For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
where are orthogonal unit vectors in arbitrary directions.
For a tensor field of any order k, the gradient is a tensor field of order k + 1.
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 = ±1 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.
When the Laplacian is equal to 0, the function is called a Harmonic Function. That is,
In Feynman subscript notation,
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.
Product rule for multiplication by a scalar
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 (or more precisely, a dyad); This may also be considered as the tensor product of two vectors, or of a covector and a vector.
Quotient rule for division by a scalar
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 .
Dot product rule
where denotes the Jacobian matrix of the vector field , and in the last expression the operations are understood not to act on the directions (which some authors would indicate by appropriate parentheses or transposes).
Alternatively, using Feynman subscript notation,
See these notes.
As a special case, when A = B,
Cross product rule
Note the difference between
Also note that the matrix is antisymmetric.
Second derivative identities
Divergence of curl is zero
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
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
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
Vector dot Del Operator
- (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):
- Exterior calculus identities
- Exterior derivative
- Del in cylindrical and spherical coordinates
- Vector algebra relations
- Comparison of vector algebra and geometric algebra
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