Practice MCQ

Unit 6 - Notes

MTH166

Unit 6: Vector calculus II

1. Line Integrals

A Line Integral generalizes the concept of a definite integral $\int_a^b f(x) dx$ to integrals over a curve $C$ in space (or a plane).

1.1 Line Integrals of Scalar Fields

If $f(x, y, z)$ is a scalar function and $C$ is a smooth curve parameterized by $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$ for $a \leq t \leq b$ , the line integral with respect to arc length $s$ is:

\int_C f(x, y, z) \, ds = \int_a^b f(\mathbf{r}(t)) \, |\mathbf{r}'(t)| \, dt

Key Components:

$ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt = |\mathbf{r}'(t)| \, dt$
Physical interpretation: If $f(x,y,z)$ represents the linear density of a wire shaped like $C$ , the integral represents the total mass of the wire.

1.2 Line Integrals of Vector Fields

If $\mathbf{F} = \langle P, Q, R \rangle$ is a vector field and $C$ is a curve parameterized by $\mathbf{r}(t)$ , 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

In differential form, this is often written as:

\int_C P \, dx + Q \, dy + R \, dz

Physical Interpretation (Work):
If $\mathbf{F}$ represents a force field, this integral calculates the Work done by the force in moving a particle along the curve $C$ .

1.3 Conservative Vector Fields and Path Independence

A vector field $\mathbf{F}$ is conservative if there exists a scalar potential function $\phi$ such that $\mathbf{F} = \nabla \phi$ .

Fundamental Theorem for Line Integrals:
If $\mathbf{F} = \nabla \phi$ and $C$ is a curve from point $A$ to point $B$ :

\int_C \nabla \phi \cdot d\mathbf{r} = \phi(B) - \phi(A)

Properties of Conservative Fields:

Path Independence: The integral depends only on the endpoints, not the path taken.
Closed Loops: The circulation around any closed curve is zero: $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$ .
Test for Conservatism (in 3D): $\text{curl } \mathbf{F} = \nabla \times \mathbf{F} = \mathbf{0}$ .

2. Green’s Theorem

Green's Theorem connects a line integral along a simple closed curve $C$ to a double integral over the plane region $D$ bounded by $C$ . It is the 2D special case of Stokes' Theorem.

2.1 Statement of the Theorem

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: It converts difficult line integrals into easier double integrals (or vice-versa).
Area Calculation: The area of region $D$ can be found using line integrals on the boundary $C$ :
$\text{Area}(D) = \oint_C x \, dy = -\oint_C y \, dx = \frac{1}{2} \oint_C (x \, dy - y \, dx)$

3. Surface Area and Surface Integral

Just as line integrals integrate over a curve, surface integrals integrate over a 2D surface in 3D space.

3.1 Parametric Surfaces

A surface $S$ is defined by a vector function of two variables $(u, v)$ in a domain $D$ :

\mathbf{r}(u, v) = x(u,v)\mathbf{i} + y(u,v)\mathbf{j} + z(u,v)\mathbf{k}

The Normal Vector:
The tangent vectors are $\mathbf{r}_u$ and $\mathbf{r}_v$ . The normal vector to the surface is:

\mathbf{N} = \mathbf{r}_u \times \mathbf{r}_v

3.2 Surface Area

The area of the parametric surface $S$ is given by:

A(S) = \iint_D |\mathbf{r}_u \times \mathbf{r}_v| \, du \, dv

Special Case (Graph of a function):
If the surface is defined explicitly as $z = g(x,y)$ , the surface area is:

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

3.3 Surface Integral of a Scalar Field

If $f(x, y, z)$ is a scalar function defined on surface $S$ :

\iint_S f(x, y, z) \, dS = \iint_D f(\mathbf{r}(u,v)) \, |\mathbf{r}_u \times \mathbf{r}_v| \, du \, dv

Application: If $f$ is density, the integral gives the mass of the surface shell.

3.4 Surface Integral of a Vector Field (Flux)

If $\mathbf{F}$ is a vector field, the surface integral (Flux) across oriented surface $S$ with unit normal vector $\mathbf{n}$ is:

\text{Flux} = \iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n} \, dS

Calculation Formula:

\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u,v)) \cdot (\mathbf{r}_u \times \mathbf{r}_v) \, du \, dv

Note: The direction of $\mathbf{r}_u \times \mathbf{r}_v$ determines the orientation (e.g., outward or upward).

4. Stokes’ Theorem

Stokes' Theorem relates a surface integral of the curl of a vector field over a surface $S$ to the line integral of the vector field over the boundary curve $C$ of that surface.

4.1 Statement of the Theorem

Let $S$ be an oriented piecewise-smooth surface bounded by a simple, closed, piecewise-smooth boundary curve $C$ (denoted $\partial S$ ) with positive orientation. Let $\mathbf{F}$ be a vector field with continuous partial derivatives. Then:

\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}

4.2 Orientation Rule (Right-Hand Rule)

The orientation of $S$ and $C$ must be compatible: If you curl the fingers of your right hand in the direction of integration around $C$ , your thumb must point in the direction of the normal vector $\mathbf{n}$ used for the surface integral.

4.3 Interpretation

Circulation: The line integral $\oint_C \mathbf{F} \cdot d\mathbf{r}$ represents the "circulation" of the fluid around the boundary.
Curl Summation: Stokes' theorem states that the macroscopic circulation around the boundary is equal to the sum of all the microscopic rotations (curls) over the surface.
Surface Independence: Since the integral depends only on the boundary $C$ , you can compute the integral over any surface $S$ that has $C$ as its boundary (often replacing a complex surface with a flat disk).

5. Gauss's Divergence Theorem

The Divergence Theorem relates a triple integral over a solid volume $V$ to a surface integral (flux) over the closed surface $S$ bounding that volume.

5.1 Statement of the Theorem

Let $V$ be a simple solid region and $S$ be the boundary surface of $V$ , given a positive (outward) orientation. Let $\mathbf{F}$ be a vector field with continuous partial derivatives. Then:

\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV

5.2 Physical Interpretation

Flux: The left side represents the net flow of the vector field (e.g., fluid, electric field) outward through the surface $S$ .
Divergence: $\nabla \cdot \mathbf{F}$ measures the rate at which "fluid" is being generated (source) or absorbed (sink) at a point.
Conservation: The theorem states that the total fluid leaving the closed surface is equal to the total fluid generated inside the volume.

5.3 Calculation Strategy

When to use: Use Divergence Theorem when calculating Flux across a closed surface (like a sphere, cube, or cylinder). It is usually much easier to integrate the divergence (a scalar) over the volume than to calculate surface integrals for every face of the solid.
Formula reminder:
If $\mathbf{F} = \langle P, Q, R \rangle$ , then $\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$ .

Summary of Integral Theorems

Theorem	Dimension	Relationship	Formula
Fund. Thm. Line Integrals	1D	Endpoints $\leftrightarrow$ Curve	$\int_C \nabla \phi \cdot d\mathbf{r} = \phi(B) - \phi(A)$
Green's Theorem	2D	Closed Curve $\leftrightarrow$ Plane Region	$\oint_C P dx + Q dy = \iint_D (Q_x - P_y) dA$
Stokes' Theorem	3D	Closed Curve $\leftrightarrow$ Surface	$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\text{curl } \mathbf{F}) \cdot d\mathbf{S}$
Divergence Theorem	3D	Closed Surface $\leftrightarrow$ Solid Volume	$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\text{div } \mathbf{F}) dV$