For a function in three-dimensional
Cartesian coordinate variables, the gradient is the vector field:
where i, j, k are the
standardunit vectors 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 mutually orthogonal unit vectors.
As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change.
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
The divergence of a
tensor field of non-zero order k is written as , a
contraction of 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 the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,
For a tensor field of order k > 1, the tensor field of order k − 1 is defined by the recursive relation
where is an arbitrary constant vector.
For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation
where is an arbitrary constant vector.
A tensor field of order greater than one may be decomposed into a sum of
outer products, and then the following identity may be used:
Specifically, for the outer product of two vectors,
The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.
When the Laplacian is equal to 0, the function is called a
harmonic function. That is,
For a
tensor field, , the Laplacian is generally written as:
and is a tensor field of the same order.
For a tensor field of order k > 0, the tensor field of order k is defined by the recursive relation
where is an arbitrary constant vector.
Special notations
In Feynman subscript notation,
where the notation ∇B means the subscripted gradient operates on only the factor B.[1][2]
Less general but similar is the Hestenesoverdot notation in
geometric algebra.[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.
First derivative identities
For scalar fields , and vector fields , , we have the following derivative identities.
Distributive properties
First derivative associative properties
Product rule for multiplication by a scalar
We have the following generalizations of the
product rule in single-variable
calculus.
Quotient rule for division by a scalar
Chain rule
Let be a one-variable function from scalars to scalars, a
parametrized curve, a function from vectors to scalars, and a vector field. We have the following special cases of the multi-variable
chain rule.
For a vector transformation we have:
Here we take the
trace of the dot product of two second-order tensors, which corresponds to the product of their matrices.
Here ∇2 is the
vector Laplacian operating on the vector field A.
Curl of divergence is not defined
The
divergence of a vector field A is a scalar, and the curl of a scalar quantity is undefined. Therefore,
Second derivative associative properties
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.
In the following surface–volume integral theorems, V denotes a three-dimensional volume with a corresponding two-dimensional
boundaryS = ∂V (a
closed surface):
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):
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 :
For a function in three-dimensional
Cartesian coordinate variables, the gradient is the vector field:
where i, j, k are the
standardunit vectors 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 mutually orthogonal unit vectors.
As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change.
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
The divergence of a
tensor field of non-zero order k is written as , a
contraction of 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 the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,
For a tensor field of order k > 1, the tensor field of order k − 1 is defined by the recursive relation
where is an arbitrary constant vector.
For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation
where is an arbitrary constant vector.
A tensor field of order greater than one may be decomposed into a sum of
outer products, and then the following identity may be used:
Specifically, for the outer product of two vectors,
The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.
When the Laplacian is equal to 0, the function is called a
harmonic function. That is,
For a
tensor field, , the Laplacian is generally written as:
and is a tensor field of the same order.
For a tensor field of order k > 0, the tensor field of order k is defined by the recursive relation
where is an arbitrary constant vector.
Special notations
In Feynman subscript notation,
where the notation ∇B means the subscripted gradient operates on only the factor B.[1][2]
Less general but similar is the Hestenesoverdot notation in
geometric algebra.[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.
First derivative identities
For scalar fields , and vector fields , , we have the following derivative identities.
Distributive properties
First derivative associative properties
Product rule for multiplication by a scalar
We have the following generalizations of the
product rule in single-variable
calculus.
Quotient rule for division by a scalar
Chain rule
Let be a one-variable function from scalars to scalars, a
parametrized curve, a function from vectors to scalars, and a vector field. We have the following special cases of the multi-variable
chain rule.
For a vector transformation we have:
Here we take the
trace of the dot product of two second-order tensors, which corresponds to the product of their matrices.
Here ∇2 is the
vector Laplacian operating on the vector field A.
Curl of divergence is not defined
The
divergence of a vector field A is a scalar, and the curl of a scalar quantity is undefined. Therefore,
Second derivative associative properties
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.
In the following surface–volume integral theorems, V denotes a three-dimensional volume with a corresponding two-dimensional
boundaryS = ∂V (a
closed surface):
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):
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 :