Unit 5 - Notes

MTH166

Unit 5: Vector Calculus I

1. Vector-Valued Functions

A vector-valued function is a rule that assigns a vector in space to each element in a domain of real numbers. It is generally denoted as $\mathbf{r}(t)$ , mapping a real parameter $t$ to a vector.

Standard Form:

\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k} = \langle f(t), g(t), h(t) \rangle

where

f(t)

g(t)

, and

h(t)

are real-valued component functions.

1.1 Limits of Vector Functions

The limit of a vector function is defined by taking the limits of its component functions.

If $\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle$ , then:

\lim_{t \to a} \mathbf{r}(t) = \left\langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \right\rangle

provided the limits of the component functions exist.

1.2 Continuity

A vector function $\mathbf{r}(t)$ is continuous at a point $t = a$ if:

$\mathbf{r}(a)$ is defined.
$\lim_{t \to a} \mathbf{r}(t)$ exists.
$\lim_{t \to a} \mathbf{r}(t) = \mathbf{r}(a)$ .

Geometric Interpretation: The curve defined by $\mathbf{r}(t)$ has no breaks or holes.

1.3 Differentiability

The derivative of a vector function $\mathbf{r}(t)$ is defined as:

\mathbf{r}'(t) = \frac{d\mathbf{r}}{dt} = \lim_{\Delta t \to 0} \frac{\mathbf{r}(t + \Delta t) - \mathbf{r}(t)}{\Delta t}

Component-wise Differentiation:

\mathbf{r}'(t) = f'(t)\mathbf{i} + g'(t)\mathbf{j} + h'(t)\mathbf{k}

Geometric Interpretation:
The vector $\mathbf{r}'(t)$ is tangent to the curve traced by $\mathbf{r}(t)$ at the point $P$ corresponding to $t$ , pointing in the direction of increasing parameter.

differentiation Rules:
Let $\mathbf{u}(t)$ and $\mathbf{v}(t)$ be differentiable vector functions, $c$ be a scalar, and $f(t)$ be a scalar function.

Sum Rule: $\frac{d}{dt}[\mathbf{u}(t) + \mathbf{v}(t)] = \mathbf{u}'(t) + \mathbf{v}'(t)$
Scalar Multiplication: $\frac{d}{dt}[c\mathbf{u}(t)] = c\mathbf{u}'(t)$
Product Rule (Scalar Function): $\frac{d}{dt}[f(t)\mathbf{u}(t)] = f'(t)\mathbf{u}(t) + f(t)\mathbf{u}'(t)$
Dot Product Rule: $\frac{d}{dt}[\mathbf{u}(t) \cdot \mathbf{v}(t)] = \mathbf{u}'(t) \cdot \mathbf{v}(t) + \mathbf{u}(t) \cdot \mathbf{v}'(t)$
Cross Product Rule: $\frac{d}{dt}[\mathbf{u}(t) \times \mathbf{v}(t)] = \mathbf{u}'(t) \times \mathbf{v}(t) + \mathbf{u}(t) \times \mathbf{v}'(t)$ (Order matters!)

2. Length of a Space Curve (Arc Length)

If a curve $C$ is defined by $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$ for $a \leq t \leq b$ , and the derivatives are continuous, the length $L$ of the curve is:

L = \int_a^b |\mathbf{r}'(t)| \, dt

Explicit Formula:

L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt

Arc Length Function

The arc length function $s(t)$ measures the length of the curve from a fixed point $t_0$ to an arbitrary point $t$ :

s(t) = \int_{t_0}^t |\mathbf{r}'(u)| \, du

Note that

\frac{ds}{dt} = |\mathbf{r}'(t)|

(speed).

3. Motion of a Body or Particle on a Curve

In kinematics, if $t$ represents time, the vector function $\mathbf{r}(t)$ represents the position of a particle.

3.1 Kinematic Quantities

Position Vector: $\mathbf{r}(t)$
Velocity Vector: $\mathbf{v}(t) = \mathbf{r}'(t)$ (Tangent to the path)
Speed: $v(t) = |\mathbf{v}(t)| = |\mathbf{r}'(t)| = \frac{ds}{dt}$ (Scalar quantity)
Acceleration Vector: $\mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t)$
Unit Tangent Vector: $\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|} = \frac{\mathbf{v}}{v}$

3.2 Tangential and Normal Components of Acceleration

Acceleration does not always point in the direction of motion. It can be resolved into two orthogonal components:

\mathbf{a} = a_T \mathbf{T} + a_N \mathbf{N}

where

\mathbf{N}

is the principal unit normal vector.

Tangential Component ( $a_T$ ):
- Responsible for changing the speed of the particle.
- Formula: $a_T = \frac{dv}{dt} = \frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{v}|}$
Normal Component ( $a_N$ ):
- Responsible for changing the direction of the particle.
- Formula: $a_N = \kappa v^2 = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|}$
- (where $\kappa$ is curvature).

4. Gradient of a Scalar Field and Directional Derivatives

We now transition from vector functions of a single variable to vector operators on fields.

4.1 The Del Operator ( $\nabla$ )

The vector differential operator $\nabla$ (nabla) is defined as:

\nabla = \frac{\partial}{\partial x}\mathbf{i} + \frac{\partial}{\partial y}\mathbf{j} + \frac{\partial}{\partial z}\mathbf{k}

4.2 Gradient

Let $\phi(x, y, z)$ be a differentiable scalar field (function). The gradient of $\phi$ , denoted $\nabla \phi$ or $\text{grad } \phi$ , is a vector field:

\nabla \phi = \frac{\partial \phi}{\partial x}\mathbf{i} + \frac{\partial \phi}{\partial y}\mathbf{j} + \frac{\partial \phi}{\partial z}\mathbf{k}

Properties of the Gradient:

Normal to Surfaces: $\nabla \phi$ is normal (perpendicular) to the level surface $\phi(x,y,z) = c$ at any point.
Maximal Increase: The vector $\nabla \phi$ points in the direction of the maximum rate of increase of the function $\phi$ .
Maximal Rate: The maximum rate of increase is $|\nabla \phi|$ .

4.3 Directional Derivatives

The derivative of a scalar field $\phi$ in the direction of a specific vector $\mathbf{a}$ is called the directional derivative.

Let $\mathbf{u}$ be the unit vector in the direction of $\mathbf{a}$ ( $\mathbf{u} = \frac{\mathbf{a}}{|\mathbf{a}|}$ ).
The directional derivative of $\phi$ in the direction of $\mathbf{u}$ , denoted $D_{\mathbf{u}}\phi$ , is:

D_{\mathbf{u}}\phi = \nabla \phi \cdot \mathbf{u}

Note: If the direction is given by vector $\mathbf{a}$ , you must normalize it to $\mathbf{u}$ before taking the dot product.

5. Divergence and Curl of a Vector Field

Let $\mathbf{F}(x,y,z) = P(x,y,z)\mathbf{i} + Q(x,y,z)\mathbf{j} + R(x,y,z)\mathbf{k}$ be a vector field.

5.1 Divergence (Scalar Result)

Divergence measures the magnitude of a vector field's source or sink at a given point (rate of expansion/compression). It maps a vector field to a scalar field.

\text{div } \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}

Physical Interpretations:

$\nabla \cdot \mathbf{F} > 0$ : The point is a source (fluid flowing out).
$\nabla \cdot \mathbf{F} < 0$ : The point is a sink (fluid flowing in).
$\nabla \cdot \mathbf{F} = 0$ : The field is Solenoidal (incompressible). No net inflow or outflow.

5.2 Curl (Vector Result)

Curl measures the rotation or angular momentum of the vector field at a point. It maps a vector field to another vector field.

\text{curl } \mathbf{F} = \nabla \times \mathbf{F}

Usually calculated via a symbolic determinant:

\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}

Expansion:

\nabla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\mathbf{k}