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How do you prove Stokes Theorem?

How do you prove Stokes Theorem?

We will prove Stokes’ theorem for a vector field of the form P (x, y, z) k . That is, we will show, with the usual notations, (3) P (x, y, z) dz = curl (P k ) · n dS . We assume S is given as the graph of z = f(x, y) over a region R of the xy-plane; we let C be the boundary of S, and C the boundary of R.

What do you mean by curl of a vector state and prove Stoke’s theorem?

Stokes Theorem Statement According to this theorem, the line integral of a vector field A vector around any closed curve is equal to the surface integral of the curl of A vector taken over any surface S of which the curve is a bounding edge.

What is the statement of Stokes theorem?

The classical Stokes’ theorem can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface. Stokes’ theorem is a special case of the generalized Stokes’ theorem.

Is Stokes theorem a surface integral?

Stokes’ theorem relates a surface integral of a the curl of the vector field to a line integral of the vector field around the boundary of the surface.

Which type of operation is used in Stokes Theorem?

2. The flux of the curl of a vector function A over any surface S of any shape is equal to the line integral of the vector field A over the boundary C of that surface i.e. It converts a line integral to a surface integral and uses the curl operation. Hence Stokes theorem uses the curl operation.

What is Stokes law in physics class 11?

Stokes Law. Stokes Law. The force that retards a sphere moving through a viscous fluid is directly ∝to the velocity and the radius of the sphere, and the viscosity of the fluid. Mathematically:-F =6πηrv where. Let retarding force F∝v where v =velocity of the sphere.

How do you know when to use Stokes theorem?

Summary

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

What is the normal vector in Stokes theorem?

The normal vector points in the positive x-direction. But we need it to point it negative x-direction. Therefore, the surface is not oriented properly if we were to choose this normal vector. To orient the surface properly, we must instead use the normal vector ∂Φ∂θ×∂Φ∂r=−ri.

Which type of operation is used in Stokes theorem?

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.

How do you prove Stokes’ theorem?

While you are walking along the curve if your head is pointing in the same direction as the unit normal vectors while the surface is on the left then you are walking in the positive direction on C C. Now that we have this curve definition out of the way we can give Stokes’ Theorem.

How to write a surface integral using Stokes’ theorem?

Using Stokes’ Theorem we can write the surface integral as the following line integral. So, it looks like we need a couple of quantities before we do this integral. Let’s first get the vector field evaluated on the curve. Remember that this is simply plugging the components of the parameterization into the vector field.

How do you reduce Stokes’ theorem to Green’s theorem?

This completes the proof of Stokes’ theorem when F = P (x, y, z)k . In the same way, if F = M(x, y, z)i and the surface is x = g(y, z), we can reduce Stokes’ theorem to Green’s theorem in the yz-plane. If F = N(x, y, z)j and y = h(x, z) is the surface, we can reduce Stokes’ theorem to Green’s theorem in the xz-plane.

How do you prove that a surface is a vector field?

Let S S be an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve C C with positive orientation. Also let →F F → be a vector field then,