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Vector Analysis

A clean reference for vector algebra (dot/cross) and vector calculus operators (gradient, divergence, curl, Laplacian) used throughout physics.

  • University Physics Year 1
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Vector analysis combines vector algebra (dot and cross products) with vector calculus (grad/div/curl). These tools appear everywhere in mechanics and electromagnetism.

1) Vector algebra

Dot product (scalar product)

Definition (geometric):

vector A · vector B = | vector A|| vector B| cos θ.

Coordinate form:

vector A · vector B = AₓBₓ + A_yB_y + A_zB_z.

Key properties:

  • Commutative: vector A · vector B = vector B · vector A
  • Distributive: vector A · (vector B + vector C) = vector A · vector B + vector A · vector C
  • vector A · vector A = | vector A|²
  • If vector A⊥ vector B then vector A · vector B = 0

Cross product (vector product)

Definition (magnitude + direction):

| vector A × vector B| = | vector A|| vector B| sin θ,

with direction given by the right-hand rule (perpendicular to the A–B plane).

Coordinate form:

vector A × vector B = | i hat, j hat, k hat; Aₓ, A_y, A_z; Bₓ, B_y, B_z | .

Key properties:

  • Anti-commutative: vector A × vector B = -(vector B × vector A)
  • Distributive: vector A × (vector B + vector C) = vector A × vector B + vector A × vector C
  • If vector A∥ vector B then vector A × vector B = vector 0

Triple products

Scalar triple product (volume):

vector A · (vector B × vector C).

Vector triple product (BAC–CAB rule):

vector A × (vector B × vector C) = vector B(vector A · vector C)- vector C(vector A · vector B).

Worked examples (vector algebra)

A. Dot product and angle

Let vector A = (1,2,2) and vector B = (2,0,1). Then:

vector A · vector B = 1 · 2 + 2 · 0 + 2 · 1 = 4.

Magnitudes:

| vector A| = square root of (1² + 2² + 2²) = 3, | vector B| = square root of (2² + 0² + 1²) = square root of 5 .

So:

cos θ = (vector A · vector B)/(| vector A|| vector B|) = 4/(3 square root of 5).

B. Cross product

Let vector A = (1,0,0) and vector B = (0,1,0). Then:

vector A × vector B = (0,0,1) = k hat .

2) Vector calculus (∇ operators in Cartesian coordinates)

Define

vector ∇ = i hat ∂/(∂ x) + j hat ∂/(∂ y) + k hat ∂/(∂ z).

Gradient (scalar → vector)

For a scalar field f(x,y,z):

vector ∇f = ((∂ f)/(∂ x),(∂ f)/(∂ y),(∂ f)/(∂ z)).

Divergence (vector → scalar)

For a vector field vector F = (Fₓ,F_y,F_z):

vector ∇ · vector F = (∂ Fₓ)/(∂ x) + (∂ F_y)/(∂ y) + (∂ F_z)/(∂ z).

Curl (vector → vector)

vector ∇ × vector F = ((∂ F_z)/(∂ y)-(∂ F_y)/(∂ z))i hat + ((∂ Fₓ)/(∂ z)-(∂ F_z)/(∂ x))j hat + ((∂ F_y)/(∂ x)-(∂ Fₓ)/(∂ y))k hat .

Laplacian (scalar → scalar)

∇² f = vector ∇ · (vector ∇f) = (∂² f)/(∂ x²) + (∂² f)/(∂ y²) + (∂² f)/(∂ z²).
Physical meaning (rule of thumb)
  • vector ∇ · vector F measures “source/sink strength” (net outflow per unit volume).
  • vector ∇ × vector F measures “swirl/rotation” (circulation density).

3) High-frequency identities (useful + safe)

These are used constantly in electromagnetism:

vector ∇ × (vector ∇f) = vector 0, vector ∇ · (vector ∇ × vector F) = 0.

Product rules:

vector ∇ · (f vector F) = f(vector ∇ · vector F) + vector F · (vector ∇f),
vector ∇ × (f vector F) = f(vector ∇ × vector F) + (vector ∇f) × vector F.
Don’t over-memorise exotic identities

Many multi-term vector identities exist, but they’re easy to misquote. If you need one, derive it carefully (or check a trusted reference).

4) One quick example

Let vector F = (-y,x,0). Then:

vector ∇ × vector F = (0,0,2).

And for vector G = (x,y,z):

vector ∇ · vector G = 3, vector ∇ × vector G = vector 0.

Coordinate systems note

The formulas above are for Cartesian coordinates. In cylindrical or spherical coordinates, grad/div/curl include scale factors. Reference: https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

Next: Vector Calculus (Integral Theorems)

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