Vector calculus identities
Part of a series of articles about  
Calculus  



Specialized 

The following identities are important in vector calculus:
Operator notations
Gradient
In the threedimensional Cartesian coordinate system, the gradient of some function is given by:
where i, j, k are the standard unit vectors.
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.
Divergence
In threedimensional Cartesian coordinates, the divergence of a continuously differentiable vector field is defined as the scalarvalued function:
The divergence of a tensor field, , of nonzero 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,
Curl
In Cartesian coordinates, for :
 curl() =
where i, j, and k are the unit vectors for the x, y, and zaxes, respectively.
For a 3dimensional vector field , curl is also a 3dimensional vector field, generally written as:
or in Einstein notation as:
where ε is the LeviCivita symbol.
Laplacian
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.
Special notations
In Feynman subscript notation,
where the notation ∇_{B} means the subscripted gradient operates on only the factor B.^{[1]}^{[2]}
A less general but similar idea is used in geometric algebra where the socalled Hestenes overdot notation is employed.^{[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.
Properties
Distributive properties
Product rule for the gradient
The gradient of the product of two scalar fields and follows the same form as the product rule in single variable calculus.
Product of a scalar and a vector
Quotient rule
Chain rule
Vector dot product
where J_{A} denotes the Jacobian of A.
Alternatively, using Feynman subscript notation,
As a special case, when A = B,
Vector cross product
Second derivatives
Curl of the gradient
The curl of the gradient of any twicedifferentiable scalar field is always the zero vector:
Divergence of the curl
The divergence of the curl of any vector field A is always zero:
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
Differentiation
Gradient
Divergence
Curl
Second derivatives
Third derivatives
Integration
Below, the curly symbol ∂ means "boundary of".
Surface–volume integrals
In the following surface–volume integral theorems, V denotes a 3d volume with a corresponding 2d boundary S = ∂V (a closed surface):
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):
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):
See also
 Exterior derivative
 Vector calculus
 Del in cylindrical and spherical coordinates
 Comparison of vector algebra and geometric algebra
References
 ↑ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lecture on Physics. AddisonWesley. Vol II, p. 27–4. ISBN 0805390499.
 ↑ Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory". arXiv:physics/0504223 [physics.classph].
 ↑ Doran, C.; Lasenby, A. (2003). Geometric algebra for physicists. Cambridge University Press. p. 169. ISBN 9780521715959.
Further reading
 Balanis, Constantine A. Advanced Engineering Electromagnetics. ISBN 0471621943.
 Schey, H. M. (1997). Div Grad Curl and all that: An informal text on vector calculus. W. W. Norton & Company. ISBN 0393969975.
 Griffiths, David J. (1999). Introduction to Electrodynamics. Prentice Hall. ISBN 013805326X.