- Vector calculus identities
The following identities are important in
vector calculus :ingle operators (summary)
This section explicitly lists what some symbols mean for clarity.
Divergence
Divergence of a vector field
For a vector field , divergence is generally written as:and is a scalar .
Divergence of a tensor
For a tensor , divergence is generally written as
:
and is a vector.
Curl
For a vector field , curl is generally written as
:
and is a vector field.
Gradient
Gradient of a vector field
For a vector field , gradient is generally written as
:
and is a tensor.
Gradient of a scalar field
For a scalar field, , the gradient is generally written as
:
and is a vector.
Combinations of multiple operators
Curl of the gradient
The curl of the
gradient of "any"scalar field is always zero::
One way to establish this identity (and most of the others listed in this article) is to use three-dimensional
Cartesian coordinates . According to the article on "curl",:
where the right hand side is a determinant, and i, j, k are unit vectors pointing in the positive axes directions, and "∂x" = "∂ / ∂ x " "etc". For example, the "x"-component of the above equation is:
:
where the left-hand side evaluates as zero assuming the order of differentiation is immaterial.
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
:
Properties
Distributive property
:
:
Vector dot product
:
In simpler form, using Feynman subscript notation:
:
where the notation ∇A means the subscripted gradient operates on only the factor A.cite book |author= R P Feynman, & Leighton & Sands |title=The Feynman Lecture on Physics |page= Vol II, p. 27-4 |publisher = Addison-Wesley |year=1964 |isbn=0805390499] [http://arxiv.org/abs/physics/0504223 Kholmetskii & Missevitch "The Faraday induction law in relativity theory", p. 4] ]
A less general but similar idea is used in "
geometric algebra " where the so-called Hestenes "overdot notation" is employed.cite book |author=C Doran & A Lasenby |title=Geometric algebra for physicists |year=2003 |publisher=Cambridge University Press |page=p. 169 |isbn=978-0-521-71595-9] The above identity is then expressed as::
where overdots define the scope of the vector derivative. In the first term it is only the first (dotted) factor that is differentiated, while the second is held constant. Likewise, in the second term it is the second (dotted) factor that is differentiated, and the first is held constant.
As a special case, when A = B::
Vector cross product
:
:
:
where the Feynman subscript notation ∇B means the subscripted gradient operates on only the factor B.In overdot notation, explained above:
:
Product of a scalar and a vector
:
:
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 variableCalculus . :ee also
*
Vector calculus
*Del in cylindrical and spherical coordinates
*List of vector identities Notes and references
Further reading
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