Vector calculus identities

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The following identities are important in vector calculus:

Operator notations

Gradient

In the three-dimensional Cartesian coordinate system, the gradient of some function is given by:

where i, j, k are the standard unit vectors.

The gradient of a tensor field, , of order n, is generally written as

and is a tensor field of order n + 1. In particular, if the tensor field has order 0 (i.e. a scalar), , the resulting gradient,

is a vector field.

Divergence

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

The divergence of a tensor field, , of non-zero order n, is generally written as

and is a contraction to a tensor field of order n − 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, thereby allowing the use of the identity,

where is the directional derivative in the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,

Curl

In Cartesian coordinates, for :

curl() =


where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.


For a 3-dimensional vector field , curl is also a 3-dimensional vector field, generally written as:

or in Einstein notation as:

where ε is the Levi-Civita symbol.

Laplacian

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.

Special notations

In Feynman subscript notation,

where the notation ∇B means the subscripted gradient operates on only the factor B.[1][2]

A less general but similar idea is used in geometric algebra where the so-called Hestenes overdot notation is employed.[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.

Properties

For scalar fields and , vector fields and , and cartesian functions and :

Distributive properties

Product rule for the gradient

The gradient of the product of two scalar fields and follows the same form as the product rule in single variable calculus.

Product of a scalar and a vector

Quotient rule

Chain rule

Vector dot product

where JA denotes the Jacobian of A. For more details, refer to these notes [4]

As a special case, when A = B,

Vector cross product

Second derivatives

Curl of the gradient

The curl of the gradient of any continuously twice-differentiable scalar field is always the zero vector:

Divergence of the curl

The divergence of the curl of any vector field A is always zero:

Divergence of the gradient

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

The result is a scalar quantity.

Curl of the curl

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

Summary of important identities

Addition and multiplication

  • (scalar triple product)
  • (vector triple product)
  • (vector triple product)
  • (Jacobi identity)
  • (Jacobi identity)

Differentiation

Gradient

Divergence

Curl

Second derivatives

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)

Third derivatives

Integration

Below, the curly symbol ∂ means "boundary of".

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):

  • \oiint (Divergence theorem)
  • \oiint
  • \oiint
  • \oiint (Green's first identity)
  • \oiint \oiint (Green's second identity)

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):

  •   (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):

\ointclockwise \ointctrclockwise


See also

References

  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" (PDF). 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.

Further reading

  • Balanis, Constantine A. 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.
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