Wikipedia / Vector calculus identities
Wikipedia / Vector calculus identities
The following are important identities involving derivatives and integrals in vector calculus.
Operator notation
Gradient
in three-dimensional Cartesian coordinate where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n, also called a scalar field, the gradient is the vector field 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.
, also called a tensor field of order 1, the gradient or total derivative is the n × n Jacobian matrix:
For a tensor field of any order k is a tensor field of order k + 1.
of order k of order k is an arbitrary constant vector.
Divergence
In Cartesian coordinates, the divergence of a continuously differentiable vector field
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, 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 is the directional derivative multiplied by its magnitude. Specifically, for the outer product of two vectors,
of order k of order k is an arbitrary constant vector.
Curl
where i, j, and k are the unit vectors 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 = ±1 or 0 is the Levi-Civita parity symbol.
of order k of order k is an arbitrary constant vector.
A tensor field of order greater than one may be decomposed into a sum of outer products Specifically, for the outer product of two vectors,
Laplacian
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
For a tensor field and is a tensor field of the same order.
of order k of order k 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.
More general but similar is the Hestenes overdot notation in geometric algebra. 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.
The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
An alternative method is to use the Cartesian components of the del operator as follows (with implicit summation over the index):
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term (i.e., the operators must be nested). The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule. For example, from the identity A⋅(B×C) = (A×B)⋅C we may derive A⋅(∇×C) = (A×∇)⋅C but not ∇⋅(B×C) = (∇×B)⋅C, nor from A⋅(B×A) = 0 may we derive A⋅(∇×A) = 0. On the other hand, a subscripted del operates on all occurrences of the subscript in the term, so that A⋅(∇A×A) = ∇A⋅(A×A) = ∇⋅(A×A) = 0. Also, from A×(A×C) = A(A⋅C) − (A⋅A)C we may derive ∇×(∇×C) = ∇(∇⋅C) − ∇2C, but from (Aψ)⋅(Aφ) = (A⋅A)(ψφ) we may not derive (∇ψ)⋅(∇φ) = ∇2(ψφ).
A subscript c on a quantity indicates that it is temporarily considered to be a constant. Since a constant is not a variable, when the substitution rule (see the preceding paragraph) is used it, unlike a variable, may be moved into or out of the scope of a del operator, as in the following example:
Another way to indicate that a quantity is a constant is to affix it as a subscript to the scope of a del operator, as follows:
For the remainder of this article, Feynman subscript notation will be used where appropriate.
First derivative identities
, 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
a parametrized a vector field. We have the following special cases of the multi-variable chain rule.
we have:
Here we take the trace of the dot product of two second-order tensors, which corresponds to the product of their matrices.
Dot product rule
denotes the Jacobian matrix.
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, which differentiates a vector field to give a vector-valued 1-form.
Cross product rule
is antisymmetric.
Second derivative identities
Divergence of curl is zero
The divergence of the curl of any continuously twice-differentiable vector field A
This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.
Divergence of gradient is Laplacian
The Laplacian The result is a scalar quantity.
Divergence of divergence is not defined
The divergence of a vector field A
Curl of gradient is zero
The curl of the gradient of any continuously twice-differentiable scalar field (i.e., differentiability class) is always the zero vector
in a Cartesian coordinate system with Schwarz's theorem (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 in the De Rham chain complex.
Curl of curl
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
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.
Summary of important identities
Differentiation
Gradient
Divergence
Curl
Vector-dot-Del Operator
Second derivatives
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 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):
Endpoint-curve integrals
In the following endpoint–curve integral theorems, P and integration along P:
Tensor integrals
A tensor form of a vector integral theorem may be obtained by replacing the vector (or one of them) by a tensor, provided that the vector is first made to appear only as the right-most vector of each integrand. For example, Stokes' theorem becomes
A scalar field may also be treated as a vector and replaced by a vector or tensor. For example, Green's first identity becomes
.
Similar rules apply to algebraic and differentiation formulas. For algebraic formulas one may alternatively use the left-most vector position.
See also
References
Further reading