सदिश कैलकुलस की सर्वसमिकाएँ का स्रोत देखें

[[सदिश कलन|सदिश कैलकुलस]] की कुछ प्रमुख [[गणितीय सर्वसमिका|सर्वसमिकाएँ]] नीचे दी गयी हैं।

== एकल ऑपरेटर वाली सर्वसमिकाएँ ==

=== किसी सदिश क्षेत्र का डाइवरजेंस (Divergence of a vector field) ===
किसी सदिश क्षेत्र <math> \mathbf{v} </math> के लिये, डाइवर्जेंस प्रायः निम्नलिखित प्रकार से लिखा जाता है:

:<math>\operatorname{div}(\mathbf{v}) = \nabla \cdot \mathbf{v}</math>
; किसी सदिश क्षेत्र का डाइवरजेंस एक अदिश राशि होती है।

=== किसी टेंसर का डाइवर्जेंस (Divergence of a tensor) ===
किसी टेंसर <math> \stackrel{\mathbf{\mathfrak{T}}}{} </math> के लिये, डाइवर्जेंस प्रायः निम्नलिखित प्रकार से लिखा जाता है:

:<math>\operatorname{div}(\mathbf{\mathfrak{T}}) = \nabla \cdot \mathbf{\mathfrak{T}}</math>

;जो कि एक सदिश राशि है।

=== कर्ल (Curl) ===

किसी सदिश क्षेत्र <math> \mathbf{v} </math> के लिये, कर्ल इस प्रकार लिखा जाता है:

:<math>\operatorname{curl}(\mathbf{v}) = \nabla \times \mathbf{v}</math>

; किसी सदिश क्षेत्र का कर्ल, एक सदिश राशि होती है।

=== ग्रेडिएन्ट (Gradient) ===

किसी सदिश क्षेत्र <math> \mathbf{v} </math> के लिये, ग्रेडिएन्ट प्रायः इस प्रकार लिखा जाता है:

:<math>\operatorname{grad}(\mathbf{v}) = \nabla \mathbf{v} </math>

; जो कि एक टेंसर है।

=== किसी अदिश क्षेत्र का ग्रेडिएंट (Gradient of a scalar field) ===
किसी अदिश क्षेत्र, <math>\psi</math> के लिये, ग्रेडिएन्ट इस प्रकार लिखा जाता है:

:<math>\operatorname{grad}(\psi) = \nabla \psi</math>

; किसी अदिश क्षेत्र का ग्रेडिएन्ट, एक सदिश राशि होती है।

== Combinations of multiple operators ==
=== Curl of the gradient ===

The [[Curl (mathematics)|curl]] of the [[gradient]] of ''any'' [[scalar field]] <math>\ \phi </math> is always zero:

:<math>\nabla \times (\nabla \phi) = 0</math>

One way to establish this identity (and most of the others listed in this article) is to use three-dimensional [[Cartesian coordinates]]. According to [[Curl (mathematics)|the article on ''curl'']],

:<math>\nabla \times \nabla \phi = \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \\
{ \partial_x } & { \partial_y } & { \partial_z } \\
 \\ \partial_x \phi & \partial_y \phi & \partial_z \phi \end{bmatrix} \, </math>

where the right hand side is a determinant, and '''i''', '''j''', '''k''' are unit vectors pointing in the positive axes directions, and ''&part;<sub>x</sub>'' = ''&part; / &part; x '' ''etc''. For example, the ''x''-component of the above equation is:

:<math> \mathbf{i} \left(\partial_y \partial_z - \partial_z \partial_y \right) \phi = 0 \, </math>

where the left-hand side evaluates as zero assuming the order of differentiation is immaterial.

=== Divergence of the curl ===
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 ===
The [[Laplacian]] of a scalar field is defined as the divergence of the gradient:
:<math> \nabla \cdot (\nabla \psi) = \nabla^2 \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>

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

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

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

=== Vector dot product ===

:<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>

In simpler form, using Feynman subscript notation:

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

where the notation '''&nabla;'''<sub>A</sub> means the subscripted gradient operates on only the factor '''A'''.<ref name=Feynman>{{cite book |author= R P Feynman, & Leighton & Sands |title=The Feynman Lecture on Physics |page= Vol II, p. 27-4 |publisher = Addison-Wesley |year=1964 |isbn=0805390499}}</ref><ref name=Missevitch>{{Cite web |url=http://arxiv.org/abs/physics/0504223 |title=Kholmetskii & Missevitch ''The Faraday induction law in relativity theory'', p. 4 |access-date=5 जुलाई 2008 |archive-url=https://web.archive.org/web/20140731005138/http://arxiv.org/abs/physics/0504223 |archive-date=31 जुलाई 2014 |url-status=live }}</ref>

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 |author=C Doran & A Lasenby |title=Geometric algebra for physicists |year=2003 |publisher=Cambridge University Press |page=p. 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>

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.

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 ===

:<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> \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>

where the Feynman subscript notation '''&nabla;'''<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> \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>

=== Product of a scalar and a vector ===
:<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} - \mathbf{A} \times \nabla\psi  </math>

=== 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 [[कलन|Calculus]]. 
: <math> \nabla (\psi \, \phi) = \phi \,\nabla \psi + \psi \,\nabla \phi </math>

== इन्हें भी देखें ==
* [[सदिश कलन|सदिश कैलकुलस]]
* [[सदिश बीजगणित]]

== टीका एवं संदर्भ ==
<references/>

[[श्रेणी:सदिश कलन]]
[[श्रेणी:गणित]]
[[श्रेणी:चित्र जोड़ें]]