Vector Calculus Formula Reference

Vector Calculus Formulas

Intermediate to Advanced

Overview

Vector calculus is the field that handles differential operators and integral theorems for scalar and vector fields. It provides the indispensable mathematical machinery for describing physical laws such as Maxwell's equations in electromagnetism and the Navier-Stokes equations in fluid dynamics.

Looking for matrix calculus? For derivatives of functions whose variables are vectors or matrices (gradient vectors, Jacobian matrices, Hessian matrices, etc.), see Matrix Calculus.

Quick-Reference Table of Vector Differential Operators (3D Cartesian Coordinates)

The differential operations most often used in vector calculus are summarized below, first in three-dimensional Cartesian coordinates. In Cartesian coordinates all of them can be expressed uniformly in terms of the nabla operator $\nabla = (\partial/\partial x,\, \partial/\partial y,\, \partial/\partial z)$. In cylindrical and spherical coordinates the same symbol $\nabla$ is used, but the component formulas change (Chapter 3).

Of these, grad, div and curl are first-order differential operators, while the Laplacian is the representative second-order operator, arising as the composition $\nabla^2 f = \nabla \cdot (\nabla f)$.

Operation Notation Input → Output Definition Physical meaning
Gradient (grad) $\nabla f$ Scalar field → Vector field In 3D Cartesian coordinates, $\left(\dfrac{\partial f}{\partial x},\, \dfrac{\partial f}{\partial y},\, \dfrac{\partial f}{\partial z}\right)$ Direction of steepest ascent and rate of change
Divergence (div) $\nabla \cdot \mathbf{A}$ Vector field → Scalar field In 3D Cartesian coordinates, $\dfrac{\partial A_x}{\partial x} + \dfrac{\partial A_y}{\partial y} + \dfrac{\partial A_z}{\partial z}$ Strength of the source/sink
Curl $\nabla \times \mathbf{A}$ Vector field → Vector field In 3D Cartesian coordinates, $\left(\dfrac{\partial A_z}{\partial y} - \dfrac{\partial A_y}{\partial z},\, \dfrac{\partial A_x}{\partial z} - \dfrac{\partial A_z}{\partial x},\, \dfrac{\partial A_y}{\partial x} - \dfrac{\partial A_x}{\partial y}\right)$ (details) Strength and axis of rotation
Laplacian $\nabla^2 f$ Scalar field → Scalar field In 3D Cartesian coordinates, $\dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2}$ Proportional to (spherical mean) − (value at the centre) in the small-sphere limit (the trace of the Hessian)

All component expressions above are those of three-dimensional Cartesian coordinates. In cylindrical and spherical coordinates the scale factors and basis vectors enter, so the formulas take a different form (Chapter 3).

Writing the curl as a $3 \times 3$ determinant is a formal notation used as a memory aid: its entries include differential operators, so it does not mean the same thing as an ordinary determinant of numbers.

The Laplacian also acts on vector fields (the vector Laplacian $\nabla^2 \mathbf{A}$). In Cartesian coordinates it reduces to applying $\nabla^2$ to each component, but that is not true in curvilinear coordinates (Chapter 3).

This series follows the coordinate notation of Chapter 3: cylindrical $(\rho,\varphi,z)$ and spherical $(r,\theta,\varphi)$ (the physics and ISO 80000-2 convention).

Important identities

  • $\nabla \times (\nabla f) = \mathbf{0}$ — the curl of a gradient is always $\mathbf{0}$ (such a field is irrotational)
  • $\nabla \cdot (\nabla \times \mathbf{A}) = 0$ — the divergence of a curl is always $0$ (such a field is solenoidal, i.e. source-free)

Both assume that the second-order partial derivatives involved exist and are continuous, so that mixed partials commute; they do not follow from vector algebra alone.

Contents

  1. Chapter 1 Gradient, Divergence, Curl, and Laplacian

    Definitions, properties, and physical meaning of the nabla operator $\nabla$, the four fundamental operations, and the directional derivative

  2. Chapter 2 Vector Calculus Identities

    Algebraic identities for dot and cross products, product rules involving $\nabla$, second-order derivative identities, position-vector identities, and the $(\mathbf{A} \cdot \nabla)$ operator — 66 formulas in all

    Individual proofs of all 66 formulas: dot and cross products, gradient, divergence, curl, second-order derivatives, position vector, the $(\mathbf{A} \cdot \nabla)$ operator

  3. Chapter 3 Vector Differential Operators by Coordinate System

    Expressions for grad, div, curl, and the Laplacian in Cartesian, cylindrical, spherical, and general orthogonal curvilinear coordinates, with a comparison table

  4. Chapter 4 Integral Theorems

    Divergence theorem (Gauss), Stokes' theorem, Green's theorem, and how the three theorems relate to one another

    Proofs: Proofs of Integral Theorems

Prerequisites

  • Partial derivatives of multivariable functions
  • Dot product and cross product of vectors
  • Expansion of $3 \times 3$ determinants (needed to read the determinant form of the curl)
  • Basics of integration of multivariable functions (multiple integrals; line and surface integrals are reviewed at the start of Chapter 4)

Reading

  • The Story of Nabla ∇ [Reading] — One character that binds gradient, divergence and curl: ∇. The story of the symbol, from Hamilton through Gibbs to Maxwell.

Frequently Asked Questions

What are the basic operators of vector calculus?

This page treats grad, div and curl, together with the Laplacian as the representative second-order operator, as its basic items. The gradient (grad, ∇f) maps a scalar field to a vector field whose direction and magnitude indicate the steepest ascent. The divergence (div, ∇·A) is a scalar measuring the net outflow of a vector field (positive for a source, negative for a sink). The curl (∇×A) is a vector representing the rotation of a vector field. The Laplacian (∇²f), applied to a scalar field, is the sum of the second-order partial derivatives, i.e. the trace of the Hessian.

What is the difference between vector calculus and matrix calculus?

Vector calculus deals with differential operators (grad, div, curl) and integral theorems, primarily for scalar and vector fields in two and three dimensions, and is widely used in electromagnetism and fluid dynamics. Matrix calculus deals with derivatives (gradients, Jacobians, Hessians) of functions whose variables are vectors or matrices in arbitrary dimensions, and is widely used in machine learning and statistics.

In which fields is vector calculus used?

It is widely used in fields that describe physical phenomena: electromagnetism (Maxwell's equations), fluid dynamics (Navier-Stokes equations), heat conduction (the heat equation), continuum mechanics, meteorology, and geophysics.

What are the main vector differentiation formulas?

There are four primary formulas: the gradient ∇f (scalar to vector), the divergence ∇·A (vector to scalar), the curl ∇×A (vector to vector), and the Laplacian ∇²f (scalar to scalar). Two important identities also hold: ∇×(∇f)=0 (the curl of a gradient is zero) and ∇·(∇×A)=0 (the divergence of a curl is zero).

Which coordinate systems are used in vector calculus?

Three coordinate systems are mainly used: Cartesian (x, y, z), cylindrical (ρ, φ, z), and spherical (r, θ, φ); angle conventions vary, so this series follows the notation of Chapter 3. Choosing a coordinate system that matches the symmetry of the problem simplifies calculations of grad, div, curl, and the Laplacian.