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Vector calculus identities ­ Wikipedia, the free encyclopedia
Operator notations
Gradient
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Main article: Gradient
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
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Main article: Divergence
The divergence of a tensor field,
, of non­zero ordern, 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
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Main article: Curl (mathematics)
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
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Main article: Laplace operator
For a tensor field,
, the laplacian is generally written as:
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and is a tensor field of the same order.
Special notations
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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
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Distributive properties
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Product rule for the gradient
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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
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Vector dot product
Vector calculus identities ­ Wikipedia, the free encyclopedia
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where JA 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
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The curl of the gradient of any twice­differentiablescalar field
Divergence of the curl
is always the zero vector:
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The divergence of the curl of any vector field A is always zero:
Divergence of the gradient
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The Laplacian of a scalar field is defined as the divergence of the gradient:
Note that the result is a scalar quantity.
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Curl of the curl
Vector calculus identities ­ Wikipedia, the free encyclopedia
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Here,∇2 is the vector Laplacian operating on the vector field A.
Summary of important identities
Addition and multiplication
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(scalar triple product)
(vector triple product)
(vector triple product)
Differentiation
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Gradient [ edit ]
Divergence [ edit ]
Curl [ edit ]
Second derivatives [ edit ]
(scalar Laplacian)
(vector Laplacian)
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DCG chart: A simple chart depicting all rules
pertaining to second derivatives. D, C, G, L and CC
stand for divergence, curl, gradient, Laplacian and curl
of curl, respectively. Arrows indicate existence of
second derivatives. Blue circle in the middle
represents curl of curl, whereas the other two red
circles(dashed) mean that DD and GG do not exist.
(Green's vector identity)
Third derivatives [ edit ]
Integration
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Below, the curly symbol ∂ means "boundary of".
Surface–volume integrals [ edit ]
In the following surface–volume integral theorems, Vdenotes a 3d volume with a corresponding 2dboundary S =
∂V (a closed surface):
(Divergence theorem)
(Green's first identity)
(Green's
second identity)
Curve–surface integrals [ edit ]
In the following curve–surface integral theorems, Sdenotes a 2d open surface with a corresponding 1d
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boundary C = ∂S (a closed curve):
(Stokes' theorem)
Integration around a closed curve in the clockwisesense is the negative of the same line integral in the
counterclockwise sense (analogous to interchanging the limits in a definite integral):
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