Unit 5: Vector Functions and Vector Calculus

Vector Functions

A **vector-valued function** (or vector function) is a function that maps a single real variable (usually t, representing time) to a vector.

It describes a curve in 3D space, where the position vector of a point on the curve is given by:

r(t) = f(t)i + g(t)j + h(t)k

Example: r(t) = cos(t)i + sin(t)j + t*k describes a helix (a spring shape).

Limit of a Vector Function

The limit of a vector function is found by taking the limit of each of its scalar component functions.

If r(t) = f(t)i + g(t)j + h(t)k, then:

lim (t→a) r(t) = [lim (t→a) f(t)]i + [lim (t→a) g(t)]j + [lim (t→a) h(t)]k

The limit exists only if the limits of all three components exist.

Continuity of a Vector Function

A vector function r(t) is continuous at t = a if:

1. r(a) is defined.
2. lim (t→a) r(t) exists.
3. lim (t→a) r(t) = r(a).

This is true if and only if the component functions f(t), g(t), and h(t) are all continuous at t = a.

Differentiation of Vector Functions

Derivative and Tangent Vector

The derivative of r(t) is defined by the same limit definition as scalar functions:

r'(t) = dr/dt = lim (Δt→0) [r(t + Δt) - r(t)] / Δt

Operationally, we just differentiate each component:

dr/dt = (df/dt)i + (dg/dt)j + (dh/dt)k

Geometric Meaning: The derivative r'(t) is a **vector tangent to the curve** r(t), pointing in the direction of increasing t.

Differentiation Rules

Let a(t) and b(t) be vector functions, and f(t) be a scalar function.

• d/dt [a + b] = a' + b' (Sum Rule)
• d/dt [f(t) * a(t)] = f'(t)a(t) + f(t)a'(t) (Scalar Product Rule)
• d/dt [a · b] = a' · b + a · b' (Dot Product Rule)
• d/dt [a × b] = (a' × b) + (a × b') (Cross Product Rule - ORDER MUST BE MAINTAINED!)
• d/dt [a(f(t))] = a'(f(t)) * f'(t) (Chain Rule)

The Del (Nabla) Operator (∇)

Vector calculus introduces the **del (or nabla) operator**, which is a vector differential operator. It is the foundation for gradient, divergence, and curl.

∇ = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k

This operator acts on scalar fields (functions f(x, y, z)) or vector fields (functions F(x, y, z)).

Gradient (grad ∇f)

• Applies to: A Scalar Field f(x, y, z).
• Result: A Vector Field.

The gradient of f is found by "multiplying" the scalar f by the vector operator ∇.

grad(f) = ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

Geometric Meaning:

1. The vector ∇f at a point (x, y, z) points in the direction of the fastest increase of the function f.
2. The magnitude |∇f| is the value of that maximum rate of change.
3. ∇f is always normal (perpendicular) to the level surface f(x, y, z) = constant.

Example: Find the gradient of f(x, y, z) = x²y + z³

∇f = (∂/∂x)(x²y + z³)i + (∂/∂y)(x²y + z³)j + (∂/∂z)(x²y + z³)k
∇f = (2xy)i + (x²)j + (3z²)k

Divergence (div ∇·F)

• Applies to: A Vector Field F = F₁i + F₂j + F₃k.
• Result: A Scalar Field.

The divergence of F is found by taking the **dot product** of ∇ with F.

div(F) = ∇ · F = (∂F₁/∂x) + (∂F₂/∂y) + (∂F₃/∂z)

Physical Meaning: Divergence measures the rate of "outflow" or "expansion" of a vector field from a point (like a source).
If ∇ · F = 0, the field is called solenoidal or incompressible (it has no sources or sinks).

Example: Find the divergence of F = (x³y)i + (yz²)j + (z³x)k

∇ · F = (∂/∂x)(x³y) + (∂/∂y)(yz²) + (∂/∂z)(z³x)
∇ · F = 3x²y + z² + 3z²x

Curl (curl ∇×F)

• Applies to: A Vector Field F = F₁i + F₂j + F₃k.
• Result: A Vector Field.

The curl of F is found by taking the **cross product** of ∇ with F. It is calculated using a determinant:

curl(F) = ∇ × F =

i	j	k
∂/∂x	∂/∂y	∂/∂z
F₁	F₂	F₃

= (∂F₃/∂y - ∂F₂/∂z)i - (∂F₃/∂x - ∂F₁/∂z)j + (∂F₂/∂x - ∂F₁/∂y)k

Physical Meaning: Curl measures the "rotation" or "circulation" of a vector field at a point (like a whirlpool).
If ∇ × F = 0, the field is called irrotational or conservative.

Vector Identities (Related Problems)

These are fundamental identities that are often the subject of "show that" problems.

Identity 1: Curl of a Gradient is Zero

For any scalar field f (with continuous second derivatives):

curl(grad f) = ∇ × (∇f) = 0

Proof (sketch):
∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k
∇ × (∇f) is the determinant:

The i-component is (∂/∂y)(∂f/∂z) - (∂/∂z)(∂f/∂y) = (∂²f/∂y∂z) - (∂²f/∂z∂y).
By Clairaut's theorem on mixed partials, fᵧz = fzᵧ, so this component is 0. The j and k components are also 0.

Identity 2: Divergence of a Curl is Zero

For any vector field F (with continuous second derivatives):

div(curl F) = ∇ · (∇ × F) = 0

Proof (sketch):
∇ · (∇ × F) is the scalar triple product of ∇, ∇, and F.
[∇ ∇ F]
This can be written as a determinant:

A determinant with two identical rows is always zero.

Unit 5: Exam Quick Tips