy - y \, dx)$

A conceptual diagram illustrating Green's Theorem. On the left, show an arbitrary irregular 2D shape... — AI-generated image — may contain inaccuracies

3. Surface Area and Surface Integrals

3.1 Surface Area

If a smooth surface $S$ is defined by $z = f(x, y)$ over a region $R$ in the $xy$ -plane, the surface area $A(S)$ is:

$A(S) = \iint_R \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} \, dA$

3.2 Surface Integrals

1. Scalar Surface Integral:
Used to calculate mass of a shell with variable density.
$\iint_S f(x, y, z) \, dS = \iint_R f(x, y, g(x,y)) \sqrt{1 + (g_x)^2 + (g_y)^2} \, dA$

2. Vector Surface Integral (Flux):
Measures the flow of a vector field $\mathbf{F}$ across a surface $S$ .
$\text{Flux} = \iint_S \mathbf{F} \cdot \mathbf{n} \, dS$

Where:

$\mathbf{n}$ is the unit normal vector to the surface.
For a surface $z = g(x,y)$ , $\mathbf{n} dS = (-\frac{\partial g}{\partial x}\mathbf{i} - \frac{\partial g}{\partial y}\mathbf{j} + \mathbf{k}) dA$ (for upward orientation).

4. Stokes' Theorem

Stokes' Theorem generalizes Green's Theorem to 3D surfaces. It relates the surface integral of the curl of a vector field to the line integral around the boundary of that surface.

4.1 Statement

Let $S$ be an oriented smooth surface bounded by a simple, closed, smooth boundary curve $C$ with positive orientation. If $\mathbf{F}$ is a vector field, then:

$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS$

4.2 Key Concepts

Orientation (Right-Hand Rule): If the fingers of your right hand curl in the direction of integration around $C$ , your thumb points in the direction of the normal vector $\mathbf{n}$ .
Independence of Surface: The flux of the curl depends only on the boundary curve $C$ . Any surface $S$ sharing the same boundary $C$ yields the same result.

A detailed 3D diagram illustrating Stokes' Theorem. Show a floating, curved surface 'S' (like a soap... — AI-generated image — may contain inaccuracies

5. Gauss's Divergence Theorem

The Divergence Theorem relates the flow (flux) across a closed surface to the sum of sources and sinks (divergence) inside the volume.

5.1 Statement

Let $V$ be a solid region bounded by a closed surface $S$ oriented by outward unit normals $\mathbf{n}$ . If $\mathbf{F}$ is a vector field defined on $V$ , then:

$\iint_S \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_V (\nabla \cdot \mathbf{F}) \, dV$

5.2 Physical Interpretation

Left Side (Surface Integral): Represents the total net flux of the fluid/field leaving the closed surface $S$ .
Right Side (Volume Integral): Represents the sum of the expansion or contraction of the fluid at every point inside the volume.
If $\nabla \cdot \mathbf{F} > 0$ , net flow is outward (Source).
If $\nabla \cdot \mathbf{F} < 0$ , net flow is inward (Sink).
If $\nabla \cdot \mathbf{F} = 0$ , the field is Solenoidal (Incompressible).

5.3 Comparison of Integral Theorems

Theorem	Relates	Dimension	Formula
Fund. Thm. Line Int.	Line $\leftrightarrow$ Endpoints	1D	$\int_A^B \nabla \phi \cdot d\mathbf{r} = \phi(B) - \phi(A)$
Green's Theorem	Line $\leftrightarrow$ Plane Area	2D (Plane)	$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D (\text{curl}_z \mathbf{F}) dA$
Stokes' Theorem	Line $\leftrightarrow$ Surface	3D (Surface)	$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\text{curl } \mathbf{F}) \cdot \mathbf{n} dS$
Divergence Theorem	Surface $\leftrightarrow$ Volume	3D (Solid)	$\iint_S \mathbf{F} \cdot \mathbf{n} dS = \iiint_V (\text{div } \mathbf{F}) dV$

A visual comparison diagram of the Divergence Theorem. The image should feature a solid 3D object, s... — AI-generated image — may contain inaccuracies

Unit 5