Vector calculus identities

The following are important identities involving derivatives and integrals in vector calculus.
Operator notation.
Gradient.
For a function formula_1 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 formula_3, also called a scalar field, the gradient is the vector field:
where formula_5 are orthogonal unit vectors in arbitrary directions. 
For a vector field formula_6 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 formula_8 of any order "k", the gradient formula_9 is a tensor field of order "k" + 1.
Divergence.
In Cartesian coordinates, the divergence of a continuously differentiable vector field formula_10 is the scalar-valued function:
The divergence of a tensor field formula_8 of non-zero order "k" is written as formula_13, 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 formula_15 is the directional derivative in the direction of formula_16 multiplied by its magnitude. Specifically, for the outer product of two vectors,
Curl.
In Cartesian coordinates, for formula_10 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 formula_20 has curl given by:
where formula_22 = ±1 or 0 is the Levi-Civita parity symbol.
Laplacian.
In Cartesian coordinates, the Laplacian of a function formula_23 is
For a tensor field, formula_8, 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. 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 formula_30, formula_31 and vector fields formula_8, formula_16, we have the following derivative identities.
Product rule for multiplication by a scalar.
We have the following generalizations of the product rule in single variable calculus.
In the second formula, the transposed gradient formula_36 is an "n" × 1 column vector, formula_8 is a 1 × "n" row vector, and their product is an "n × n" matrix (or more precisely, a dyad); This may also be considered as the tensor product formula_38 of two vectors, or of a covector and a vector"."
Chain rule.
Let formula_40 be a one-variable function from scalars to scalars, formula_41 a parametrized curve, and formula_42 a function from vectors to scalars. We have the following special cases of the multi-variable chain rule.
For a coordinate parametrization formula_44 we have:
Here we take the trace of the product of two "n × n" matrices: the gradient of A and the Jacobian of formula_46.
Dot product rule.
where formula_48 denotes the Jacobian matrix of the vector field formula_49, and in the last expression the formula_50 operations are understood not to act on the formula_51 directions (which some authors would indicate by appropriate parentheses or transposes).
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.
Note the difference between
and
Also note that the matrix formula_57 is antisymmetric.
Second derivative identities.
Divergence of curl is zero.
The divergence 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 in the De Rham chain complex.
Divergence of gradient is Laplacian.
The Laplacian 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 of the gradient of "any" continuously twice-differentiable scalar field formula_61 is always the zero vector:
This 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 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:
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.
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):