What are the basic operators of vector calculus?

There are four basic operators. 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. The curl (∇×A) is a vector representing the rotation of a vector field. The Laplacian (∇²) is the second-order derivative of a scalar field.

What is the difference between vector calculus and matrix calculus?

Vector calculus deals with differential operators (grad, div, curl) and integral theorems for 3D vector fields, 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), thermodynamics (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 (r,φ,z), and spherical (r,θ,φ). Choosing a coordinate system that matches the symmetry of the problem simplifies calculations of grad, div, curl, and the Laplacian.

Vector Calculus Formula Reference

Vector Calculus Formulas

Introductory to Intermediate

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

The four fundamental operations of vector calculus are summarized below. All can be expressed uniformly in terms of the nabla operator $\nabla = (\partial/\partial x,\, \partial/\partial y,\, \partial/\partial z)$.

Operation	Notation	Input → Output	Definition	Physical meaning
Gradient (grad)	$\nabla f$	Scalar field → Vector field	$\left(\dfrac{\partial f}{\partial x},\, \dfrac{\partial f}{\partial y},\, \dfrac{\partial f}{\partial z}\right)$	Direction of steepest ascent and rate of change
Divergence (div)	$\nabla \cdot \mathbf{A}$	Vector field → Scalar field	$\dfrac{\partial A_x}{\partial x} + \dfrac{\partial A_y}{\partial y} + \dfrac{\partial A_z}{\partial z}$	Strength of the source/sink
Curl	$\nabla \times \mathbf{A}$	Vector field → Vector field	Determinant form (details)	Strength and axis of rotation
Laplacian	$\nabla^2 f$	Scalar field → Scalar field	$\dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2}$	Difference from the surrounding average

Important identities

$\nabla \times (\nabla f) = \mathbf{0}$ — A gradient field is always irrotational
$\nabla \cdot (\nabla \times \mathbf{A}) = 0$ — A curl field is always solenoidal (source-free)

Chapter 1 Gradient, Divergence, Curl, and Laplacian
Definitions, properties, and physical meaning of the nabla operator $\nabla$ and the four fundamental operations
Chapter 2 Vector Calculus Identities
Algebraic identities for dot and cross products, product rules involving $\nabla$, and second-order derivative identities
Chapter 3 Coordinate-System Formulas
Expressions for grad, div, curl, and the Laplacian in Cartesian, cylindrical, and spherical coordinates
Chapter 4 Integral Theorems
Divergence theorem (Gauss), Stokes' theorem, and Green's theorem

Prerequisites

Partial derivatives of multivariable functions
Dot product and cross product of vectors
Basics of multiple integrals, line integrals, and surface integrals

Overview

Quick-Reference Table of Vector Differential Operators

Important identities

Contents

Prerequisites

Related Pages