Vector calculus identities

Short description: Mathematical identities

Part of a series of articles about

Calculus

* Fundamental theorem

* Limits of functions
* Continuity

* Mean value theorem
* Rolle's theorem

Differential

|  Definitions

* Derivative (generalizations)

* Differential
* infinitesimal
* of a function
* total

Concepts

* Differentiation notation
* Second derivative
* Third derivative
* Change of variables
* Implicit differentiation
* Related rates
* Taylor's theorem

Rules and identities

* Sum
* Product
* Chain
* Power
* Quotient
* Inverse
* General Leibniz
* Faà di Bruno's formula

Integral

* Lists of integrals

Definitions

* Antiderivative
* Integral (improper)
* Riemann integral
* Lebesgue integration
* Contour integration

Integration by

* Parts
* Discs
* Cylindrical shells
* Substitution (trigonometric)
* Partial fractions
* Order
* Reduction formulae

Series

* Geometric (arithmetico-geometric)
* Harmonic
* Alternating
* Power
* Binomial
* Taylor

Convergence tests

* Summand limit (term test)
* Ratio
* Root
* Integral
* Direct comparison

Limit comparison
* Alternating series
* Cauchy condensation
* Dirichlet
* Abel

Vector

* Gradient
* Divergence
* Curl
* Laplacian
* Directional derivative
* Identities

Theorems

* Divergence
* Gradient
* Green's
* Kelvin–Stokes
* Stokes

Multivariable

|  Formalisms

* Matrix
* Tensor
* Exterior
* Geometric

Definitions

* Partial derivative
* Multiple integral
* Line integral
* Surface integral
* Volume integral
* Jacobian
* Hessian

Specialized

* Fractional
* Malliavin
* Stochastic
* Variations

Glossary of calculus

* Glossary of calculus

* v
* t
* e

The following are important identities involving derivatives and integrals in vector calculus.

## Contents

* 1 Operator notation
* 1.1 Gradient
* 1.2 Divergence
* 1.3 Curl
* 1.4 Laplacian
* 1.5 Special notations
* 2 First derivative identities
* 2.1 Distributive properties
* 2.2 First derivative associative properties
* 2.3 Product rule for multiplication by a scalar
* 2.4 Quotient rule for division by a scalar
* 2.5 Chain rule
* 2.6 Dot product rule
* 2.7 Cross product rule
* 3 Second derivative identities
* 3.1 Divergence of curl is zero
* 3.2 Divergence of gradient is Laplacian
* 3.3 Divergence of divergence is not defined
* 3.4 Curl of gradient is zero
* 3.5 Curl of curl
* 3.6 Curl of divergence is not defined
* 3.7 Second derivative associative properties
* 3.8 A mnemonic
* 4 Summary of important identities
* 4.1 Differentiation
* 4.1.1 Gradient
* 4.1.2 Divergence
* 4.1.3 Curl
* 4.1.4 Vector-dot-Del Operator
* 4.1.5 Second derivatives
* 4.1.6 Third derivatives
* 4.2 Integration
* 4.2.1 Surface–volume integrals
* 4.2.2 Curve–surface integrals
* 4.2.3 Endpoint-curve integrals
* 4.2.4 Tensor integrals
* 5 See also
* 6 References
* 7 Further reading

## Operator notation

### Gradient

Main article: Gradient

For a function f(x,y,z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field: grad⁡(f)=∇f=(∂∂x, ∂∂y, ∂∂z)f=∂f∂x𝐢+∂f∂y𝐣+∂f∂z𝐤 where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables ψ(x1,…,xn), also called a scalar field, the gradient is the vector field: ∇ψ=(∂∂x1,…,∂∂xn)ψ=∂ψ∂x1𝐞1+…+∂ψ∂xn𝐞n where 𝐞i(i=1,2,...,n) are mutually orthogonal unit vectors.

As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change.

For a vector field 𝐀=(A1,…,An), also called a tensor field of order 1, the gradient or total derivative is the n × n Jacobian matrix:[1] 𝐉𝐀=d𝐀=(∇𝐀)T=(∂Ai∂xj)ij.

For a tensor field 𝐓 of any order k, the gradient grad⁡(𝐓)=d𝐓=(∇𝐓)T is a tensor field of order k \+ 1.

For a tensor field 𝐓 of order k > 0, the tensor field ∇𝐓 of order k \+ 1 is defined by the recursive relation (∇𝐓)⋅𝐂=∇(𝐓⋅𝐂) where 𝐂 is an arbitrary constant vector.

### Divergence

Main article: Divergence

In Cartesian coordinates, the divergence of a continuously differentiable vector field 𝐅=Fx𝐢+Fy𝐣+Fz𝐤 is the scalar-valued function: div⁡𝐅=∇⋅𝐅=(∂∂x, ∂∂y, ∂∂z)⋅(Fx, Fy, Fz)=∂Fx∂x+∂Fy∂y+∂Fz∂z.

As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.

The divergence of a tensor field 𝐓 of non-zero order k is written as div⁡(𝐓)=∇⋅𝐓, a contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar. The divergence of a higher-order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity, ∇⋅(𝐀⊗𝐓)=𝐓(∇⋅𝐀)+(𝐀⋅∇)𝐓 where 𝐀⋅∇ is the directional derivative in the direction of 𝐀 multiplied by its magnitude. Specifically, for the outer product of two vectors,[2] ∇⋅(𝐀𝐁T)=𝐁(∇⋅𝐀)+(𝐀⋅∇)𝐁.

For a tensor field 𝐓 of order k > 1, the tensor field ∇⋅𝐓 of order k − 1 is defined by the recursive relation (∇⋅𝐓)⋅𝐂=∇⋅(𝐓⋅𝐂) where 𝐂 is an arbitrary constant vector.

### Curl

Main article: Curl (mathematics)

In Cartesian coordinates, for 𝐅=Fx𝐢+Fy𝐣+Fz𝐤 the curl is the vector field: curl⁡𝐅=∇×𝐅=(∂∂x, ∂∂y, ∂∂z)×(Fx, Fy, Fz)=|𝐢𝐣𝐤∂∂x∂∂y∂∂zFxFyFz|=(∂Fz∂y−∂Fy∂z)𝐢+(∂Fx∂z−∂Fz∂x)𝐣+(∂Fy∂x−∂Fx∂y)𝐤 where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.

As the name implies the curl is a measure of how much nearby vectors tend in a circular direction.

In Einstein notation, the vector field 𝐅=(F1, F2, F3) has curl given by: ∇×𝐅=εijk𝐞i∂Fk∂xj where ε = ±1 or 0 is the Levi-Civita parity symbol.

For a tensor field 𝐓 of order k > 1, the tensor field ∇×𝐓 of order k is defined by the recursive relation (∇×𝐓)⋅𝐂=∇×(𝐓⋅𝐂) where 𝐂 is an arbitrary constant vector.

A tensor field of order greater than one may be decomposed into a sum of outer products, and then the following identity may be used: ∇×(𝐀⊗𝐓)=(∇×𝐀)⊗𝐓−𝐀×(∇𝐓). Specifically, for the outer product of two vectors,[3] ∇×(𝐀𝐁T)=(∇×𝐀)𝐁T−𝐀×(∇𝐁).

### Laplacian

Main article: Laplace operator

In Cartesian coordinates, the Laplacian of a function f(x,y,z) is Δf=∇2f=(∇⋅∇)f=∂2f∂x2+∂2f∂y2+∂2f∂z2.

The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.

When the Laplacian is equal to 0, the function is called a harmonic function. That is, Δf=0.

For a tensor field, 𝐓, the Laplacian is generally written as: Δ𝐓=∇2𝐓=(∇⋅∇)𝐓 and is a tensor field of the same order.

For a tensor field 𝐓 of order k > 0, the tensor field ∇2𝐓 of order k is defined by the recursive relation (∇2𝐓)⋅𝐂=∇2(𝐓⋅𝐂) where 𝐂 is an arbitrary constant vector.

### Special notations

In Feynman subscript notation, ∇𝐁(𝐀⋅𝐁)=𝐀×(∇×𝐁)+(𝐀⋅∇)𝐁 where the notation ∇B means the subscripted gradient operates on only the factor B.[4][5][6]

More general but similar is the Hestenes overdot notation in geometric algebra.[7][8] The above identity is then expressed as: ∇˙(𝐀⋅𝐁˙)=𝐀×(∇×𝐁)+(𝐀⋅∇)𝐁 where overdots define the scope of the vector derivative. The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant.

The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:

∇⋅(𝐀×𝐁)=∇𝐀⋅(𝐀×𝐁)+∇𝐁⋅(𝐀×𝐁)=(∇𝐀×𝐀)⋅𝐁+(∇𝐁×𝐀)⋅𝐁=(∇𝐀×𝐀)⋅𝐁−(𝐀×∇𝐁)⋅𝐁=(∇𝐀×𝐀)⋅𝐁−𝐀⋅(∇𝐁×𝐁)=(∇×𝐀)⋅𝐁−𝐀⋅(∇×𝐁)

An alternative method is to use the Cartesian components of the del operator as follows (with implicit summation over the index i):

∇⋅(𝐀×𝐁)=𝐞i∂i⋅(𝐀×𝐁)=𝐞i⋅∂i(𝐀×𝐁)=𝐞i⋅(∂i𝐀×𝐁+𝐀×∂i𝐁)=𝐞i⋅(∂i𝐀×𝐁)+𝐞i⋅(𝐀×∂i𝐁)=(𝐞i×∂i𝐀)⋅𝐁+(𝐞i×𝐀)⋅∂i𝐁=(𝐞i×∂i𝐀)⋅𝐁−(𝐀×𝐞i)⋅∂i𝐁=(𝐞i×∂i𝐀)⋅𝐁−𝐀⋅(𝐞i×∂i𝐁)=(𝐞i∂i×𝐀)⋅𝐁−𝐀⋅(𝐞i∂i×𝐁)=(∇×𝐀)⋅𝐁−𝐀⋅(∇×𝐁)

Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term (i.e., the operators must be nested). The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule. For example, from the identity A⋅(B×C) = (A×B)⋅C we may derive A⋅(∇×C) = (A×∇)⋅C but not ∇⋅(B×C) = (∇×B)⋅C, nor from A⋅(B×A) = 0 may we derive A⋅(∇×A) = 0. On the other hand, a subscripted del operates on all occurrences of the subscript in the term, so that A⋅(∇A×A) = ∇A⋅(A×A) = ∇⋅(A×A) = 0. Also, from A×(A×C) = A(A⋅C) − (A⋅A)C we may derive ∇×(∇×C) = ∇(∇⋅C) − ∇2C, but from (Aψ)⋅(Aφ) = (A⋅A)(ψφ) we may not derive (∇ψ)⋅(∇φ) = ∇2(ψφ).

A subscript c on a quantity indicates that it is temporarily considered to be a constant. Since a constant is not a variable, when the substitution rule (see the preceding paragraph) is used it, unlike a variable, may be moved into or out of the scope of a del operator, as in the following example:[9]

∇⋅(𝐀×𝐁)=∇⋅(𝐀×𝐁c)+∇⋅(𝐀c×𝐁)=∇⋅(𝐀×𝐁c)−∇⋅(𝐁×𝐀c)=(∇×𝐀)⋅𝐁c−(∇×𝐁)⋅𝐀c=(∇×𝐀)⋅𝐁−(∇×𝐁)⋅𝐀

Another way to indicate that a quantity is a constant is to affix it as a subscript to the scope of a del operator, as follows:[10] ∇(𝐀⋅𝐁)𝐀=𝐀×(∇×𝐁)+(𝐀⋅∇)𝐁

For the remainder of this article, Feynman subscript notation will be used where appropriate.

## First derivative identities

For scalar fields ψ, ϕ and vector fields 𝐀, 𝐁, we have the following derivative identities.

### Distributive properties

* ∇(ψ+ϕ)=∇ψ+∇ϕ
* ∇(𝐀+𝐁)=∇𝐀+∇𝐁
* ∇⋅(𝐀+𝐁)=∇⋅𝐀+∇⋅𝐁
* ∇×(𝐀+𝐁)=∇×𝐀+∇×𝐁

### First derivative associative properties

* (𝐀⋅∇)ψ=𝐀⋅(∇ψ)
* (𝐀⋅∇)𝐁=𝐀⋅(∇𝐁)
* (𝐀×∇)ψ=𝐀×(∇ψ)
* (𝐀×∇)𝐁=𝐀×(∇𝐁)

### Product rule for multiplication by a scalar

We have the following generalizations of the product rule in single-variable calculus.

* ∇(ψϕ)=ϕ∇ψ+ψ∇ϕ
* ∇(ψ𝐀)=(∇ψ)𝐀T+ψ∇𝐀=∇ψ⊗𝐀+ψ∇𝐀
* ∇⋅(ψ𝐀)=ψ∇⋅𝐀+(∇ψ)⋅𝐀
* ∇×(ψ𝐀)=ψ∇×𝐀+(∇ψ)×𝐀
* ∇2(ψϕ)=ψ∇2ϕ+2∇ψ⋅∇ϕ+ϕ∇2ψ

### Quotient rule for division by a scalar

* ∇(ψϕ)=ϕ∇ψ−ψ∇ϕϕ2
* ∇(𝐀ϕ)=ϕ∇𝐀−∇ϕ⊗𝐀ϕ2
* ∇⋅(𝐀ϕ)=ϕ∇⋅𝐀−∇ϕ⋅𝐀ϕ2
* ∇×(𝐀ϕ)=ϕ∇×𝐀−∇ϕ×𝐀ϕ2
* ∇2(ψϕ)=ϕ∇2ψ−2ϕ∇(ψϕ)⋅∇ϕ−ψ∇2ϕϕ2

### Chain rule

Let f(x) be a one-variable function from scalars to scalars, 𝐫(t)=(x1(t),…,xn(t)) a parametrized curve, ϕ:ℝn->ℝ a function from vectors to scalars, and 𝐀:ℝn->ℝn a vector field. We have the following special cases of the multi-variable chain rule.

* ∇(f∘ϕ)=(f′∘ϕ)∇ϕ
* (𝐫∘f)′=(𝐫′∘f)f′
* (ϕ∘𝐫)′=(∇ϕ∘𝐫)⋅𝐫′
* (𝐀∘𝐫)′=𝐫′⋅(∇𝐀∘𝐫)
* ∇(ϕ∘𝐀)=(∇𝐀)⋅(∇ϕ∘𝐀)
* ∇⋅(𝐫∘ϕ)=∇ϕ⋅(𝐫′∘ϕ)
* ∇×(𝐫∘ϕ)=∇ϕ×(𝐫′∘ϕ)
* ∇(𝐫∘ϕ)=∇ϕ⊗(𝐫′∘ϕ)

For a vector transformation 𝐮:ℝn->ℝn we have:

∇⋅(𝐀∘𝐮)=tr((∇𝐮)⋅(∇𝐀∘𝐮))

Here we take the trace of the dot product of two second-order tensors, which corresponds to the product of their matrices.

### Dot product rule

∇(𝐀⋅𝐁) = (𝐀⋅∇)𝐁+(𝐁⋅∇)𝐀+𝐀×(∇×𝐁)+𝐁×(∇×𝐀) = 𝐀⋅𝐉𝐁+𝐁⋅𝐉𝐀 = (∇𝐁)⋅𝐀+(∇𝐀)⋅𝐁

where 𝐉𝐀=(∇𝐀)T=(∂Ai/∂xj)ij denotes the Jacobian matrix of the vector field 𝐀=(A1,…,An).

Alternatively, using Feynman subscript notation,

∇(𝐀⋅𝐁)=∇𝐀(𝐀⋅𝐁)+∇𝐁(𝐀⋅𝐁) .

See these notes.[11]

As a special case, when A = B,

12∇(𝐀⋅𝐀) = 𝐀⋅𝐉𝐀 = (∇𝐀)⋅𝐀 = (𝐀⋅∇)𝐀+𝐀×(∇×𝐀) = A∇A.

The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form.

### Cross product rule

* ∇(𝐀×𝐁) = (∇𝐀)×𝐁−(∇𝐁)×𝐀
* ∇⋅(𝐀×𝐁) = (∇×𝐀)⋅𝐁−𝐀⋅(∇×𝐁)
* ∇×(𝐀×𝐁) = 𝐀(∇⋅𝐁)−𝐁(∇⋅𝐀)+(𝐁⋅∇)𝐀−(𝐀⋅∇)𝐁 = 𝐀(∇⋅𝐁)+(𝐁⋅∇)𝐀−(𝐁(∇⋅𝐀)+(𝐀⋅∇)𝐁) = ∇⋅(𝐁𝐀T)−∇⋅(𝐀𝐁T) = ∇⋅(𝐁𝐀T−𝐀𝐁T)
* 𝐀×(∇×𝐁) = ∇𝐁(𝐀⋅𝐁)−(𝐀⋅∇)𝐁 = 𝐀⋅𝐉𝐁−(𝐀⋅∇)𝐁 = (∇𝐁)⋅𝐀−𝐀⋅(∇𝐁) = 𝐀⋅(𝐉𝐁−𝐉𝐁T)
* (𝐀×∇)×𝐁 = (∇𝐁)⋅𝐀−𝐀(∇⋅𝐁) = 𝐀×(∇×𝐁)+(𝐀⋅∇)𝐁−𝐀(∇⋅𝐁)
* (𝐀×∇)⋅𝐁 = 𝐀⋅(∇×𝐁)

Note that the matrix 𝐉𝐁−𝐉𝐁T is antisymmetric.

## Second derivative identities

### Divergence of curl is zero

The divergence of the curl of any continuously twice-differentiable vector field A is always zero: ∇⋅(∇×𝐀)=0

This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.

### Divergence of gradient is Laplacian

The Laplacian of a scalar field is the divergence of its gradient: Δψ=∇2ψ=∇⋅(∇ψ) The result is a scalar quantity.

### Divergence of divergence is not defined

The divergence of a vector field A is a scalar, and the divergence of a scalar quantity is undefined. Therefore, ∇⋅(∇⋅𝐀) is undefined.

### Curl of gradient is zero

The curl of the gradient of any continuously twice-differentiable scalar field φ (i.e., differentiability class C2) is always the zero vector: ∇×(∇φ)=𝟎.

It can be easily proved by expressing ∇×(∇φ) in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). This result is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.

### Curl of curl

∇×(∇×𝐀) = ∇(∇⋅𝐀)−∇2𝐀

Here ∇2 is the vector Laplacian operating on the vector field A.

### Curl of divergence is not defined

The divergence of a vector field A is a scalar, and the curl of a scalar quantity is undefined. Therefore, ∇×(∇⋅𝐀) is undefined.

### Second derivative associative properties

* (∇⋅∇)ψ=∇⋅(∇ψ)=∇2ψ
* (∇⋅∇)𝐀=∇⋅(∇𝐀)=∇2𝐀
* (∇×∇)ψ=∇×(∇ψ)=𝟎
* (∇×∇)𝐀=∇×(∇𝐀)=𝟎

DCG chart: Some rules for second derivatives.

### A mnemonic

The figure to the right is a mnemonic for some of these identities. The abbreviations used are:

* D: divergence,
* C: curl,
* G: gradient,
* L: Laplacian,
* CC: curl of curl.

Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist.

## Summary of important identities

### Differentiation

#### Gradient

* ∇(ψ+ϕ)=∇ψ+∇ϕ
* ∇(ψϕ)=ϕ∇ψ+ψ∇ϕ
* ∇(ψ𝐀)=∇ψ⊗𝐀+ψ∇𝐀
* ∇(𝐀⋅𝐁)=(𝐀⋅∇)𝐁+(𝐁⋅∇)𝐀+𝐀×(∇×𝐁)+𝐁×(∇×𝐀)

#### Divergence

* ∇⋅(𝐀+𝐁)=∇⋅𝐀+∇⋅𝐁
* ∇⋅(ψ𝐀)=ψ∇⋅𝐀+𝐀⋅∇ψ
* ∇⋅(𝐀×𝐁)=(∇×𝐀)⋅𝐁−(∇×𝐁)⋅𝐀

#### Curl

* ∇×(𝐀+𝐁)=∇×𝐀+∇×𝐁
* ∇×(ψ𝐀)=ψ(∇×𝐀)−(𝐀×∇)ψ=ψ(∇×𝐀)+(∇ψ)×𝐀
* ∇×(ψ∇ϕ)=∇ψ×∇ϕ
* ∇×(𝐀×𝐁)=𝐀(∇⋅𝐁)−𝐁(∇⋅𝐀)+(𝐁⋅∇)𝐀−(𝐀⋅∇)𝐁 [12]

#### Vector-dot-Del Operator

* (𝐀⋅∇)𝐁=−12[∇(𝐀⋅𝐁)−𝐁(∇⋅𝐀)+𝐀(∇⋅𝐁)]−12[∇×(𝐀×𝐁)+𝐁×(∇×𝐀)+𝐀×(∇×𝐁)] [13]
* (𝐀⋅∇)𝐀=12∇|𝐀|2−𝐀×(∇×𝐀)=12∇|𝐀|2+(∇×𝐀)×𝐀
* 𝐀⋅∇(𝐁⋅𝐂)=𝐁⋅(𝐀⋅∇)𝐂+𝐂⋅(𝐀⋅∇)𝐁

#### Second derivatives

* ∇⋅(∇×𝐀)=0
* ∇×(∇ψ)=𝟎
* ∇⋅(∇ψ)=∇2ψ (scalar Laplacian)
* ∇(∇⋅𝐀)−∇×(∇×𝐀)=∇2𝐀 (vector Laplacian)
* ∇⋅[∇𝐀+(∇𝐀)T]=∇2𝐀+∇(∇⋅𝐀)
* ∇⋅(ϕ∇ψ)=ϕ∇2ψ+∇ϕ⋅∇ψ
* ψ∇2ϕ−ϕ∇2ψ=∇⋅(ψ∇ϕ−ϕ∇ψ)
* ∇2(ϕψ)=ϕ∇2ψ+2(∇ϕ)⋅(∇ψ)+(∇2ϕ)ψ
* ∇2(ψ𝐀)=𝐀∇2ψ+2(∇ψ⋅∇)𝐀+ψ∇2𝐀
* ∇⋅[(𝐀⋅∇)𝐁]=(𝐀⋅∇)(∇⋅𝐁)+(∇𝐀):_(∇𝐁)
* ∇×[(𝐀⋅∇)𝐀]+[(∇×𝐀)⋅∇]𝐀=[(𝐀⋅∇)+(∇⋅𝐀)](∇×𝐀)
* ∇2(𝐀⋅𝐁)=𝐀⋅∇2𝐁−𝐁⋅∇2𝐀+2∇⋅((𝐁⋅∇)𝐀+𝐁×(∇×𝐀)) (Green's vector identity)

#### Third derivatives

* ∇2(∇ψ)=∇(∇⋅(∇ψ))=∇(∇2ψ)
* ∇2(∇⋅𝐀)=∇⋅(∇(∇⋅𝐀))=∇⋅(∇2𝐀)
* ∇2(∇×𝐀)=−∇×(∇×(∇×𝐀))=∇×(∇2𝐀)

### Integration

Below, the curly symbol ∂ means "boundary of" a surface or solid.

#### Surface–volume integrals

In the following surface–volume integral theorems, V denotes a three-dimensional volume with a corresponding two-dimensional boundary S = ∂V (a closed surface):

* ∂V ψd𝐒 = ∭V∇ψdV
* ∂V 𝐀⋅d𝐒 = ∭V∇⋅𝐀dV (divergence theorem)
* ∂V 𝐀×d𝐒 = −∭V∇×𝐀dV
* ∂V ψ∇φ⋅d𝐒 = ∭V(ψ∇2φ+∇φ⋅∇ψ)dV (Green's first identity)
* ∂V (ψ∇φ−φ∇ψ)⋅d𝐒 = ∂V (ψ∂φ∂n−φ∂ψ∂n)dS = ∭V(ψ∇2φ−φ∇2ψ)dV (Green's second identity)
* ∭V𝐀⋅∇ψdV = ∂V ψ𝐀⋅d𝐒−∭Vψ∇⋅𝐀dV (integration by parts)
* ∭Vψ∇⋅𝐀dV = ∂V ψ𝐀⋅d𝐒−∭V𝐀⋅∇ψdV (integration by parts)
* ∭V𝐀⋅(∇×𝐁)dV = − ∂V (𝐀×𝐁)⋅d𝐒+∭V(∇×𝐀)⋅𝐁dV (integration by parts)
* ∂V 𝐀×(d𝐒⋅(𝐁𝐂T)) = ∭V𝐀×(∇⋅(𝐁𝐂T))dV+∭V𝐁⋅(∇𝐀)×𝐂dV [14]
* ∭V(∇⋅𝐁+𝐁⋅∇)𝐀dV = ∂V (𝐁⋅d𝐒)𝐀 [15]

#### Curve–surface integrals

In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ∂S (a closed curve):

* ∮∂S𝐀⋅dℓ = ∬S(∇×𝐀)⋅d𝐒 (Stokes' theorem)
* ∮∂Sψdℓ = −∬S∇ψ×d𝐒
* ∮∂S𝐀×dℓ = −∬S(∇𝐀−(∇⋅𝐀)𝟏)⋅d𝐒 = −∬S(d𝐒×∇)×𝐀
* ∮∂S𝐀×(𝐁×dℓ) = ∬S(∇×(𝐀𝐁T))⋅d𝐒+∬S(∇⋅(𝐁𝐀T))×d𝐒 [16]
* ∮∂S(𝐁⋅dℓ)𝐀=∬S(d𝐒⋅[∇×𝐁−𝐁×∇])𝐀 [17]

Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a definite integral):

∂S 𝐀⋅dℓ=− ∂S 𝐀⋅dℓ.

#### Endpoint-curve integrals

In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points 𝐪−𝐩=∂P and integration along P is from 𝐩 to 𝐪:

* ψ|∂P=ψ(𝐪)−ψ(𝐩)=∫P∇ψ⋅dℓ (gradient theorem)
* 𝐀|∂P=𝐀(𝐪)−𝐀(𝐩)=∫P(dℓ⋅∇)𝐀
* 𝐀|∂P=𝐀(𝐪)−𝐀(𝐩)=∫P(∇𝐀)⋅dℓ+∫P(∇×𝐀)×dℓ

#### Tensor integrals

A tensor form of a vector integral theorem may be obtained by replacing the vector (or one of them) by a tensor, provided that the vector is first made to appear only as the right-most vector of each integrand. For example, Stokes' theorem becomes[18]

∮∂Sdℓ⋅𝐓 = ∬Sd𝐒⋅(∇×𝐓).

A scalar field may also be treated as a vector and replaced by a vector or tensor. For example, Green's first identity becomes

∂V ψd𝐒⋅∇𝐀 = ∭V(ψ∇2𝐀+∇ψ⋅∇𝐀)dV.

Similar rules apply to algebraic and differentiation formulas. For algebraic formulas one may alternatively use the left-most vector position.

## See also

* Comparison of vector algebra and geometric algebra
* Del in cylindrical and spherical coordinates - Mathematical gradient operator in certain coordinate systems
* Differentiation rules - Rules for computing derivatives of functions
* Exterior calculus identities
* Exterior derivative - Operation on differential forms
* List of limits
* Vector algebra relations - Formulas about vectors in three-dimensional Euclidean space

## References

1. ↑ Wilson, p. 404.
2. ↑ Wilson, p. 407.
3. ↑ Wilson, p. 407.
4. ↑ Coffin, Joseph George (1911). Vector Analysis. New York: John Wiley & Sons, Inc.. pp. 105–106, 120–123. https://books.google.com/books?id=9mgGAQAAIAAJ.
5. ↑ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lectures on Physics. Addison-Wesley. pp. Vol II, pp. 27–4, 5. ISBN 0-8053-9049-9. https://archive.org/details/the-feynman-lectures-on-physics-vol-2/The_Feynman_Lectures_on_Physics_VOL2/.
6. ↑ Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory". p. 4\. arXiv:physics/0504223.
7. ↑ Coffin, pp. 227–228.
8. ↑ Doran, C.; Lasenby, A. (2003). Geometric algebra for physicists. Cambridge University Press. p. 169. ISBN 978-0-521-71595-9. https://archive.org/details/doran-lasenby-geometric-algebra-for-physicists-2003.
9. ↑ Borisenko, A. I.; Tarapov, I. E. (1968). Vector and Tensor Analysis. New York: Dover Publications, Inc.. pp. 170, 180. https://archive.org/details/borisenko-tarapov-vector-and-tensor-analysis-with-applications.
10. ↑ Wilson, Edwin Bidwell (1901). Vector Analysis. New York: Charles Scribner's Sons. pp. 159, 161–162. https://archive.org/details/0479777.0001.001.umich.edu.
11. ↑ Kelly, P. (2013). "Chapter 1.14 Tensor Calculus 1: Tensor Fields". Mechanics Lecture Notes Part III: Foundations of Continuum Mechanics. University of Auckland. http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf. Retrieved 7 December 2017.
12. ↑ "lecture15.pdf". https://www2.ph.ed.ac.uk/~mevans/mp2h/VTF/lecture15.pdf.
13. ↑ Kuo, Kenneth K.; Acharya, Ragini (2012). Applications of turbulent and multi-phase combustion. Hoboken, N.J.: Wiley. p. 520. doi:10.1002/9781118127575.app1. ISBN 9781118127575. https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781118127575.app1. Retrieved 19 April 2020.
14. ↑ Page and Adams, pp. 65–66.
15. ↑ Wangsness, Roald K.; Cloud, Michael J. (1986). Electromagnetic Fields (2nd ed.). Wiley. ISBN 978-0-471-81186-2.
16. ↑ Page, Leigh; Adams, Norman Ilsley, Jr. (1940). Electrodynamics. New York: D. Van Nostrand Company, Inc.. pp. 44–45, Eq. (18-3). https://archive.org/details/in.ernet.dli.2015.523590.
17. ↑ Pérez-Garrido, Antonio (2024). "Recovering seldom-used theorems of vector calculus and their application to problems of electromagnetism". American Journal of Physics 92 (5): 354–359. doi:10.1119/5.0182191. Bibcode: 2024AmJPh..92e.354P.
18. ↑ Wilson, p. 409.

## Further reading

* Balanis, Constantine A. (23 May 1989). Advanced Engineering Electromagnetics. ISBN 0-471-62194-3.
* Schey, H. M. (1997). Div Grad Curl and all that: An informal text on vector calculus. W. W. Norton & Company. ISBN 0-393-96997-5.
* Griffiths, David J. (1999). Introduction to Electrodynamics. Prentice Hall. ISBN 0-13-805326-X. https://archive.org/details/introductiontoel00grif_0.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Vector calculus identities. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Vector_calculus_identities&oldid=4566019"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template