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The following identities are important in [[vector calculus]]:
The following identities are important in [[vector calculus]]:


==Operator definitions==
==Operator notations==
===Gradient===
===Gradient===
{{main|Gradient}}
{{main|Gradient}}
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and is also a 3-dimensional vector field.
and is also a 3-dimensional vector field.


===Laplacian===
==Combinations of multiple operators==
{{main|Laplace operator}}
===Curl of the gradient===
For a [[tensor]], <math> \mathbf{\mathfrak{T}} </math>, the laplacian is generally written as:


:<math>\Delta\mathbf{\mathfrak{T}} = \nabla^2 \mathbf{\mathfrak{T}} = (\nabla \cdot \nabla) \mathbf{\mathfrak{T}}</math>
The [[Curl (mathematics)|curl]] of the [[gradient]] of ''any'' [[scalar field]] <math>\ \phi </math> is always the [[zero vector]]:


and is a tensor of the same order.
:<math>\nabla \times ( \nabla \phi ) = \vec{0}</math>


===Divergence of the curl===
===Integrals===
{{Incomplete|section}}
The [[divergence]] of the curl of ''any'' [[vector field]] '''A''' is always zero:
:<math>\nabla \cdot ( \nabla \times \mathbf{A} ) = 0 </math>


===Divergence of the gradient===
===Special notations===
In ''Feynman subscript notation'',
The [[Laplacian]] of a scalar field is defined as the divergence of the gradient:
:<math> \nabla^2 \psi = \nabla \cdot (\nabla \psi) </math>
: <math> \nabla_B \left( \mathbf{A \cdot B} \right) = \mathbf{A \ \times } \left( \mathbf{ \nabla \times B} \right) + \left( \mathbf{A \cdot \nabla } \right) \mathbf{ B} </math>
Note that the result is a scalar quantity.


where the notation '''∇'''<sub>B</sub> means the subscripted gradient operates on only the factor '''B'''.<ref name=Feynman>{{cite book |first1=R. P. |last1=Feynman |first2=R. B. |last2=Leighton |first3=M. |last3=Sands |title=The Feynman Lecture on Physics |publisher = Addison-Wesley |year=1964 |isbn=0805390499 |nopp= true |pages= Vol II, p. 27–4}}</ref><ref name=Missevitch>{{cite arXiv |eprint=physics/0504223 |first1=A. L. |last1=Kholmetskii |first2=O. V. |last2=Missevitch |title=The Faraday induction law in relativity theory |year=2005 |page=4 |class=physics.class-ph }}</ref>
===Curl of the curl===

:<math> \nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}</math>
A less general but similar idea is used in ''[[geometric algebra]]'' where the so-called Hestenes ''overdot notation'' is employed.<ref name=Doran>{{cite book |first1=C. |last1=Doran |first2=A. |last2=Lasenby |title=Geometric algebra for physicists |year=2003 |publisher=Cambridge University Press |page=169 |isbn=978-0-521-71595-9}}</ref> The above identity is then expressed as:
Here, ∇<sup>2</sup> is the [[vector Laplacian]] operating on the vector field '''A'''.

: <math> \dot{\nabla} \left( \mathbf{A} \cdot \dot{\mathbf{B}} \right) = \mathbf{A \ \times } \left( \mathbf{ \nabla \times B} \right) + \left( \mathbf{A \cdot \nabla } \right) \mathbf{ B} </math>

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==
==Properties==
===Distributive property===
===Distributive properties===

:<math> \nabla ( \psi + \phi ) = \nabla \psi + \nabla \phi </math>


:<math> \nabla \cdot ( \mathbf{A} + \mathbf{B} ) = \nabla \cdot \mathbf{A} + \nabla \cdot \mathbf{B} </math>
:<math> \nabla \cdot ( \mathbf{A} + \mathbf{B} ) = \nabla \cdot \mathbf{A} + \nabla \cdot \mathbf{B} </math>
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:<math> \nabla \times ( \mathbf{A} + \mathbf{B} ) = \nabla \times \mathbf{A} + \nabla \times \mathbf{B} </math>
:<math> \nabla \times ( \mathbf{A} + \mathbf{B} ) = \nabla \times \mathbf{A} + \nabla \times \mathbf{B} </math>


===Vector dot product===
===Product rule for the gradient===
The gradient of the product of two scalar fields <math>\psi</math> and <math>\phi</math> follows the same form as the [[product rule]] in single variable [[calculus]].
: <math> \nabla (\psi \, \phi) = \phi \,\nabla \psi + \psi \,\nabla \phi </math>


===Product of a scalar and a vector===
:<math> \nabla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) </math>
:<math> \nabla \cdot (\psi\mathbf{A}) = \mathbf{A} \cdot\nabla\psi + \psi\nabla \cdot \mathbf{A} </math>


:<math> \nabla \times (\psi\mathbf{A}) = \psi\nabla \times \mathbf{A} + \nabla\psi \times \mathbf{A} </math>
In simpler form, using Feynman subscript notation:


===Vector dot product===
:<math> \nabla(\mathbf{A} \cdot \mathbf{B})= \nabla_A(\mathbf{A} \cdot \mathbf{B}) + \nabla_B (\mathbf{A} \cdot \mathbf{B}) \ , </math>


:<math> \nabla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) \ . </math>
where the notation '''∇'''<sub>A</sub> means the subscripted gradient operates on only the factor '''A'''.<ref name=Feynman>{{cite book |first1=R. P. |last1=Feynman |first2=R. B. |last2=Leighton |first3=M. |last3=Sands |title=The Feynman Lecture on Physics |publisher = Addison-Wesley |year=1964 |isbn=0805390499 |nopp= true |pages= Vol II, p. 27–4}}</ref><ref name=Missevitch>{{cite arXiv |eprint=physics/0504223 |first1=A. L. |last1=Kholmetskii |first2=O. V. |last2=Missevitch |title=The Faraday induction law in relativity theory |year=2005 |page=4 |class=physics.class-ph }}</ref>


Alternatively, using Feynman subscript notation,
A less general but similar idea is used in ''[[geometric algebra]]'' where the so-called Hestenes ''overdot notation'' is employed.<ref name=Doran>{{cite book |first1=C. |last1=Doran |first2=A. |last2=Lasenby |title=Geometric algebra for physicists |year=2003 |publisher=Cambridge University Press |page=169 |isbn=978-0-521-71595-9}}</ref> The above identity is then expressed as:


:<math> \nabla(\mathbf{A} \cdot \mathbf{B})={\dot \nabla}(\dot{\mathbf{A} } \cdot \mathbf{B}) + \dot{ \nabla }(\mathbf{A} \cdot \dot{ \mathbf{B}}) \ , </math>
:<math> \nabla(\mathbf{A} \cdot \mathbf{B})= \nabla_A(\mathbf{A} \cdot \mathbf{B}) + \nabla_B (\mathbf{A} \cdot \mathbf{B}) \ . </math>


As a special case, when '''A''' = '''B''',
where overdots define the scope of the vector derivative. In the first term it is only the first (dotted) factor that is differentiated, while the second is held constant. Likewise, in the second term it is the second (dotted) factor that is differentiated, and the first is held constant.
:<math> \frac{1}{2} \nabla \left( \mathbf{A}\cdot\mathbf{A} \right) = \mathbf{A} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{A} \ . </math>

As a special case, when '''A''' = '''B''':
:<math> \frac{1}{2} \nabla \left( \mathbf{A}\cdot\mathbf{A} \right) = \mathbf{A} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{A}. </math>


===Vector cross product===
===Vector cross product===


:<math> \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) </math>
:<math> \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) \ . </math>


:<math> \nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} </math>
:<math> \nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} \ . </math>


==Second derivatives==
:<math> \mathbf{A \ \times } \left( \mathbf{ \nabla \times B} \right) =\nabla_B \left( \mathbf{A \cdot B} \right) - \left( \mathbf{A \cdot \nabla } \right) \mathbf{ B} \ , </math>
===Curl of the gradient===


The [[Curl (mathematics)|curl]] of the [[gradient]] of ''any'' [[scalar field]] <math>\ \phi </math> is always the [[zero vector]]:
where the Feynman subscript notation '''∇'''<sub>B</sub> means the subscripted gradient operates on only the factor '''B'''.<ref name=Feynman/><ref name=Missevitch/>
In overdot notation, explained above:<ref name=Doran/>


:<math>\nabla \times ( \nabla \phi ) = \vec{0}</math>
:<math> \mathbf{A \ \times } \left( \mathbf{ \nabla \times B} \right) =\dot{\nabla} \left( \mathbf{A \cdot } \dot{\mathbf{B}} \right) - \left( \mathbf{A \cdot \nabla } \right) \mathbf{ B} \ . </math><ref>{{cite book | first1=Robert A. |last1=Adams |first2=Christopher |last2=Essex |title=Calculus: Several Variables |edition=7th |publisher=Pearson Canada|location=Toronto |year=2008 |page=897 | isbn=0201798026}}</ref>


===Product of a scalar and a vector===
===Divergence of the curl===
The [[divergence]] of the curl of ''any'' [[vector field]] '''A''' is always zero:
:<math> \nabla \cdot (\psi\mathbf{A}) = \mathbf{A} \cdot\nabla\psi + \psi\nabla \cdot \mathbf{A} </math>
:<math>\nabla \cdot ( \nabla \times \mathbf{A} ) = 0 </math>


===Divergence of the gradient===
:<math> \nabla \times (\psi\mathbf{A}) = \psi\nabla \times \mathbf{A} + \nabla\psi \times \mathbf{A} </math>
The [[Laplacian]] of a scalar field is defined as the divergence of the gradient:
:<math> \nabla^2 \psi = \nabla \cdot (\nabla \psi) </math>
Note that the result is a scalar quantity.

===Curl of the curl===
:<math> \nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}</math>
Here, ∇<sup>2</sup> is the [[vector Laplacian]] operating on the vector field '''A'''.


===Product rule for the gradient===
The gradient of the product of two scalar fields <math>\psi</math> and <math>\phi</math> follows the same form as the [[product rule]] in single variable [[calculus]].
: <math> \nabla (\psi \, \phi) = \phi \,\nabla \psi + \psi \,\nabla \phi </math>


==Summary of all identities==
==Summary of important identities==
===Addition and multiplication===
===Addition and multiplication===
*<math> \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A} </math>
*<math> \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A} </math>

Revision as of 15:40, 19 September 2011

The following identities are important in vector calculus:

Operator notations

Gradient

Gradient of a tensor, , of order n, is generally written as

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

is a vector field.

Divergence

Divergence of a tensor, , of non-zero order n, is generally written as

and is a contraction to a tensor of order n-1. Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor may be found by decomposing the tensor 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

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

and is also a 3-dimensional vector field.

Laplacian

For a tensor, , the laplacian is generally written as:

and is a tensor of the same order.

Integrals

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

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

Vector dot product

Alternatively, using Feynman subscript notation,

As a special case, when A = B,

Vector cross product

Second derivatives

Curl of the gradient

The curl of the gradient of any 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 defined as the divergence of the gradient:

Note that 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)

Differentiation

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.

Gradient

Divergence

Curl

Second derivatives

  • (scalar Laplacian)
  • (vector Laplacian)

Integration

  • (Divergence theorem)
  • (Green's first identity)
  • (Green's second identity)
  • (Stokes' theorem)

See also

Notes and references

  1. ^ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lecture on Physics. Addison-Wesley. Vol II, p. 27–4. ISBN 0805390499. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)
  2. ^ Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory". p. 4. arXiv:physics/0504223. {{cite arXiv}}: |class= ignored (help)
  3. ^ Doran, C.; Lasenby, A. (2003). Geometric algebra for physicists. Cambridge University Press. p. 169. ISBN 978-0-521-71595-9.

Further reading

  • Balanis, Constantine A. Advanced Engineering Electromagnetics. ISBN 0471621943.
  • 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.