Vector calculus identities
|Part of a series of articles about|
For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
For a tensor field of any order k, the gradient is a tensor field of order k+1.
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 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.
In Feynman subscript notation,
Less general but similar is the Hestenes overdot notation in geometric algebra. 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.
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: this may also be considered as the tensor product of two vectors, or of a covector and a vector.
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 Φ.
Dot product rule
where denotes the Jacobian matrix of the vector field .
Alternatively, using Feynman subscript notation,
As a special case, when A = B,
Cross product rule
Second derivative identities
Curl of gradient is zero
Divergence of curl is zero
Divergence of gradient
The Laplacian of a scalar field is the divergence of its gradient:
The result is a scalar quantity.
Curl of curl
Here ∇2 is the vector Laplacian operating on the vector field A.
Summary of important identities
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.
In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ∂S (a closed curve):
- 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.
- Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory" (PDF). p. 4. arXiv:physics/0504223.
- Doran, C.; Lasenby, A. (2003). Geometric algebra for physicists. Cambridge University Press. p. 169. ISBN 978-0-521-71595-9.
- 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.
- Balanis, Constantine A. 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.