What is the proof of Stokes theorem?

The Stoke’s theorem states that “the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface.” Where, C = A closed curve. S = Any surface bounded by C.

What is curl of a vector state and prove Stokes theorem?

Stoke’s theorem statement is “the surface integral of the curl of a function over the surface bounded by a closed surface will be equal to the line integral of the particular vector function around it.” Stokes theorem gives a relation between line integrals and surface integrals.

How do you solve problems with Stokes theorem?

1. For F(x,y,z)=(y,z,x), compute ∫CF⋅ds.
2. Solution: Since we are given a line integral and told to use Stokes’ theorem, we need to compute a surface integral ∬ScurlF⋅dS,
3. We’ll use the fact that ∫CF⋅ds=∫C1F⋅ds+∫C2F⋅ds+∫C3F⋅ds.
4. The integral for C3 is similar. ∫C3F⋅ds=0.
5. Therefore, ∫CF⋅ds=π4.
6. ∫CF⋅ds=∬ScurlF⋅dS.

Which is the mathematical form of Stokes theorem?

To use Stokes’s Theorem, we pick a surface with C as the boundary; the simplest such surface is that portion of the plane y+z=2 inside the cylinder. This has vector equation r=⟨vcosu,vsinu,2−vsinu⟩. We compute ru=⟨−vsinu,vcosu,−vcosu⟩, rv=⟨cosu,sinu,−sinu⟩, and ru×rv=⟨0,−v,−v⟩.

What is Stokes law in physics class 11?

Stoke’s Law states that the force that retards a sphere moving through a viscous fluid is directly proportional to the velocity and the radius of the sphere, and the viscosity of the fluid.

Who discovered Stokes theorem?

It is named after Sir George Gabriel Stokes (1819–1903), although the first known statement of the theorem is by William Thomson (Lord Kelvin) and appears in a letter of his to Stokes in July 1850. The theorem acquired its name from Stokes’s habit of including it in the Cambridge prize examinations.

What is the boundary of a surface in Stokes theorem?

The Stokes boundary If S is a 2-dimensional surface in R3, and if F is a C1 vector field, then Stokes’ Theorem relates the integral over S of curlF with the integral of F over ∂S, the boundary of S.

What is divergence theorem and Stokes theorem?

Long story short, Stokes’ Theorem evaluates the flux going through a single surface, while the Divergence Theorem evaluates the flux going in and out of a solid through its surface(s). Think of Stokes’ Theorem as “air passing through your window”, and of the Divergence Theorem as “air going in and out of your room”.

What is the curl of F?

The curl of F is ∇×F=|ijk∂∂x∂∂y∂∂zfgh|=⟨∂h∂y−∂g∂z,∂f∂z−∂h∂x,∂g∂x−∂f∂y⟩.

What is the significance of Stokes theorem?

As a matter of fact, Stokes’ theorem provides insight into a physical interpretation of the curl. In a vector field, the rotation of the vector field is at a maximum when the curl of the vector field and the normal vector have the same direction.

How do planes use the Stokes theorem?

To use Stokes’s Theorem, we pick a surface with C as the boundary; the simplest such surface is that portion of the plane y+z=2 inside the cylinder. This has vector equation r=⟨vcosu,vsinu,2−vsinu⟩. We compute ru=⟨−vsinu,vcosu,−vcosu⟩, rv=⟨cosu,sinu,−sinu⟩, and ru×rv=⟨0,−v,−v⟩.

What is the difference between Gauss theorem and Stokes theorem?

Comparison between Stokes’s Theorem and Gauss’s Theorem : Both theorems can be used to evaluate certain surface integrals, but there are some significant differences: Gauss’s Theorem applies only to surface integrals over closed surfaces; Stokes’s Theorem applies to any surface integrals satisfying the above basic …

What is green and Stokes theorem?

Green and Stokes’ Theorems are generalizations of the Fundamental Theorem of Calculus, letting us relate double integrals over 2 dimensional regions to single integrals over their boundary; as you study this section, it’s very important to try to keep this idea in mind.

What is called Stokes law?

Stokes’ law describes the settling of spheres in a Newtonian fluid. A spherical particle placed in a Newtonian fluid will sink if the buoyant force does not match or exceed the gravitational force on the sphere. The net downward force on a sphere is the difference between the settling force and the buoyant force.

What is Stokes method physics?

Stoke’s law was established by an English scientist Sir George G Stokes (1819-1903). When a spherical body moves down through an infinite column of highly viscous liquid, it drags the layer of the liquid in contact with it. As a result, the body experiences a retarding force.

What are the four conditions of Stokes law?

Conditions under which Stoke’s law is valid are: The fluid through which the body moves must have infinite extension. The body is perfectly rigid and smooth. There is no slip between the body and the fluid. The motion of the body does not give rise to turbulent motion.

Who was Stokes theorem named after?

history of mathematics The Gauss-Green-Stokes theorem, named after Gauss and two leading English applied mathematicians of the 19th century (George Stokes and George Green), generalizes the fundamental theorem of the calculus to functions of several variables.…

Does Stokes theorem calculate flux?

Stokes’ theorem says we can calculate the flux of curl F across surface S by knowing information only about the values of F along the boundary of S. Conversely, we can calculate the line integral of vector field F along the boundary of surface S by translating to a double integral of the curl of F over S.

What does it mean when Stokes theorem equals zero?

If F = ∇ f , the line integral of F along any curve is the difference of the values of f at the endpoints. For a closed curve, this is always zero. Stokes’ Theorem then says that the surface integral of its curl is zero for every surface, so it is not surprising that the curl itself is zero.

Can Stokes theorem be applied to closed surfaces?

Of course it is. If a surface has no boundary, the line-integral term of the classical Stokes’ theorem (or curl theorem, or Kelvin-Stokes ) vanishes, so the surface integral is zero. The same holds for the general Stokes’ theorem , in any dimension.

What is curl of a vector field?

In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation.

How do you orient in Stokes theorem?

The curve’s orientation should follow the right-hand rule, in the sense that if you stick the thumb of your right hand in the direction of a unit normal vector near the edge of the surface, and curl your fingers, the direction they point on the curve should match its orientation.

What is a total divergence?

The divergence theorem is employed in any conservation law which states that the total volume of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume’s boundary.