Vector calculus identities explained

See also: Vector algebra relations. The following are important identities involving derivatives and integrals in vector calculus.

Operator notation

Gradient

See main article: Gradient.

For a function

f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field:\operatorname{grad}(f) = \nabla f = \begin{pmatrix}\displaystyle (\partial)/(, \partial x) (\partial)/(, \partial y) (\partial)/(\partial z) \end{pmatrix} f = (\partial f)/(\partial x) i + (\partial f)/(\partial y) j + (\partial f)/(\partial z) k where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables \psi(x _{1,} \ldots, x _{n)} , also called a scalar field, the gradient is the vector field:$$\backslash nabla\backslash psi\; =\; \backslash begin\backslash displaystyle\backslash frac,\; \backslash ldots,\; \backslash frac\; \backslash end\backslash psi\; =\; \backslash frac\; \backslash mathbf\_1\; +\; \backslash dots\; +\; \backslash frac\backslash mathbf\_n$$where e _{i} (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 = \left(A _{1,} \ldots, A _{n\right)} , also called a tensor field of order 1, the gradient or total derivative is the n × n Jacobian matrix:$$\backslash mathbf\_\; =\; d\backslash mathbf\; =\; (\backslash nabla\; \backslash !\backslash mathbf)^\backslash textsf\; =\; \backslash left(\backslash frac\backslash right)\_.$$T of any order k, the gradient \operatorname{grad}( T ) = d T = (\nabla T ) ^{sf{T}} is a tensor field of order k + 1.

For a tensor field

T of order k > 0, the tensor field \nabla T of order k + 1 is defined by the recursive relation$$(\backslash nabla\; \backslash mathbf)\; \backslash cdot\; \backslash mathbf\; =\; \backslash nabla\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)$$where C is an arbitrary constant vector.

Divergence

See main article: Divergence.

F = F _{xi} + F _{yj} + F _{zk} is the scalar-valued function:$$\backslash operatorname\backslash mathbf\; =\; \backslash nabla\backslash cdot\backslash mathbf\; =\; \backslash begin\backslash displaystyle\backslash frac,\backslash \; \backslash frac,\backslash \; \backslash frac\backslash end\; \backslash cdot\; \backslash beginF\_,\backslash \; F\_,\backslash \; F\_\backslash end\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac.$$

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

T of non-zero order k is written as \operatorname{div}( T ) = \nabla ⋅ 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,$$\backslash nabla\; \backslash cdot\; \backslash left(\backslash mathbf\; \backslash otimes\; \backslash mathbf\backslash right)\; =\; \backslash mathbf\; (\backslash nabla\; \backslash cdot\; \backslash mathbf)\; +\; (\backslash mathbf\; \backslash cdot\; \backslash nabla)\; \backslash mathbf$$where A ⋅ \nabla is the directional derivative in the direction of A multiplied by its magnitude. Specifically, for the outer product of two vectors,$$\backslash nabla\; \backslash cdot\; \backslash left(\backslash mathbf\; \backslash mathbf^\backslash textsf\backslash right)\; =\; \backslash mathbf\; (\backslash nabla\; \backslash cdot\; \backslash mathbf)\; +\; (\backslash mathbf\; \backslash cdot\; \backslash nabla)\; \backslash mathbf.$$

For a tensor field

T of order k > 1, the tensor field \nabla ⋅ T of order k − 1 is defined by the recursive relation$$(\backslash nabla\; \backslash cdot\; \backslash mathbf)\; \backslash cdot\; \backslash mathbf\; =\; \backslash nabla\; \backslash cdot\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)$$where C is an arbitrary constant vector.

Curl

See main article: Curl (mathematics).

In Cartesian coordinates, for

F = F _{xi} + F _{yj} + F _{zk} the curl is the vector field: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 = \begin{pmatrix}F _{1, F} _{2, F} _{3\end{pmatrix}} has curl given by:$$\backslash nabla\; \backslash times\; \backslash mathbf\; =\; \backslash varepsilon^\backslash mathbf\_i\; \backslash frac$$where \varepsilon = ±1 or 0 is the Levi-Civita parity symbol.

For a tensor field

T of order k > 1, the tensor field \nabla x T of order k is defined by the recursive relation$$(\backslash nabla\; \backslash times\; \backslash mathbf)\; \backslash cdot\; \backslash mathbf\; =\; \backslash nabla\; \backslash times\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)$$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:$$\backslash nabla\; \backslash times\; \backslash left(\backslash mathbf\; \backslash otimes\; \backslash mathbf\backslash right)\; =\; (\backslash nabla\; \backslash times\; \backslash mathbf)\; \backslash otimes\; \backslash mathbf\; -\; \backslash mathbf\; \backslash times\; (\backslash nabla\; \backslash mathbf).$$Specifically, for the outer product of two vectors,$$\backslash nabla\; \backslash times\; \backslash left(\backslash mathbf\; \backslash mathbf^\backslash textsf\backslash right)\; =\; (\backslash nabla\; \backslash times\; \backslash mathbf)\; \backslash mathbf^\backslash textsf\; -\; \backslash mathbf\; \backslash times\; (\backslash nabla\; \backslash mathbf).$$

Laplacian

See main article: Laplace operator.

In Cartesian coordinates, the Laplacian of a function

f(x,y,z) is$$\backslash Delta\; f\; =\; \backslash nabla^2\backslash !\; f\; =\; (\backslash nabla\; \backslash cdot\; \backslash nabla)\; f\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac.$$

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,$$\backslash Delta\; f\; =\; 0.$$

For a tensor field,

T , the Laplacian is generally written as:$$\backslash Delta\backslash mathbf\; =\; \backslash nabla^2\; \backslash mathbf\; =\; (\backslash nabla\; \backslash cdot\; \backslash nabla)\; \backslash mathbf$$and is a tensor field of the same order.

For a tensor field

T of order k > 0, the tensor field \nabla ^{2} T of order k is defined by the recursive relation$$\backslash left(\backslash nabla^2\; \backslash mathbf\backslash right)\; \backslash cdot\; \backslash mathbf\; =\; \backslash nabla^2\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)$$where C is an arbitrary constant vector.

Special notations

In Feynman subscript notation,$$\backslash nabla\_\backslash mathbf\backslash !\; \backslash left(\backslash mathbf\; \backslash right)\; =\; \backslash mathbf\; \backslash !\; \backslash left(\backslash nabla\; \backslash mathbf\; \backslash right)\; +\; \backslash left(\backslash mathbf\; \backslash nabla\; \backslash right)\; \backslash mathbf$$where the notation ∇_B means the subscripted gradient operates on only the factor B.^{[1] [2]}

Less general but similar is the Hestenes overdot notation in geometric algebra.^[3] The above identity is then expressed as:$$\backslash dot\; \backslash left(\backslash mathbf\; \backslash dot\; \backslash right)\; =\; \backslash mathbf\; \backslash !\; \backslash left(\backslash nabla\; \backslash mathbf\; \backslash right)\; +\; \backslash left(\backslash mathbf\; \backslash nabla\; \backslash right)\; \backslash mathbf$$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.

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

First derivative identities

For scalar fields

\psi , \phi and vector fields A , B , we have the following derivative identities.

Distributive properties

\begin{align} \nabla ( \psi + \phi ) & = \nabla \psi + \nabla \phi \\ \nabla ( A + B ) & = \nabla A + \nabla B \\ \nabla ⋅ ( A + B ) & = \nabla ⋅ A + \nabla ⋅ B \\ \nabla x ( A + B ) & = \nabla x A + \nabla x B \end{align}

First derivative associative properties

\begin{align} ( A ⋅ \nabla ) \psi & = A ⋅ ( \nabla \psi ) \\ ( A ⋅ \nabla ) B & = A ⋅ ( \nabla B ) \\ ( A x \nabla ) \psi & = A x ( \nabla \psi ) \\ ( A x \nabla ) B & = A x ( \nabla B ) \end{align}

Product rule for multiplication by a scalar

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

\begin{align} \nabla ( \psi \phi ) & = \phi \nabla \psi + \psi \nabla \phi \\ \nabla ( \psi A ) & = (\nabla \psi) A ^{sf{T}+} \psi \nabla A = \nabla \psi ⊗ A + \psi \nabla A \\ \nabla ⋅ ( \psi A ) & = \psi \nabla { ⋅ } A + ( \nabla \psi ) { ⋅ } A \\ \nabla { x } ( \psi A ) & = \psi \nabla { x } A + ( \nabla \psi ) { x } A \\ \nabla ^{2} (\psi \phi) & = \psi\nabla ^{2} \phi + 2 \nabla \psi ⋅ \nabla \phi + \phi \nabla ^{2} \psi \end{align}

Quotient rule for division by a scalar

\begin{align} \nabla\left( (\psi)/(\phi) \right) & = (\phi\nabla \psi - \psi\nabla\phi)/(\phi ^{2}) \\[ 1 em] \nabla\left( (A)/(\phi) \right) & = (\phi\nabla A - \nabla\phi ⊗ A)/(\phi ^{2}) \\[ 1 em] \nabla ⋅ \left( (A)/(\phi) \right) & = (\phi \nabla{ ⋅)/(A) - \nabla\phi ⋅ A } \\[1em] \nabla \times \left(\frac\right) &= \frac \\[1em] \nabla^2 \left(\frac\right) &= \frac\end

Chain rule

Let

f(x) be a one-variable function from scalars to scalars, r (t) = (x _{1(t),} \ldots, x _{n(t))} a parametrized curve, \phi: R ^{n} \to R a function from vectors to scalars, and A : R ^{n} \to R ^{n} a vector field. We have the following special cases of the multi-variable chain rule.\begin{align} \nabla(f \circ \phi) & = \left(f' \circ \phi\right) \nabla \phi \\ ( r \circ f)' & = ( r ' \circ f) f' \\ (\phi \circ r )' & = (\nabla \phi \circ r ) ⋅ r ' \\ ( A \circ r )' & = r ' ⋅ (\nabla A \circ r ) \\ \nabla(\phi \circ A ) & = (\nabla A ) ⋅ (\nabla \phi \circ A ) \\ \nabla ⋅ ( r \circ \phi) & = \nabla \phi ⋅ ( r ' \circ \phi) \\ \nabla x ( r \circ \phi) & = \nabla \phi x ( r ' \circ \phi) \end{align}

For a vector transformation

x : R ^{n} \to R ^{n} we have:\nabla ⋅ ( A \circ x ) = tr \left((\nabla x ) ⋅ (\nabla A \circ x )\right)

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

\begin{align} \nabla( A ⋅ B ) & = ( A ⋅ \nabla) B + ( B ⋅ \nabla) A + A { x } (\nabla { x } B ) + B { x } (\nabla { x } A ) \\ & = A ⋅ J _{B} + B ⋅ J _{A} = (\nabla B ) ⋅ A + (\nabla A ) ⋅ B \end{align}

where

J _{A} = (\nabla A ) ^{sf{T}=} (\partial A _{i/\partial} x _{j)} _{ij} denotes the Jacobian matrix of the vector field A = (A _{1,\ldots,A} _{n)} .

Alternatively, using Feynman subscript notation,

\nabla( A ⋅ B ) = \nabla _{A(A} ⋅ B ) + \nabla _{B} ( A ⋅ B ) .

See these notes.^[4]

As a special case, when,

\tfrac{ 1 }{ 2 } \nabla \left( A ⋅ A \right) = A ⋅ J _{A} = (\nabla A ) ⋅ A = ( A { ⋅ } \nabla) A + A { x } (\nabla { x } A ) = A \nabla 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

\begin{align} \nabla ⋅ ( A x B ) & = (\nabla { x } A ) ⋅ B - A ⋅ (\nabla { x } B ) \\[ 5 pt] \nabla x ( A x B ) & = A (\nabla { ⋅ } B ) - B (\nabla { ⋅ } A ) + ( B { ⋅ } \nabla) A - ( A { ⋅ } \nabla) B \\[ 2 pt] & = A (\nabla { ⋅ } B ) + ( B { ⋅ } \nabla) A - ( B (\nabla { ⋅ } A ) + ( A { ⋅ } \nabla) B ) \\[ 2 pt] & = \nabla { ⋅ } \left( B A ^{sf{T}\right)} - \nabla { ⋅ } \left( A B ^{sf{T}\right)\\[2pt]} & = \nabla { ⋅ } \left( B A ^{sf{T}-} A B ^{sf{T}\right)\\[5pt]} A x (\nabla x B ) & = \nabla _{B} ( A { ⋅ } B ) - ( A { ⋅ } \nabla) B \\[ 2 pt] & = A ⋅ J _{B} - ( A { ⋅ } \nabla) B \\[ 2 pt] & = (\nabla B ) ⋅ A - A ⋅ (\nabla B ) \\[ 2 pt] & = A ⋅ ( J _{B} - J ^{sf{T}) \\[ 5 pt]} _{B} ( A x \nabla) x B & = (\nabla B ) ⋅ A - A (\nabla { ⋅ } B ) \\[ 2 pt] & = A x (\nabla x B ) + ( A { ⋅ } \nabla) B - A (\nabla { ⋅ } B ) \\[ 5 pt] ( A x \nabla) ⋅ B & = A ⋅ (\nabla { x } B ) \end{align}

Note that the matrix

J _{B} - J ^{sf{T}} _{B} 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:$$\backslash nabla\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash mathbf)\; =\; 0$$

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:$$\backslash Delta\; \backslash psi\; =\; \backslash nabla^2\; \backslash psi\; =\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; \backslash psi)$$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,$$\backslash nabla\; \backslash cdot\; (\backslash nabla\; \backslash cdot\; \backslash mathbf)\; \backslash text$$

Curl of gradient is zero

\varphi (i.e., differentiability class C ^{2} ) is always the zero vector:$$\backslash nabla\; \backslash times\; (\backslash nabla\; \backslash varphi)\; =\; \backslash mathbf.$$

It can be easily proved by expressing

\nabla x ( \nabla \varphi ) 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

$$\backslash nabla\; \backslash times\; \backslash left(\backslash nabla\; \backslash times\; \backslash mathbf\; \backslash right)\; \backslash \; =\backslash \; \backslash nabla(\backslash nabla\; \backslash mathbf)\; \backslash ,-\backslash ,\; \backslash nabla^\backslash mathbf$$

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,$$\backslash nabla\; \backslash times\; (\backslash nabla\; \backslash cdot\; \backslash mathbf)\; \backslash text$$

Second derivative associative properties

\begin{align} ( \nabla ⋅ \nabla ) \psi & = \nabla ⋅ ( \nabla \psi ) = \nabla ^{2} \psi \\ ( \nabla ⋅ \nabla ) A & = \nabla ⋅ ( \nabla A ) = \nabla ^{2} A \\ ( \nabla x \nabla ) \psi & = \nabla x ( \nabla \psi ) = 0 \\ ( \nabla x \nabla ) A & = \nabla x ( \nabla A ) = 0 \end{align}

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

\nabla(\psi + \phi) = \nabla\psi + \nabla\phi \nabla(\psi \phi) = \phi\nabla \psi + \psi \nabla \phi \nabla(\psi A ) = \nabla \psi ⊗ A + \psi \nabla A \nabla( A ⋅ B ) = ( A ⋅ \nabla) B + ( B ⋅ \nabla) A + A x (\nabla x B ) + B x (\nabla x A )

Divergence

\nabla ⋅ ( A + B ) = \nabla ⋅ A + \nabla ⋅ B \nabla ⋅ \left(\psi A \right) = \psi\nabla ⋅ A + A ⋅ \nabla \psi \nabla ⋅ \left( A x B \right) = (\nabla x A ) ⋅ B - (\nabla x B ) ⋅ A

Curl

\nabla x ( A + B ) = \nabla x A + \nabla x B \nabla x \left(\psi A \right) = \psi(\nabla x A ) - ( A x \nabla)\psi = \psi(\nabla x A ) + (\nabla\psi) x A \nabla x \left(\psi\nabla\phi\right) = \nabla \psi x \nabla \phi \nabla x \left( A x B \right) = A \left(\nabla ⋅ B \right) - B \left( \nabla ⋅ A \right) + \left( B ⋅ \nabla\right) A - \left( A ⋅ \nabla\right) B ^[5]

Vector-dot-Del Operator

( A ⋅ \nabla) B = (1)/(2) [ \nabla( A ⋅ B ) - \nabla x ( A x B ) - B x (\nabla x A ) - A x (\nabla x B ) - B (\nabla ⋅ A ) + A (\nabla ⋅ B ) ] ^[6] ( A ⋅ \nabla) A = (1)/(2) \nabla | A | ^{2-A x (\nabla x A)} = (1)/(2) \nabla | A | ^{2} + (\nabla x A ) x A

Second derivatives

\nabla ⋅ (\nabla x A ) = 0 \nabla x (\nabla\psi) = 0 \nabla ⋅ (\nabla\psi) = \nabla ^{2\psi} (scalar Laplacian)\nabla\left(\nabla ⋅ A \right) - \nabla x \left(\nabla x A \right) = \nabla ^{2A} (vector Laplacian)\nabla ⋅ (\phi\nabla\psi) = \phi\nabla ^{2\psi} + \nabla\phi ⋅ \nabla\psi \psi\nabla ^{2\phi} - \phi\nabla ^{2\psi} = \nabla ⋅ \left(\psi\nabla\phi - \phi\nabla\psi\right) \nabla ^{2(\phi\psi)} = \phi\nabla ^{2\psi} + 2 (\nabla\phi) ⋅ (\nabla\psi) + \left(\nabla ^{2\phi\right)\psi} \nabla ^{2(\psiA)} = A \nabla ^{2\psi} + 2 (\nabla\psi ⋅ \nabla) A + \psi\nabla ^{2A} \nabla ^{2(A} ⋅ B ) = A ⋅ \nabla ^{2B} - B ⋅ \nabla ^{2A} + 2 \nabla ⋅ (( B ⋅ \nabla) A + B x (\nabla x A )) (Green's vector identity)

Third derivatives

\nabla ^{2(\nabla\psi)} = \nabla(\nabla ⋅ (\nabla\psi)) = \nabla\left(\nabla ^{2\psi\right)} \nabla ^{2(\nabla} ⋅ A ) = \nabla ⋅ (\nabla(\nabla ⋅ A )) = \nabla ⋅ \left(\nabla ^{2A\right)} \nabla ^{2} (\nabla x A ) = - \nabla x (\nabla x (\nabla x A )) = \nabla x \left(\nabla ^{2A\right)}

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):

• • (divergence theorem)
  • (Green's first identity)
• (Green's second identity)
• (integration by parts)
• (integration by parts)
• (integration by parts)

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):

\oint _{\partial} A ⋅ d\boldsymbol{\ell} = \iint _{S} \left(\nabla x A \right) ⋅ d S (Stokes' theorem)\oint _{\partial} \psi d\boldsymbol{\ell} = - \iint _{S} \nabla\psi x d S \oint _{\partial} A x d\boldsymbol{\ell} = - \iint _{S} \left(\nabla A - (\nabla ⋅ A ) 1 \right) ⋅ d S = - \iint _{S} \left(d S x \nabla\right) x A

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):

Endpoint-curve integrals

In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points

q - p = \partial P and integration along P is from p to q :\psi| _{\partial} = \psi( q ) - \psi( p ) = \int _{P} \nabla\psi ⋅ d\boldsymbol{\ell} (gradient theorem)A | _{\partial} = A ( q ) - A ( p ) = \int _{P} \left(d\boldsymbol{\ell} ⋅ \nabla\right) A

Further reading

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

Notes and References

1. Book: R. P. . Feynman . R. B. . Leighton . M. . Sands . The Feynman Lectures on Physics . Addison-Wesley . 1964 . 0-8053-9049-9 . true . Vol II, p. 27–4.
2. physics/0504223 . A. L. . Kholmetskii . O. V. . Missevitch . The Faraday induction law in relativity theory . 2005 . 4 .
3. Book: C. . Doran . Chris J. L. Doran . A. . Lasenby . Geometric algebra for physicists . 2003 . Cambridge University Press . 169 . 978-0-521-71595-9.
4. Book: Kelly . P. . 2013 . Mechanics Lecture Notes Part III: Foundations of Continuum Mechanics . University of Auckland . Chapter 1.14 Tensor Calculus 1: Tensor Fields . http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf . 7 December 2017.
5. Web site: lecture15.pdf .
6. Book: Kuo . Kenneth K. . Acharya . Ragini . Applications of turbulent and multi-phase combustion . 2012 . Wiley . Hoboken, N.J. . 9781118127575 . 520 . 10.1002/9781118127575.app1 . 19 April 2020 . https://web.archive.org/web/20210419221120/https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781118127575.app1. 19 April 2021 . live .