Vector calculus identities

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

Operator notation

Gradient

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.

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 , also called a tensor field of order 1, the gradient or total 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.

For a tensor field of order k > 0, the tensor field of order k + 1 is defined by the recursive relation

where is an arbitrary constant vector.

Divergence

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

As the name implies the divergence is a measure of how much vectors are diverging.

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,

For a tensor field of order k > 1, the tensor field of order k − 1 is defined by the recursive relation

where is an arbitrary constant vector.

Curl

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.

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 has curl given by:

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

For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation

where 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:

Specifically, for the outer product of two vectors,

Laplacian

In Cartesian coordinates, the Laplacian of a function is

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,

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

and is a tensor field of the same order.

For a tensor field of order k > 0, the tensor field of order k is defined by the recursive relation

where is an arbitrary constant vector.

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

First derivative associative properties

Product rule for multiplication by a scalar

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

Quotient rule for division by a scalar

Chain rule

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

For a vector transformation we have:

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

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 continuously twice-differentiable 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 (i.e., differentiability class ) is always the zero vector:

It can be easily proved by expressing 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

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

Second derivative associative properties

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

Gradient

Divergence

Curl

  • [5]

Vector dot Del Operator

  • [6]

Second derivatives

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

Third derivatives

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):

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

Endpoint-curve integrals

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

  • (gradient theorem)

See also

References

  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. "lecture15.pdf" (PDF).
  6. 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 2021. Retrieved 19 April 2020.

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.
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