Vector calculus identities

The following are important identities involving derivatives and integrals in vector calculus.

Operator notationEdit

GradientEdit

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:

 

where   are orthogonal unit vectors in arbitrary directions.

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.

DivergenceEdit

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,

 

CurlEdit

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   = ±1 or 0 is the Levi-Civita parity symbol.

LaplacianEdit

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,

 

Special notationsEdit

In Feynman subscript notation,

 

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.[3] 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 identitiesEdit

For scalar fields  ,   and vector fields  ,  , we have the following derivative identities.

Distributive propertiesEdit

 

Product rule for multiplication by a scalarEdit

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 (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 scalarEdit

 

Chain ruleEdit

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  .

Dot product ruleEdit

 

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.[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.

Cross product ruleEdit

 

Note the difference between

 

and

 


Also note that the matrix   is antisymmetric.

Second derivative identitiesEdit

Divergence of curl is zeroEdit

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 LaplacianEdit

The Laplacian of a scalar field is the divergence of its gradient:

 

The result is a scalar quantity.

Divergence of divergence is NOT definedEdit

Divergence of a vector field A is a scalar, and you cannot take the divergence of a scalar quantity. Therefore:

 

Curl of gradient is zeroEdit

The curl of the gradient of any continuously twice-differentiable scalar field   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 curlEdit

 

Here ∇2 is the vector Laplacian operating on the vector field A.

Curl of divergence is not definedEdit

The divergence of a vector field A is a scalar, and you cannot take curl of a scalar quantity. Therefore

 
 
DCG chart: Some rules for second derivatives.

A mnemonicEdit

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 identitiesEdit

DifferentiationEdit

GradientEdit

  •  
  •  
  •  
  •  

DivergenceEdit

  •  
  •  
  •  

CurlEdit

  •  
  •  
  •  
  •  

Vector dot Del OperatorEdit

  •  [5]
  •  

Second derivativesEdit

  •  
  •  
  •   (scalar Laplacian)
  •   (vector Laplacian)
  •  
  •  
  •  
  •  
  •   (Green's vector identity)

Third derivativesEdit

 

IntegrationEdit

Below, the curly symbol ∂ means "boundary of" a surface or solid.

Surface–volume integralsEdit

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)

Curve–surface integralsEdit

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):

         

See alsoEdit

ReferencesEdit

  1. ^ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lectures on Physics. Addison-Wesley. Vol II, p. 27–4. ISBN 0-8053-9049-9.
  2. ^ Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory". p. 4. arXiv:physics/0504223.
  3. ^ Doran, C.; Lasenby, A. (2003). Geometric algebra for physicists. Cambridge University Press. p. 169. ISBN 978-0-521-71595-9.
  4. ^ Kelly, P. (2013). "Chapter 1.14 Tensor Calculus 1: Tensor Fields" (PDF). Mechanics Lecture Notes Part III: Foundations of Continuum Mechanics. University of Auckland. Retrieved 7 December 2017.
  5. ^ Kuo, Kenneth K.; Acharya, Ragini (2012). Applications of turbulent and multi-phase combustion. Hoboken, N.J.: Wiley. p. 520. doi:10.1002/9781118127575.app1. ISBN 9781118127575. Archived from the original on 19 April 2020. Retrieved 19 April 2020.

Further readingEdit