# Stokes Theorem

Published on August 1, 2014

Author: Saruuu

Source: authorstream.com

Stokes Theorem And it Applications: Stokes Theorem And it Applications Sir George Gabriel Stokes: Sir George Gabriel Stokes History Of Stokes Theorem : History Of Stokes Theorem The theorem is named after the Irish mathematician physicist Sir George Gabriel Stokes (1819–1903). What we call Stokes Theorem was actually discovered by the Scottish physicist Sir William Thomson (1824–1907, known as Lord Kelvin) Stokes learned of it in a letter from Thomson in 1850 Stokes Theorem: Stokes Theorem Statement: Let S be a piecewise smooth oriented surface in space and let boundary of S be a piecewise smooth simple closed curve C . let F (x, y, z) be a continuous vector function that has continuous first partial derivatives in a domain in space containing S . Then This theorem relates the surface integral of the curl of a vector field F over a surface S in Euclidean three-space to the line integral of the vector field over its boundary C . Applications Of Stokes Theorem : Applications Of Stokes Theorem Stokes theorem plays astonishing role in Fluid Mechanics , Electrodynamics and in Multivariable Calculus in 3D. Let us discuss some of its application one by one. In electromagnetism: In electromagnetism Faraday’s Law Of Induction in Maxwell equation: Faraday’s Law Of Induction in Maxwell equation The electromotive force, in a circuit is equal to the circulation of the electric field, E around the circuit: Faraday discovered that in a stationary circuit an electromotive force is induced by changing magnetic flux: where , remember S is any capping surface of C . Continued: Continued Applying Stoke’s theorem to the definition of , we have Next using the definition of we have, Note: the last equality is possible since S is not time varying (stationary circuit) , this argument is a little deep. Continued: Continued Comparing the last two equations above we have, The above is one of Maxwell’s equation of electro-magnetic theory. In Mathematics: In Mathematics Finding Area: Finding Area Example Evaluate by Stokes theorem, where and C is the boundary of the triangle with vertices at (0, 0, 0), (1, 0, 0) and (1, 1, 0). Solution We have Therefore, Continued: Continued We note that the z -coordinate of each vertex of the triangle is zero. Therefore, the triangle lies in the xy -plane. Hence, . Thus, The equation of OB is y=x . Therefore, by Stokes theorem, we have Reference : Reference Mathematical and physical papers (Vol. 5) by Sir George Gabriel Stokes Advanced Engineering Mathematics (9 th edition) by Erwin Kreyszig All you wanted to know about Mathematics but were afraid to ask (Vol. 2) by Louis Lyon Engineering Mathematics by Babu Ram

01. 08. 2014
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