To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

Vector calculus identities

From Wikipedia, the free encyclopedia

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

Operator notation


For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field:

where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables , also called a scalar field, the gradient is the vector field:

where are orthogonal unit vectors in arbitrary directions.

For a vector field written as a 1 × n row vector, also called a tensor field of order 1, the gradient or covariant derivative is the n × n Jacobian matrix:

For a tensor field of any order k, the gradient is a tensor field of order k + 1.


In Cartesian coordinates, the divergence of a continuously differentiable vector field is the scalar-valued function:

The divergence of a tensor field of non-zero order k is written as , a contraction to 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,

where is the directional derivative in the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,


In Cartesian coordinates, for the curl is the vector field:

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. In Einstein notation, the vector field has curl given by:

where = ±1 or 0 is the Levi-Civita parity symbol.


In Cartesian coordinates, the Laplacian of a function is

For a tensor field, , the Laplacian is generally written as:

and is a tensor field of the same order.

When the Laplacian is equal to 0, the function is called a Harmonic Function. That is,

Special notations

In Feynman subscript notation,

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:

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 , and vector fields , , we have the following derivative identities.

Distributive properties

Product rule for multiplication by a scalar

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

In the second formula, the transposed gradient is an n × 1 column vector, is a 1 × n row vector, and their product is an n × n matrix (or more precisely, a dyad); This may also be considered as the tensor product of two vectors, or of a covector and a vector.

Quotient rule for division by a scalar

Chain rule

Let be a one-variable function from scalars to scalars, a parametrized curve, and a function from vectors to scalars. We have the following special cases of the multi-variable chain rule.

For a coordinate parametrization we have:

Here we take the trace of the product of two n × n matrices: the gradient of A and the Jacobian of .

Dot product rule

where denotes the Jacobian matrix of the vector field .

Alternatively, using Feynman subscript notation,

See these notes.[4]

As a special case, when A = B,

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

Note that the matrix is antisymmetric.

Second derivative identities

Divergence of curl is zero

The divergence of the curl of any vector field A is always zero:

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:

The result is a scalar quantity.

Divergence of divergence is not defined

Divergence of a vector field A is a scalar, and you cannot take the divergence of a scalar quantity. Therefore:

Curl of gradient is zero

The curl of the gradient of any continuously twice-differentiable scalar field is always the zero vector:

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

Curl of curl

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 you cannot take curl of a scalar quantity. Therefore

DCG chart:  Some rules for second derivatives.
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





Vector dot Del Operator

  • [5]

Second derivatives

  • (scalar Laplacian)
  • (vector Laplacian)
  • (Green's vector identity)

Third derivatives


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):

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

  • (Stokes' theorem)

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):

\ointclockwise \ointctrclockwise

See also


  1. ^ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lectures on Physics. Addison-Wesley. Vol II, p. 27–4. ISBN 0-8053-9049-9.
  2. ^ Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory". p. 4. arXiv:physics/0504223.
  3. ^ Doran, C.; Lasenby, A. (2003). Geometric algebra for physicists. Cambridge University Press. p. 169. ISBN 978-0-521-71595-9.
  4. ^ Kelly, P. (2013). "Chapter 1.14 Tensor Calculus 1: Tensor Fields" (PDF). Mechanics Lecture Notes Part III: Foundations of Continuum Mechanics. University of Auckland. Retrieved 7 December 2017.
  5. ^ 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. Archived from the original on 19 April 2020. Retrieved 19 April 2020.

Further reading

This page was last edited on 8 November 2021, at 14:43
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.