Vector Calculus and Differential Forms glossary

Clear, one-line definitions of the Vector Calculus and Differential Forms terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

circulation

The line integral ∮_C F·dr of a vector field F around a closed curve C; measures the net 'swirl' of F along C.

curl

For a vector field F in ℝ³, the vector field ∇ × F whose magnitude and direction encode the local rotation of F.

divergence

For a vector field F in ℝⁿ, the scalar ∇·F = ∂F₁/∂x₁ + ... + ∂Fₙ/∂xₙ; measures net outflow per unit volume.

See: The Divergence Theorem

flux

The surface integral ∫∫_S F·n dA of a vector field F across an oriented surface S; the rate at which F flows through S.

gradient

The vector ∇f of partial derivatives of a scalar field f; points in the direction of steepest increase of f.

See: Gradient descent by hand, ML for FWI gradient acceleration

hodge-star

On an oriented inner-product space, the linear map * sending each k-form to its complementary (n−k)-form; encodes duality between forms.

k-form

A smooth alternating multilinear map taking k tangent vectors to a real number; the object integrated over a k-dimensional surface.

laplacian

The second-order differential operator Δf = ∇·∇f = ∂²f/∂x₁² + ... + ∂²f/∂xₙ²; measures how f deviates from its local average.

See: The Laplacian and Harmonic Functions

scalar field

A function f: U → ℝ on a region U of space assigning a real number (the scalar) to each point.

wedge-product

The antisymmetric product ∧ on differential forms: α ∧ β = (−1)^{kℓ} β ∧ α for a k-form α and ℓ-form β.

zero-form

A 0-form on a manifold is simply a smooth real-valued function; integrating it 'over a 0-chain' means evaluating at points.