Vector calculus identities
The following are important identities involving derivatives and integrals in vector calculus.
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:
In Cartesian coordinates, the divergence of a continuously differentiable vector field
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 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.
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:
The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.
where the notation ∇B means the subscripted gradient operates on only the factor B.
[1][2] Less 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: 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 the remainder of this article, Feynman subscript notation will be used where appropriate.
We have the following generalizations of the product rule in single-variable calculus.
We have the following special cases of the multi-variable chain rule.
Alternatively, using Feynman subscript notation, See these notes.
[4] As a special case, when A = B, 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.
The divergence of the curl of any continuously twice-differentiable vector field A is always zero:
This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.
The curl of the gradient of any continuously twice-differentiable scalar field
This result is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.
The abbreviations used are: 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.
Below, the curly symbol ∂ means "boundary of" a surface or solid.
In the following surface–volume integral theorems, V denotes a three-dimensional volume with a corresponding two-dimensional boundary S = ∂V (a closed surface): 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): In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points
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.
