Vector calculus identities

Summary

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

Operator notation

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Gradient

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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 mutually orthogonal unit vectors.

As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change.

For a vector field  , also called a tensor field of order 1, the gradient or total 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.

For a tensor field   of order k > 0, the tensor field   of order k + 1 is defined by the recursive relation

 
where   is an arbitrary constant vector.

Divergence

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In Cartesian coordinates, the divergence of a continuously differentiable vector field   is the scalar-valued function:

 

As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.

The divergence of a tensor field   of non-zero order k is written as  , a contraction of 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,
 

For a tensor field   of order k > 1, the tensor field   of order k − 1 is defined by the recursive relation

 
where   is an arbitrary constant vector.

Curl

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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.

As the name implies the curl is a measure of how much nearby vectors tend in a circular direction.

In Einstein notation, the vector field   has curl given by:

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

For a tensor field   of order k > 1, the tensor field   of order k is defined by the recursive relation

 
where   is an arbitrary constant vector.

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:

 
Specifically, for the outer product of two vectors,
 

Laplacian

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In Cartesian coordinates, the Laplacian of a function   is

 

The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.

When the Laplacian is equal to 0, the function is called a harmonic function. That is,

 

For a tensor field,  , the Laplacian is generally written as:

 
and is a tensor field of the same order.

For a tensor field   of order k > 0, the tensor field   of order k is defined by the recursive relation

 
where   is an arbitrary constant vector.

Special notations

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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 identities

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

Distributive properties

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First derivative associative properties

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Product rule for multiplication by a scalar

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We have the following generalizations of the product rule in single-variable calculus.

 

Quotient rule for division by a scalar

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Chain rule

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Let   be a one-variable function from scalars to scalars,   a parametrized curve,   a function from vectors to scalars, and   a vector field. We have the following special cases of the multi-variable chain rule.

 

For a vector transformation   we have:

 

Here we take the trace of the dot product of two second-order tensors, which corresponds to the product of their matrices.

Dot product rule

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where   denotes the Jacobian matrix of the vector field  .

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 rule

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Note that the matrix   is antisymmetric.

Second derivative identities

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Divergence of curl is zero

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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.

Divergence of gradient is Laplacian

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

 
The result is a scalar quantity.

Divergence of divergence is not defined

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The divergence of a vector field A is a scalar, and the divergence of a scalar quantity is undefined. Therefore,

 

Curl of gradient is zero

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The curl of the gradient of any continuously twice-differentiable scalar field   (i.e., differentiability class  ) is always the zero vector:

 

It can be easily proved by expressing   in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). This result is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.

Curl of curl

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

Curl of divergence is not defined

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The divergence of a vector field A is a scalar, and the curl of a scalar quantity is undefined. Therefore,

 

Second derivative associative properties

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DCG chart: Some rules for second derivatives.

A mnemonic

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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

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Differentiation

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Gradient

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  •  
  •  
  •  
  •  

Divergence

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  •  
  •  
  •  

Curl

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  •  
  •  
  •  
  •  [5]

Vector-dot-Del Operator

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  •  [6]
  •  

Second derivatives

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

Third derivatives

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  •  
  •  
  •  

Integration

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

Surface–volume integrals

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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)
  •        (integration by parts)

Curve–surface integrals

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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):

         

Endpoint-curve integrals

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In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points   and integration along P is from   to  :

  •   (gradient theorem)
  •  

See also

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References

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  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. ^ "lecture15.pdf" (PDF).
  6. ^ 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 2021. Retrieved 19 April 2020.

Further reading

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  • Balanis, Constantine A. (23 May 1989). Advanced Engineering Electromagnetics. ISBN 0-471-62194-3.
  • Schey, H. M. (1997). Div Grad Curl and all that: An informal text on vector calculus. W. W. Norton & Company. ISBN 0-393-96997-5.
  • Griffiths, David J. (1999). Introduction to Electrodynamics. Prentice Hall. ISBN 0-13-805326-X.