Stokes Theorem: Statement, Formula, Applications & Sample Questions

Stokes' theorem connects the surface integral of the vector field's curl to a line integral of the vector field around a surface boundary. George Gabriel Stokes is the one who gave their name to this theorem. Stokes' theorem is a higher-dimensional extension of Green's theorem. Unlike Green's theorem, which equates a two-dimensional area integral with a corresponding line integral, Stokes' theorem reduces an integral over an n-dimensional area to an integral over a dimensional boundary, including the 1-dimensional case, where it is known as the Fundamental Theorem of Calculus. This provides inductive proof. Here, we will be learning more about Stokes’ theorem, formula, statement, applications, and discussing some important questions. 

Key Takeaways: Formula, Meaning, Gauss Divergence Theorem, Applications, Stokes Law Derivation, Assumptions

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Stokes Theorem Meaning

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The surface integral of the vector field's curl is related to a line integral of the vector field around a surface boundary by Stokes' theorem. George Gabriel Stokes is the one who gave it his name. Although William Thomson is credited with the first known statement of the theorem, which appears in a letter to Stokes. The surface integral of the curl of a function across the surface bounded by a closed surface will be equal to the line integral of the particular vector function around it, according to Stoke's theorem. The Stokes theorem establishes a relationship between line and surface integrals. One integral can be computed in terms of the other if it is more convenient.

In many applications, "Stokes' theorem" refers to the classical Stokes' theorem, especially Stokes' theorem, which equals an integral over a two-dimensional surface with an integral over a one-dimensional boundary curve. The classical Stokes' theorem is the subject of this article, which follows the convention. The generalized theorem is discussed in the references section at the end of this article. Many elements of classical physics, most notably Maxwell's equations regulating electromagnetic, rely on Stokes' theorem to create different equivalent formulations of physical principles.

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Stokes Theorem Formula

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We suppose that there is a right-hand rule-related orientation on both the surface and the curve. The surface would always be on your left if you walked around the curve in its preferred direction with your head facing in the same direction as the normal vector to the surface. 

Where,

A closed curve is denoted by the letter C.

Any surface that is circumscribed by C is referred to as S.

F = A vector field whose components in an open region of R3 containing S have continuous derivatives.

This classical proclamation, along with the classical divergence theorem, the fundamental theorem of calculus, and Green's theorem, are exceptional situations of the above-mentioned broad formulation.

That is to say: The surface will always be on your left if you walk around C in a positive direction with your head looking in the direction of n.

S is a positive-oriented oriented smooth surface bounded by a simple, closed smooth-boundary curve C.

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Assumptions made in Stokes Theorem

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In the Stokes Theorem, some assumptions were made. The Stokes Theorem is based on the following assumptions:

• Laminar flow is a type of flow that occurs.
• Particles have a spherical shape.
• The material's composition is homogeneous or uniform.
• The surfaces are very smooth.
• Particles do not collide with one another.

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Applications for Stokes Theorem

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Stoke's law is useful in a variety of situations. The following are some examples of Stokes law applications:

• It aids in the detection of silt settling in freshwater.
• It's also used to figure out how viscous a fluid is.

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Gauss Divergence Theorem

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The volume integral of the divergence over the area within the surface is equal to the vector's outward flow through a closed surface, according to the Gauss divergence theorem. To put it another way, the net flow of a region is the sum of all sources minus the sum of all sinks. The Gauss divergence theorem defines the flow of a vector field over a surface as well as the behavior of the vector field within the surface.

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Things to Remember

• The Stokes formula is used to determine the viscosity of oils by letting a sphere of known diameter fall freely in the liquid. Following the initial acceleration. When the external drag on the surface and buoyancy, both act upwards and in opposite directions to the motions.
• Stokes' Theorem is one of a group of mathematical conclusions that connects a volume's property to its boundary property. That could be compared to holography on some levels, but in its most basic form, it works with fluids or fluid-like substances.
• The Stokes theorem has nothing to do with N-S equations. The surface integral of the curl of a function over any surface limited by a closed path is equivalent to the line integral of a particular vector function around that path, according to Stoke's theorem. The use of Stoke’s theorem in fluid mechanics yields an important result.
• The stokes theorem has numerous applications in physics. It aids in the creation of numerous useful formulae and equations. The stokes theorem in electromagnetic theory, for example, is well-known in Physics.

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