In vector calculus , and more generally differential geometry , Stokes' theorem sometimes spelled Stokes's theorem , and also called the generalized Stokes theorem or the Stokes—Cartan theorem  is a statement about the integration of differential forms on manifolds , which both simplifies and generalizes several theorems from vector calculus.
This modern form of Stokes' theorem is a vast generalization of a classical result that Lord Kelvin communicated to George Stokes in a letter dated July 2, It was first published by Hermann Hankel in This classical statement, along with the classical divergence theorem , the fundamental theorem of calculus , and Green's theorem are simply special cases of the general formulation stated above. Green's Theorem is a special case of Stoke's Theorem in the 2d plane.
Stoke's Theorem relates line integral to surface integral. On the other hand, Gauss Divergence Theorem relates surface integral to volume integral.
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In even simpler terms, one can consider the points as boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds. This quantity is well-defined; that is, it does not depend on the choice of the coordinate charts, nor the partition of unity.
The right-hand side of the equation is often used to formulate integral laws; the left-hand side then leads to equivalent differential formulations see below.
Let M be a smooth manifold. A smooth singular k -simplex in M is defined as a smooth map from the standard simplex in R k to M. The group C k M , Z of singular k - chains on M is defined to be the free abelian group on the set of singular k -simplices in M.
The corresponding homology resp. On the other hand, the differential forms, with exterior derivative, d , as the connecting map, form a cochain complex, which defines the de Rham cohomology groups H k dR M , R.
Differential k -forms can be integrated over a k -simplex in a natural way, by pulling back to R k. Extending by linearity allows one to integrate over chains.
Stokes’ Theorem Formula
This gives a linear map from the space of k -forms to the k th group of singular cochains, C k M , Z , the linear functionals on C k M , Z. This gives a homomorphism from de Rham cohomology to singular cohomology. On the level of forms, this means:. De Rham's theorem shows that this homomorphism is in fact an isomorphism. So the converse to 1 and 2 above hold true.
Stokes' theorem on smooth manifolds can be derived from Stokes' theorem for chains in smooth manifolds, and vice versa. The essential idea can be understood by the diagram on the left, which shows that, in an oriented tiling of a manifold, the interior paths are traversed in opposite directions; their contributions to the path integral thus cancel each other pairwise. As a consequence, only the contribution from the boundary remains.
It thus suffices to prove Stokes' theorem for sufficiently fine tilings or, equivalently, simplices , which usually is not difficult.
For example, if the domain of integration is defined as the plane region between two x -coordinates and the graphs of two functions, it will often happen that the domain has corners. Nevertheless, it is possible to check that the conclusion of Stokes' theorem is still true.
A version of Stokes' theorem that extends to rough domains [ definition needed ] was proved by Whitney.
Gauss Divergence theorem
The study of measure-theoretic properties of rough sets leads to geometric measure theory. Even more general versions of Stokes' theorem have been proved by Federer and by Harrison. The general form of the Stokes theorem using differential forms is more powerful and easier to use than the special cases.
The traditional versions can be formulated using Cartesian coordinates without the machinery of differential geometry, and thus are more accessible. Further, they are older and their names are more familiar as a result.
The traditional forms are often considered more convenient by practicing scientists and engineers but the non-naturalness of the traditional formulation becomes apparent when using other coordinate systems, even familiar ones like spherical or cylindrical coordinates. There is potential for confusion in the way names are applied, and the use of dual formulations. This special case is often just referred to as Stokes' theorem in many introductory university vector calculus courses and is used in physics and engineering.
It is also sometimes known as the curl theorem. One consequence of the Kelvin—Stokes theorem is that the field lines of a vector field with zero curl cannot be closed contours.
The formula can be rewritten as:. Green's theorem is immediately recognizable as the third integrand of both sides in the integral in terms of P , Q , and R cited above. Two of the four Maxwell equations involve curls of 3-D vector fields, and their differential and integral forms are related by the Kelvin—Stokes theorem.
Caution must be taken to avoid cases with moving boundaries: the partial time derivatives are intended to exclude such cases.
If moving boundaries are included, interchange of integration and differentiation introduces terms related to boundary motion not included in the results below see Differentiation under the integral sign :. The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed in SI units.
In other systems of units, such as CGS or Gaussian units , the scaling factors for the terms differ. Likewise, the Ostrogradsky—Gauss theorem also known as the divergence theorem or Gauss's theorem. Working out the divergence of the product gives. From Wikipedia, the free encyclopedia.
This article is about the generalized theorem. For the classical theorem, see Kelvin—Stokes theorem. For the equation governing viscous drag in fluids, see Stokes' law. Limits of functions Continuity. Mean value theorem Rolle's theorem. Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem. Fractional Malliavin Stochastic Variations.
Glossary of calculus. Paris: Hermann. Mathematics Magazine. Differential Forms". In James, I. History of Topology.
Amsterdam: Elsevier. May Wilson, David B. Cambridge, England: Cambridge University Press. Neither Thomson nor Stokes published a proof of the theorem. The first published proof appeared in in: Hankel, Hermann Hankel doesn't mention the author of the theorem. In a footnote, Larmor mentions earlier researchers who had integrated, over a surface, the curl of a vector field. See: Stokes, George Gabriel Cambridge, England: University of Cambridge Press. Oxford, England. Maxwell took this examination and tied for first place with Edward John Routh.
See: Clerk Maxwell, James Harman, P. See also Smith's prize or the Clerk Maxwell Foundation.
LEC-22 PROOF OF STOKE'S THEOREM
Clerk Maxwell, James A Treatise on Electricity and Magnetism. Oxford, England: Clarendon Press. In a footnote on page 27, Maxwell mentions that Stokes used the theorem as question 8 in the Smith's Prize Examination of This footnote appears to have been the cause of the theorem's being known as "Stokes' theorem".
Essential Calculus: Early Transcendentals. Robert Scheichl University of Bath , U. Manifolds, Tensors, and Forms. Introduction to Smooth Manifolds.
New York: Springer. October Bulletin of the American Mathematical Society. New Series. Classical Electrodynamics 2nd ed.
New York, NY: Wiley. Principles of Optics 6th ed. Categories : Differential topology Differential forms Duality theories Integration on manifolds Theorems in calculus Theorems in differential geometry.