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| {{See also|Vector algebra relations}}
| | Marvella is what you can contact her but it's not the most female name out there. For years I've been operating as a payroll clerk. Doing ceramics is what adore performing. South Dakota is her birth location but she needs to move simply because of her family.<br><br>My web blog ... [http://www.neweracinema.com/tube/blog/181510 home std test] |
| The following [[Identity (mathematics)|identities]] are important in [[vector calculus]]:
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| ==Operator notations==
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| ===Gradient===
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| {{main|Gradient}}
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| Gradient of an [[tensor field]], <math>\mathbf{\mathfrak{T}}</math>, of order ''n'', is generally written as
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| :<math>\operatorname{grad}(\mathbf{\mathfrak{T}}) = \nabla \mathbf{\mathfrak{T}} </math>
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| and is a tensor field of order {{nowrap|''n'' + 1}}. In particular, if the tensor field has order 0 (i.e. a scalar), <math>\psi</math>, the resulting gradient,
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| :<math>\operatorname{grad}(\psi) = \nabla \psi</math>
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| is a [[vector field]].
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| ===Divergence===
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| {{main|Divergence}}
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| The divergence of a [[tensor field]], <math>\mathbf{\mathfrak{T}}</math>, of non-zero order ''n'', is generally written as
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| :<math>\operatorname{div}(\mathbf{\mathfrak{T}}) = \nabla \cdot \mathbf{\mathfrak{T}}</math>
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| and is a [[Tensor contraction|contraction]] to a tensor field of order {{nowrap|''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,
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| :<math>\nabla \cdot (\mathbf{a} \otimes \hat{\mathbf{\mathfrak{T}}}) = \hat{\mathbf{\mathfrak{T}}}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \hat{\mathbf{\mathfrak{T}}}</math>
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| where <math> \mathbf{a}\cdot\nabla </math> is the [[directional derivative]] in the direction of <math> \mathbf{a} </math> multiplied by its magnitude. Specifically, for the outer product of two vectors,
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| :<math>\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b} \ .</math>
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| ===Curl===
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| {{main|Curl (mathematics)}}
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| For a 3-dimensional vector field <math> \mathbf{v} </math>, curl is generally written as:
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| :<math>\operatorname{curl}(\mathbf{v}) = \nabla \times \mathbf{v}</math>
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| and is also a 3-dimensional vector field.
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| ===Laplacian===
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| {{main|Laplace operator}}
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| For a [[tensor field]], <math> \mathbf{\mathfrak{T}} </math>, the laplacian is generally written as:
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| :<math>\Delta\mathbf{\mathfrak{T}} = \nabla^2 \mathbf{\mathfrak{T}} = (\nabla \cdot \nabla) \mathbf{\mathfrak{T}}</math>
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| and is a tensor field of the same order.
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| ===Special notations===
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| In ''Feynman subscript notation'',
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| : <math> \nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) = \mathbf{A} \times \left( \nabla \times \mathbf{B} \right) + \left( \mathbf{A} \cdot \nabla \right) \mathbf{B} </math>
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| 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=0-8053-9049-9 |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>
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| 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:
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| : <math> \dot{\nabla} \left( \mathbf{A} \cdot \dot{\mathbf{B}} \right) = \mathbf{A} \times \left( \nabla \times \mathbf{B} \right) + \left( \mathbf{A} \cdot \nabla \right) \mathbf{B} </math>
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| 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.
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| For the remainder of this article, Feynman subscript notation will be used where appropriate.
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| ==Properties==
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| ===Distributive properties===
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| :<math> \nabla ( \psi + \phi ) = \nabla \psi + \nabla \phi </math>
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| :<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>
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| ===Product rule for the gradient===
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| 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]].
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| : <math> \nabla (\psi \, \phi) = \phi \,\nabla \psi + \psi \,\nabla \phi </math>
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| ===Product of a scalar and a vector===
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| :<math> \nabla \cdot (\psi\mathbf{A}) = \mathbf{A} \cdot\nabla\psi + \psi\nabla \cdot \mathbf{A} </math>
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| :<math> \nabla \times (\psi\mathbf{A}) = \psi\nabla \times \mathbf{A} + \nabla\psi \times \mathbf{A} </math>
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| ===Vector dot product===
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| :<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>
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| Alternatively, using Feynman subscript notation,
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| :<math> \nabla(\mathbf{A} \cdot \mathbf{B})= \nabla_\mathbf{A}(\mathbf{A} \cdot \mathbf{B}) + \nabla_\mathbf{B} (\mathbf{A} \cdot \mathbf{B}) \ . </math>
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| As a special case, when '''A''' = '''B''',
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| :<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>
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| ===Vector cross product===
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| :<math> \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) \ . </math>
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| :<math> \begin{align}\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} \\
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| &= (\nabla \cdot \mathbf{B} + \mathbf{B} \cdot \nabla)\mathbf{A} -(\nabla \cdot \mathbf{A} + \mathbf{A} \cdot \nabla )\mathbf{B} \\
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| &= \nabla \cdot (\mathbf{B} \mathbf{A}^\mathrm{T}) - \nabla \cdot (\mathbf{A} \mathbf{B}^\mathrm{T}) \\
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| &= \nabla \cdot (\mathbf{B} \mathbf{A}^\mathrm{T} - \mathbf{A} \mathbf{B}^\mathrm{T} ) \ .
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| \end{align}
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| </math>
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| ==Second derivatives==
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| ===Curl of the gradient===
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| The [[Curl (mathematics)|curl]] of the [[gradient]] of ''any'' [[scalar field]] <math>\ \phi </math> is always the [[zero vector]]:
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| :<math>\nabla \times ( \nabla \phi ) = \mathbf{0}</math>
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| ===Divergence of the curl===
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| The [[divergence]] of the curl of ''any'' [[vector field]] '''A''' is always zero:
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| :<math>\nabla \cdot ( \nabla \times \mathbf{A} ) = 0 </math>
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| ===Divergence of the gradient===
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| The [[Laplacian]] of a scalar field is defined as the divergence of the gradient:
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| :<math> \nabla^2 \psi = \nabla \cdot (\nabla \psi) </math> | |
| Note that the result is a scalar quantity.
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| ===Curl of the curl===
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| :<math> \nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}</math>
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| Here,∇<sup>2</sup> is the [[vector Laplacian]] operating on the vector field '''A'''.
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| ==Summary of important identities==
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| ===Addition and multiplication===
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| *<math> \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A} </math>
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| *<math> \mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A} </math>
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| *<math> \mathbf{A}\times\mathbf{B}=\mathbf{-B}\times\mathbf{A} </math>
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| *<math> \left(\mathbf{A}+\mathbf{B}\right)\cdot\mathbf{C}=\mathbf{A}\cdot\mathbf{C}+\mathbf{B}\cdot\mathbf{C} </math>
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| *<math> \left(\mathbf{A}+\mathbf{B}\right)\times\mathbf{C}=\mathbf{A}\times\mathbf{C}+\mathbf{B}\times\mathbf{C} </math>
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| *<math> \mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)=\mathbf{B}\cdot\left(\mathbf{C}\times\mathbf{A}\right)=\mathbf{C}\cdot\left(\mathbf{A}\times\mathbf{B}\right)</math> ([[scalar triple product]])
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| *<math> \mathbf{A}\times\left(\mathbf{B}\times\mathbf{C}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\mathbf{B}-\left(\mathbf{A}\cdot\mathbf{B}\right)\mathbf{C} </math> ([[vector triple product]])
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| *<math> \left(\mathbf{A}\times\mathbf{B}\right)\cdot\left(\mathbf{C}\times\mathbf{D}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\left(\mathbf{B}\cdot\mathbf{D}\right)-\left(\mathbf{B}\cdot\mathbf{C}\right)\left(\mathbf{A}\cdot\mathbf{D}\right) </math>
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| *<math>
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| \left(\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)\right)\mathbf{D}=\left(\mathbf{A}\cdot\mathbf{D}\right)\left(\mathbf{B}\times\mathbf{C}\right)+\left(\mathbf{B}\cdot\mathbf{D}\right)\left(\mathbf{C}\times\mathbf{A}\right)+\left(\mathbf{C}\cdot\mathbf{D}\right)\left(\mathbf{A}\times\mathbf{B}\right) </math>
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| *<math>
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| \left(\mathbf{A}\times\mathbf{B}\right)\times\left(\mathbf{C}\times\mathbf{D}\right)
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| =\left(\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{D}\right)\right)\mathbf{C}-\left(\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)\right)\mathbf{D}</math>
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| ===Differentiation===
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| [[File:DCG chart.svg|right|thumb|300px|DCG chart:
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| A simple chart depicting all rules pertaining to second derivatives.
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| D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively.
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| 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.
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| ]]
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| ====Gradient====
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| *<math> \nabla(\psi+\phi)=\nabla\psi+\nabla\phi </math>
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| *<math> \nabla (\psi \, \phi) = \phi \,\nabla \psi + \psi \,\nabla \phi </math>
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| *<math> \nabla\left(\mathbf{A}\cdot\mathbf{B}\right)=\left(\mathbf{A}\cdot\nabla\right)\mathbf{B}+\left(\mathbf{B}\cdot\nabla\right)\mathbf{A}+\mathbf{A}\times\left(\nabla\times\mathbf{B}\right)+\mathbf{B}\times\left(\nabla\times\mathbf{A}\right) </math>
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| ====Divergence====
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| *<math> \nabla\cdot(\mathbf{A}+\mathbf{B})=\nabla\cdot\mathbf{A}+\nabla\cdot\mathbf{B} </math>
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| *<math> \nabla\cdot\left(\psi\mathbf{A}\right)=\psi\nabla\cdot\mathbf{A}+\mathbf{A}\cdot\nabla \psi </math>
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| *<math> \nabla\cdot\left(\mathbf{A}\times\mathbf{B}\right)=\mathbf{B}\cdot (\nabla\times\mathbf{A})-\mathbf{A}\cdot(\nabla\times\mathbf{B}) </math>
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| ====Curl====
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| *<math> \nabla\times(\mathbf{A}+\mathbf{B})=\nabla\times\mathbf{A}+\nabla\times\mathbf{B} </math>
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| *<math> \nabla\times\left(\psi\mathbf{A}\right)=\psi\nabla\times\mathbf{A}-\mathbf{A}\times\nabla \psi </math>
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| *<math> \nabla\times\left(\mathbf{A}\times\mathbf{B}\right)=\mathbf{A}\left(\nabla\cdot\mathbf{B}\right)-\mathbf{B}\left(\nabla\cdot\mathbf{A}\right)+\left(\mathbf{B}\cdot\nabla\right)\mathbf{A}-\left(\mathbf{A}\cdot\nabla\right)\mathbf{B} </math>
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| ====Second derivatives====
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| *<math> \nabla\cdot(\nabla\times\mathbf{A})=0 </math>
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| *<math> \nabla\times(\nabla\psi)= \mathbf{0} </math>
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| *<math> \nabla\cdot(\nabla\psi)=\nabla^{2}\psi </math> ([[Laplace operator|scalar Laplacian]])
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| *<math> \nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla^{2}\mathbf{A} </math> ([[vector Laplacian]])
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| *<math> \nabla^{2}(\nabla\cdot\mathbf{A})=\nabla\cdot(\nabla^{2}\mathbf{A})
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| </math>
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| *<math> \nabla\cdot(\phi\nabla\psi)=\phi\nabla^{2}\psi + \nabla\phi\cdot\nabla\psi </math>
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| *<math> \psi\nabla^2\phi-\phi\nabla^2\psi= \nabla\cdot\left(\psi\nabla\phi-\phi\nabla\psi\right)</math>
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| *<math> \nabla^2(\phi\psi)=\phi\nabla^2\psi+2\nabla\phi\cdot\nabla\psi+\psi\nabla^2\phi</math>
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| ===Integration===
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| Below, the curly symbol ∂ means "[[boundary (topology)|boundary of]]".
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| ====Surface–volume integrals====
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| In the following surface–volume integral theorems, ''V'' denotes a 3d volume with a corresponding 2d [[Boundary (topology)|boundary]] ''S'' = ∂''V'' (a [[closed surface]]):
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| *{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\mathbf{A}\cdot d\mathbf{s}=\iiint_V \left(\nabla \cdot \mathbf{A}\right)dV </math>}} ([[Divergence theorem]])
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| *{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\psi d \mathbf{s} = \iiint_V \nabla \psi\, dV</math>}}
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| *{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\left(\hat{\mathbf{n}}\times\mathbf{A}\right)dS=\iiint _{V}\left(\nabla\times\mathbf{A}\right)dV</math>}}
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| *{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\psi\left(\nabla\varphi\cdot\hat{\mathbf{n}}\right)dS = \iiint _{V}\left(\psi\nabla^{2}\varphi+\nabla\varphi\cdot\nabla\psi\right)dV</math>}} ([[Green's first identity]])
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| *{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\left[\left(\psi\nabla\varphi-\varphi\nabla\psi\right)\cdot\hat{\mathbf{n}}\right]dS=\,\!</math>{{oiint|intsubscpt=<math>\scriptstyle \partial V</math>|integrand=<math>\left[\psi\frac{\partial\varphi}{\partial n}-\varphi\frac{\partial\psi}{\partial n}\right]dS</math>}} <math>=\iiint_{V}\left(\psi\nabla^{2}\varphi-\varphi\nabla^{2}\psi\right)dV\,\!</math>}} ([[Green's second identity]])
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| ====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]]):
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| *<math> \oint_{\partial S}\mathbf{A}\cdot d\boldsymbol{\ell}=\iint_{S}\left(\nabla\times\mathbf{A}\right)\cdot d\mathbf{s} </math> {{pad|2em}} ([[Stokes' theorem]])
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| *<math> \oint_{\partial S}\psi d\boldsymbol{\ell}=\iint_{S}\left(\hat{\mathbf{n}}\times\nabla\psi\right)dS </math>
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| 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]]):
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| :{{intorient|
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| | preintegral = {{intorient|
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| | preintegral =
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| |symbol=oint
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| | intsubscpt = <math>{\scriptstyle \partial S}</math>
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| | integrand = <math>\mathbf{A}\cdot{\rm d}\boldsymbol{\ell}=-</math>
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| }}
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| |symbol=ointctr
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| | intsubscpt = <math>{\scriptstyle \partial S}</math>
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| | integrand = <math>\mathbf{A}\cdot{\rm d}\boldsymbol{\ell}.</math>
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| }}
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| ==See also==
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| * [[Exterior derivative]]
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| * [[Vector calculus]]
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| * [[Del in cylindrical and spherical coordinates]]
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| * [[Comparison of vector algebra and geometric algebra]]
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| ==Notes and references==
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| {{reflist}}
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| ==Further reading==
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| {{Refbegin}}
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| * {{cite book | title = Advanced Engineering Electromagnetics | first = Constantine A. | last = Balanis | isbn = 0-471-62194-3 }}
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| * {{cite book | first = H. M. | last = Schey | title = Div Grad Curl and all that: An informal text on vector calculus | publisher=W. W. Norton & Company | year= 1997 | isbn = 0-393-96997-5 }}
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| * {{cite book | first = David J. | last = Griffiths | title = Introduction to Electrodynamics | publisher=Prentice Hall|year=1999|isbn= 0-13-805326-X}}
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| {{Refend}}
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| <!--List of vector identities, oldid=343957081-->
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| [[Category:Vector calculus]]
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| [[Category:Mathematical identities]]
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| [[Category:Mathematics-related lists]]
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| [[bs:Spisak vektorskih identiteta]]
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| [[eo:Vektoraj identoj]]
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| [[zh:向量恆等式列表]]
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