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The following identities are important in vector calculus:

Contents

Operator notationsEdit

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:

 

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   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.

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.

PropertiesEdit

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

Distributive propertiesEdit

 
 
 
 

Product ruleEdit

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: this may also be considered as the tensor product of two vectors, or of a covector and a vector.

Quotient ruleEdit

 
 
 

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 Φ.

Dot product ruleEdit

 

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 ruleEdit

 
 

Second derivativesEdit

Curl of gradient is zeroEdit

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

 

Divergence of curl is zeroEdit

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

 

The above two vanishing properties are a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.

Divergence of gradientEdit

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

 

The result is a scalar quantity.

Curl of curlEdit

 

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

Summary of important identitiesEdit

Addition and multiplicationEdit

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

DifferentiationEdit

GradientEdit

  •  
  •  
  •  
  •  

DivergenceEdit

  •  
  •  
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CurlEdit

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  •  
  •  
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Second derivativesEdit

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

Third derivativesEdit

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  •  
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IntegrationEdit

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

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)

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" (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 readingEdit

  • 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.