What are the applications of Stokes theorem?

Stokes’ theorem can be used to turn surface integrals through a vector field into line integrals. This only works if you can express the original vector field as the curl of some other vector field. Make sure the orientation of the surface’s boundary lines up with the orientation of the surface itself.

How do you solve problems with Stokes theorem?

Use Stokes’ Theorem to evaluate ∫C→F⋅d→r ∫ C F → ⋅ d r → where →F=−yz→i+(4y+1)→j+xy→k F → = − y z i → + ( 4 y + 1 ) j → + x y k → and C is is the circle of radius 3 at y=4 and perpendicular to the y -axis.

What is Stokes theorem simple words?

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.

Which equation is governed by Stokes theorem?

∫C⇀F⋅d⇀r=∬Scurl⇀F⋅d⇀S. Figure 16.7. 1: Stokes’ theorem relates the flux integral over the surface to a line integral around the boundary of the surface.

How do you verify a triangle using Stokes theorem?

How do you find unit vector in Stokes theorem?

What is difference between divergence 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”.

Who invented 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 vector field example?

Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

Which of the following is correct for Stokes theorem?

It states that the line integral of a vector field around any closed surface C is equal to the surface integral of the normal component of curl of vector over an unclosed surface ‘S’.

How do you solve Green’s theorem problems?

Green’s theorem is used to integrate the derivatives in a particular plane. If a line integral is given, it is converted into a surface integral or the double integral or vice versa using this theorem.
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Therefore, we can write the area formulas as:
  • A = − ∫ c y d x.
  • A = ∫ c x d y.
  • A = 1 2 ∫ c ( x d y − y d x )

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.

How do you parameterize Stokes theorem?

To use Stokes’ Theorem, we need to think of a surface whose boundary is the given curve C. First, let’s try to understand C a little better. We are given a parameterization r(t) of C. In this parameterization, x = cos t, y = sin t, and z = 8 − cos2 t − sin t.

What is the difference between Green theorem and Stokes theorem?

Green’s theorem applies only to two-dimensional vector fields and to regions in the two-dimensional plane. Stokes’ theorem generalizes Green’s theorem to three dimensions.

What is the application of Green theorem?

Green’s theorem converts the line integral to a double integral of the microscopic circulation. The double integral is taken over the region D inside the path. Only closed paths have a region D inside them. The idea of circulation makes sense only for closed paths.

What is curl and divergence?

Roughly speaking, divergence measures the tendency of the fluid to collect or disperse at a point, and curl measures the tendency of the fluid to swirl around the point. Divergence is a scalar, that is, a single number, while curl is itself a vector.

What is difference between divergence 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”.

Who invented Stoke 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.

Is Green’s theorem is a special case of Stokes theorem?

We see that Green’s theorem is really just a special case of Stokes’ theorem, where our surface is flattened out, and it’s in the xy plane.

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 is meant by flux of a vector?

For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.

What is divergence theorem formula?

The divergence theorem states that the surface integral of the normal component of a vector point function “F” over a closed surface “S” is equal to the volume integral of the divergence of. F → taken over the volume “V” enclosed by the surface S. Thus, the divergence theorem is symbolically denoted as: ∬ v ∫ ▽ F → .

What are the limitations of Stokes law?

Limitations of the Stoke’s Law. Negative density difference in Stoke’s equation: Stokes’ equation is invalid if the density difference in the equation is negative that is when the particles are lighter than the dispersion medium.