सदिश कलन

सदिश कलन या सदिश कैल्कुलस या सदिश विश्लेषण (Vector calculus / vector analysis) गणित की वह विधा है जो सदिश राशियों के वास्तविक विश्लेषण (real analysis) से सम्बन्ध रखती है।

इसके अन्तर्गत बहुत सी समस्याएं हल करने की विधियाँ एवं सूत्र आते हैं जो कि प्रौद्योगिकी एवं विज्ञान में बहुत उपयोगी हैं। अमेरिकी वैज्ञानिक एवं इंजीनियर विलार्ड गिब्स (J. Willard Gibbs) तथा ब्रिटिश इंजीनियर हेवीसाइड (Oliver Heaviside) ने इस क्षेत्र के अग्रदूत रहे।

सदिश विश्लेषण अदिश क्षेत्र तथा सदिश क्षेत्र के साथ गहरा सम्बन्ध है।

अदिश क्षेत्रः (scalar field) के प्रत्येक बिन्दु के साथ एक अदिश राशि सम्बन्धित होती है। जबकि

सदिश क्षेत्र (vector field) के प्रत्येक बिन्दु पर एक सदिश राशि जुड़ी होती है।

उदाहरण

किसी तालाब का तापमान एक अदिश क्षेत्र है क्योंकि इसके अन्तर्गत प्रत्येक बिन्दु पर एक अदिश राशि - तापमान का अस्तित्व है। इसके विपरीत यदि तालाब का पानी गतिशील है तो इसके हरेक बिन्दु पर जल का वेग एक सदिश क्षेत्र है।

सदिश विश्लेषण की चार प्रमुख संक्रियाएं (कार्तीय निर्देशांक में) नीचे दी गयीं हैं। ये संक्रियाएं सदिश या अदिश क्षेत्र के उपर डेल् ऑपरेटर (${\displaystyle \nabla }$) के प्रयोग से की जातीं हैं।

${\displaystyle \operatorname {grad} ~\varphi ={\vec {\nabla }}\varphi ={\begin{pmatrix}{\frac {\partial \varphi }{\partial x}}\\[0.2cm]{\frac {\partial \varphi }{\partial y}}\\[0.2cm]{\frac {\partial \varphi }{\partial z}}\end{pmatrix}}}$

${\displaystyle \operatorname {div} ~{\vec {F}}={\vec {\nabla }}\cdot {\vec {F}}={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}}$

${\displaystyle \operatorname {rot} ~{\vec {F}}={\vec {\nabla }}\times {\vec {F}}={\begin{pmatrix}{\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\\[0.2cm]{\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\\[0.2cm]{\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\end{pmatrix}}}$

यह एक द्वितीय ऑर्डर का डिफरेंसिअल संक्रिया है। त्रि-बिमीय कार्तीय निर्देशांक तंत्र में इसे इस प्रकार परिभाषित किया गया है:

${\displaystyle \Delta =\nabla ^{2}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}$.

प्वासों का समीकरण (Poisson's equation)

${\displaystyle {\nabla }^{2}\varphi =f}$

तनी हुई डोरी का कम्पन

${\displaystyle {\nabla }^{2}\varphi (x,y,z,t)={\frac {1}{c^{2}}}\cdot {\frac {\partial ^{2}\varphi (x,y,z,t)}{\partial t^{2}}}}$

संक्रिया (Operation)	प्रकट करने का तरीका	व्याख्या	क्षेत्र/परास (Domain/Range)
ग्रेडिएन्ट (Gradient)	${\displaystyle \operatorname {grad} (f)=\nabla f}$	Measures the rate and direction of change in a scalar field.	Maps scalar fields to vector fields.
कर्ल (Curl)	${\displaystyle \operatorname {curl} (\mathbf {F} )=\nabla \times \mathbf {F} }$	Measures the tendency to rotate about a point in a vector field.	Maps vector fields to vector fields.
डाइवरजेंस (Divergence)	${\displaystyle \operatorname {div} (\mathbf {F} )=\nabla \cdot \mathbf {F} }$	Measures the magnitude of a source or sink at a given point in a vector field.	Maps vector fields to scalar fields.
लाप्लास का ऑपरेटर (Laplace operator)	${\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}$	A composition of the divergence and gradient operations.	Maps scalar fields to scalar fields.

Theorem	Statement	Description
ग्रेडिएन्ट प्रमेय (Gradient theorem)	${\displaystyle \varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)=\int _{L}\nabla \varphi \cdot d\mathbf {r} .}$	The line integral through a gradient (vector) field equals the difference in its scalar field at the endpoints of the curve.
ग्रीन का प्रमेय (Green's theorem)	${\displaystyle \int _{C}L\,dx+M\,dy=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dA}$	The integral of the scalar curl of a vector field over some region in the plane equals the line integral of the vector field over the curve bounding the region.
स्टोक का प्रमेय (Stokes' theorem)	${\displaystyle \int _{\Sigma }\nabla \times \mathbf {F} \cdot d\mathbf {\Sigma } =\oint _{\partial \Sigma }\mathbf {F} \cdot d\mathbf {r} ,}$	The integral of the curl of a vector field over a surface equals the line integral of the vector field over the curve bounding the surface.
डाइवर्जेंस प्रमेय (Divergence theorem)	${\displaystyle \iiint \limits _{V}\left(\nabla \cdot \mathbf {F} \right)dV=\iint \limits _{\partial V}\mathbf {F} \cdot d\mathbf {S} ,}$	The integral of the divergence of a vector field over some solid equals the integral of the flux through the surface bounding the solid.

${\displaystyle {\vec {\mathrm {grad} }}f={\frac {\partial f}{\partial r}}{\vec {u_{r}}}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\vec {u_{\theta }}}+{\frac {\partial f}{\partial z}}{\vec {u_{z}}}}$

${\displaystyle \mathrm {div} {\vec {A}}={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rA_{r}\right)+{\frac {1}{r}}{\frac {\partial A_{\theta }}{\partial \theta }}+{\frac {\partial A_{z}}{\partial z}}}$

${\displaystyle {\vec {\mathrm {curl} }}{\vec {A}}=\left({\frac {1}{r}}{\frac {\partial A_{z}}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial z}}\right){\vec {u_{r}}}+\left({\frac {\partial A_{r}}{\partial z}}-{\frac {\partial A_{z}}{\partial r}}\right){\vec {u_{\theta }}}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}(rA_{\theta })-{\frac {\partial A_{r}}{\partial \theta }}\right){\vec {u_{z}}}}$

${\displaystyle \Delta f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}}$

${\displaystyle {\vec {\mathrm {grad} }}f={\frac {\partial f}{\partial r}}{\vec {u_{r}}}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\vec {u_{\theta }}}+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}{\vec {u_{\varphi }}}}$

${\displaystyle \mathrm {div} {\vec {A}}={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}A_{r})+{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \theta }}(\sin \theta A_{\theta })+{\frac {1}{r\sin \theta }}{\frac {\partial A_{\varphi }}{\partial \varphi }}}$

${\displaystyle {\vec {\mathrm {curl} }}{\vec {A}}={\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}(\sin \theta A_{\varphi })-{\frac {\partial A_{\theta }}{\partial \varphi }}\right){\vec {u_{r}}}+\left({\frac {1}{r\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {1}{r}}{\frac {\partial }{\partial r}}(rA_{\varphi })\right){\vec {u_{\theta }}}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}(rA_{\theta })-{\frac {\partial A_{r}}{\partial \theta }}\right){\vec {u_{\varphi }}}}$

${\displaystyle \Delta f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}}$

• ${\displaystyle {\vec {\nabla }}\times {\big (}{\vec {\nabla }}\Psi {\big )}={\vec {0}}}$
• ${\displaystyle {\vec {\nabla }}\cdot {\big (}{\vec {\nabla }}\times {\vec {A}}{\big )}=0}$
• ${\displaystyle {\vec {\nabla }}\times {\big (}{\vec {\nabla }}\times {\vec {A}}{\big )}={\vec {\nabla }}{\big (}{\vec {\nabla }}\cdot {\vec {A}}{\big )}-\nabla ^{2}{\vec {A}}}$

• ${\displaystyle {\vec {\nabla }}\cdot {\vec {x}}=3}$
• ${\displaystyle {\vec {\nabla }}\times {\vec {x}}={\vec {0}}}$

यदि ${\displaystyle {\vec {\nabla }}\cdot {\vec {A}}=0}$ तो ${\displaystyle {\vec {A}}={\vec {\nabla }}\times {\vec {B}}}$ जहाँ ${\displaystyle {\vec {B}}}$ कोई सदिश क्षेत्र है।

यदि ${\displaystyle {\vec {\nabla }}\times {\vec {A}}=0}$ तो ${\displaystyle {\vec {A}}={\vec {\nabla }}\Psi }$ जहाँ ${\displaystyle \Psi }$ कोई अदिश क्षेत्र है।

• सदिश कैलकुलस की सर्वसमिकाएँ

• Michael J. Crowe (1994). A History of Vector Analysis : The Evolution of the Idea of a Vectorial System. Dover Publications; Reprint edition. ISBN 0-486-67910-1. (Summary)
• H. M. Schey (2005). Div Grad Curl and all that: An informal text on vector calculus. W. W. Norton & Company. ISBN 0-393-92516-1.
• J.E. Marsden (1976). Vecor Calculus. W. H. Freeman & Company. ISBN 0-7167-o462-5.

• Chen-To Tai (1995). A historical study of vector analysis. Technical Report RL 915, Radiation Laboratory, University of Michigan.

• Expanding vector analysis to a non-orthogonal space
• Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, (based upon the lectures of Willard Gibbs) by Edwin Bidwell Wilson, published 1902.