sadish kalan

sadish kalan ya sadish kailkulas ya sadish vishleshan (Vector calculus / vector analysis) ganit ki vah vidha hai jo sadish raashiyon ke vaastavik vishleshan (real analysis) se sambandh rakhati hai.

iske antargat bahut si samasyaaen hal karne ki vidhiyaaain evam sootr aate hain jo ki praudyogiki evam vigyaan mein bahut upayogi hain. ameriki vaigyaanik evam engineer vilaard Gibbs (J. Willard Gibbs) tatha british engineer heveesaaid (Oliver Heaviside) ne is kshetr ke agradoot rahe.

sadish vishleshan adish kshetr tatha sadish kshetr ke saath gahra sambandh hai.

adish kshetr: (scalar field) ke pratyek bindu ke saath ek adish raashi sambandhit hoti hai. jabki

sadish kshetr (vector field) ke pratyek bindu par ek sadish raashi judi hoti hai.

udaaharan

kisi taalaab ka taapamaan ek adish kshetr hai kyonki iske antargat pratyek bindu par ek adish raashi - taapamaan ka astitv hai. iske vipreet yadi taalaab ka paani gatisheel hai to iske harek bindu par jal ka veg ek sadish kshetr hai.

anukram

sadish sankriyaaeain (vector operations)

sadish vishleshan ki chaar pramukh sankriyaaen (kaarteeya nirdeshaank mein) neeche di gayeen hain. ye sankriyaaen sadish ya adish kshetr ke upar del oparetar () ke prayog se ki jaateen hain.

grediyent (Gradient)

daaivarjeins (Divergence)

karl ya rotation (curl)

laaplaasian (Laplacian)

yeh ek dviteeya order ka difreinsial sankriya hai. tri-bimeeya kaarteeya nirdeshaank tantr mein ise is prakaar paribhaashit kiya gaya hai:

.

laapalaasian ke kuchh upayog

pvaason ka sameekaran (Poisson's equation)

tani hui dori ka kampan

sankriyaaon ka bhautik jagat mein arth

sankriya (Operation) prakat karne ka tareeka vyaakhya kshetr/paraas (Domain/Range)
grediyent (Gradient) Measures the rate and direction of change in a scalar field. Maps scalar fields to vector fields.
karl (Curl) Measures the tendency to rotate about a point in a vector field. Maps vector fields to vector fields.
daaivarajeins (Divergence) Measures the magnitude of a source or sink at a given point in a vector field. Maps vector fields to scalar fields.
laaplaas ka oparetar (Laplace operator) A composition of the divergence and gradient operations. Maps scalar fields to scalar fields.

pramukh prameya

Theorem Statement Description
grediyent prameya (Gradient theorem) The line integral through a gradient (vector) field equals the difference in its scalar field at the endpoints of the curve.
green ka prameya (Green's theorem) 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.
stok ka prameya (Stokes' theorem) 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.
daaivarjeins prameya (Divergence theorem) The integral of the divergence of a vector field over some solid equals the integral of the flux through the surface bounding the solid.

anya nirdeshaankon mein sadish sankriyaaeain

belani nirdeshaank mein

goleeya nirdeshaank mein

sarvasamikaaeain

yadi to jahaaain koi sadish kshetr hai.

yadi to jahaaain koi adish kshetr hai.

inhein bhi dekhein

  • sadish kailakulas ki sarvasamikaaeain

sandarbh

  • 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.

baahari kadiyaaain