Vector calculus identities
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In the three-dimensional Cartesian coordinate system, the gradient of some function is given by:
The gradient of a tensor field, , of order n, is generally written as
and is a tensor field of order n + 1. In particular, if the tensor field has order 0 (i.e. a scalar), , the resulting gradient,
is a vector field.
The divergence of a tensor field, , of non-zero order n, is generally written as
and is a contraction to a tensor field of order n − 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, thereby allowing the use of 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 :
- curl() =
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.
For a 3-dimensional vector field , curl is also a 3-dimensional vector field, generally written as:
or in Einstein notation as:
where ε is the Levi-Civita 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,
A less general but similar idea is used in geometric algebra where the so-called Hestenes overdot notation is employed. 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.
Product rule for the gradient
Product of a scalar and a vector
Vector dot product
where JA denotes the Jacobian of A.
Alternatively, using Feynman subscript notation,
As a special case, when A = B,
Vector cross product
Curl of the gradient
Divergence of the curl
Divergence of the gradient
The Laplacian of a scalar field is defined as the divergence of the gradient:
Note that the result is a scalar quantity.
Curl of the curl
Here,∇2 is the vector Laplacian operating on the vector field A.
Summary of important identities
Addition and multiplication
Below, the curly symbol ∂ means "boundary of".
In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ∂S (a closed curve):
- Exterior derivative
- Vector calculus
- Del in cylindrical and spherical coordinates
- Comparison of vector algebra and geometric algebra
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- Doran, C.; Lasenby, A. (2003). Geometric algebra for physicists. Cambridge University Press. p. 169. ISBN 978-0-521-71595-9.
- 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.