The following are important identities involving derivatives and integrals in vector calculus.
For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
where i, j, k are the standard unit 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:
For a vector field written as a 1 × n row vector, also called a tensor field of order 1, the gradient or covariant derivative is the n × n Jacobian matrix:
For a tensor field of any order k, the gradient is a tensor field of order k+1.
In Cartesian coordinates, the divergence of a continuously differentiable vector field is the scalar-valued function:
The divergence of a tensor field of non-zero order k is written as , a contraction 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 the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,
In Cartesian coordinates, for the curl is the vector field:
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. In Einstein notation, the vector field has curl given by:
where or 0 is the Levi-Civita parity symbol.
In Cartesian coordinates, the Laplacian of a function is
For a tensor field, , the Laplacian is generally written as:
and is a tensor field of the same order.
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. 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.
We have the following generalizations of the product rule in single variable calculus.
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: this may also be considered as the tensor product of two vectors, or of a covector and a vector.
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.
For a coordinate parametrization we have:
Here we take the trace of the product of two n × n matrices: the gradient of A and the Jacobian of Φ.
Dot product rule
where denotes the Jacobian matrix of the vector field .
Alternatively, using Feynman subscript notation,
See these notes.
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.
Cross product rule
Second derivative identities
Curl of gradient is zero
The curl of the gradient of any continuously twice-differentiable scalar field is always the zero vector:
Divergence of curl is zero
The divergence of the curl of any vector field A is always zero:
The above two vanishing properties are a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.
Divergence of gradient
The Laplacian of a scalar field is the divergence of its gradient:
The result is a scalar quantity.
Curl of curl
Here ∇2 is the vector Laplacian operating on the vector field A.
Summary of important identities
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.
- (scalar Laplacian)
- (vector Laplacian)
- (Green's vector identity)
Below, the curly symbol ∂ means "boundary of".
In the following surface–volume integral theorems, V denotes a three-dimensional volume with a corresponding two-dimensional boundary S = ∂V (a closed surface):
- (Divergence theorem)
- (Green's first identity)
- (Green's second identity)
- (Integration by parts)
- (Integration by parts)
In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ∂S (a closed curve):
- (Stokes' theorem)
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):