The following are important
identities involving derivatives and integrals in
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
.
Operator notation
Gradient
For a function
in three-dimensional
Cartesian coordinate
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
variables, the gradient is the vector field:
where i, j, k are the
standard unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
s for the ''x'', ''y'', ''z''-axes. More generally, for a function of ''n'' variables
, also called a
scalar field, the gradient is the
vector field:
where
are orthogonal unit vectors in arbitrary directions.
As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change.
For a vector field
written as a 1 × ''n'' row vector, also called a tensor field of order 1, the gradient or
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...
is the ''n × n''
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
:
For a
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
of any order ''k'', the gradient
is a tensor field of order ''k'' + 1.
Divergence
In Cartesian coordinates, the divergence of a
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
vector field is the scalar-valued function:
As the name implies the divergence is a measure of how much vectors are diverging.
The divergence of a
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
of non-zero order ''k'' is written as
, a
contraction
Contraction may refer to:
Linguistics
* Contraction (grammar), a shortened word
* Poetic contraction, omission of letters for poetic reasons
* Elision, omission of sounds
** Syncope (phonology), omission of sounds in a word
* Synalepha, merged ...
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 mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
in the direction of
multiplied by its magnitude. Specifically, for the outer product of two vectors,
Curl
In Cartesian coordinates, for
the curl is the vector field:
where i, j, and k are the
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
s for the ''x''-, ''y''-, and ''z''-axes, respectively.
As the name implies the curl is a measure of how much nearby vectors tend in a circular direction.
In
Einstein notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
, the vector field
has curl given by:
where
= ±1 or 0 is the
Levi-Civita parity symbol.
Laplacian
In
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, the Laplacian of a function
is
The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.
For a
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
,
, the Laplacian is generally written as:
and is a tensor field of the same order.
When the Laplacian is equal to 0, the function is called a
harmonic function. That is,
Special notations
In ''Feynman subscript notation'',
where the notation ∇
B means the subscripted gradient operates on only the factor B.
Less general but similar is the ''Hestenes'' ''overdot notation'' in
geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...
.
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.
Distributive properties
:
Product rule for multiplication by a scalar
We have the following generalizations of the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
in single variable
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
.
:
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 (or more precisely, a
dyad
Dyad or dyade may refer to:
Arts and entertainment
* Dyad (music), a set of two notes or pitches
* ''Dyad'' (novel), by Michael Brodsky, 1989
* ''Dyad'' (video game), 2012
* ''Dyad 1909'' and ''Dyad 1929'', ballets by Wayne McGregor
Other uses ...
); This may also be considered as the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of two vectors, or of a covector and a vector''.''
Quotient rule for division by a scalar
:
Chain rule
Let
be a one-variable function from scalars to scalars,
a
parametrized curve, and
a function from vectors to scalars. We have the following special cases of the multi-variable
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
.
:
For a
coordinate parametrization we have:
:
Here we take the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of the product of two ''n × n'' matrices: the gradient of A and the Jacobian of
.
Dot product rule
:
where
denotes the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of the vector field
.
Alternatively, using Feynman subscript notation,
:
See these notes.
As a special case, when ,
:
The generalization of the dot product formula to Riemannian manifolds is a defining property of a
Riemannian connection
In mathematics, a metric connection is a connection (vector bundle), connection in a vector bundle ''E'' equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are p ...
, which differentiates a vector field to give a vector-valued
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...
.
Cross product rule
:
Note that the matrix
is antisymmetric.
Second derivative identities
Divergence of curl is zero
The
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of the curl of ''any''
vector field A is always zero:
This is a special case of the vanishing of the square of the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
in the
De Rham chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
.
Divergence of gradient is Laplacian
The
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
of a scalar field is the divergence of its gradient:
The result is a scalar quantity.
Divergence of divergence is not defined
Divergence of a vector field A is a scalar, and you cannot take the divergence of a scalar quantity. Therefore:
Curl of gradient is zero
The
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
of the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of ''any'' continuously twice-differentiable
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
(i.e.,
differentiability class
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
) is always the
zero vector
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identi ...
:
It can be easily proved by expressing
in a
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
with
Schwarz's theorem
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function
:f\left(x_1,\, x_2,\, \ldots,\, x_n\right)
of ''n'' ...
(also called Clairaut's theorem on equality of mixed partials). This result is a special case of the vanishing of the square of the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
in the
De Rham chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
.
Curl of curl
Here ∇
2 is the
vector Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
operating on the vector field A.
Curl of divergence is not defined
The
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of a vector field A is a scalar, and you cannot take curl of a scalar quantity. Therefore
A mnemonic
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.
Summary of important identities
Differentiation
Gradient
*
*
*
*
Divergence
*
*
*
Curl
*
*
*
*
Vector dot Del Operator
*
*
Second derivatives
*
*
*
(
scalar Laplacian)
*
(
vector Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
)
*
*
*
*
*
(
Green's vector identity)
Third derivatives
*
*
*
Integration
Below, the
curly symbol ∂ means "
boundary of" a surface or solid.
Surface–volume integrals
In the following surface–volume integral theorems, ''V'' denotes a three-dimensional volume with a corresponding two-dimensional
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
''S'' = ∂''V'' (a
closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...
):
*
* (
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
)
*
* (
Green's first identity)
* (
Green's second identity
In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's ...
)
* (
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
)
* (
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
)
* (
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
)
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
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
):
*
(
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
)
*
Integration around a closed curve in the
clockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a
definite integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
):
:
Endpoint-curve integrals
In the following endpoint–curve integral theorems, ''P'' denotes a 1d open path with signed 0d boundary points
and integration along ''P'' is from
to
:
*
(
Gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
).
See also
*
*
*
*
*
*
*
*
References
Further reading
*
*
*
{{Refend
Mathematical identities
Mathematics-related lists
Vector calculus
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