What is the use of Stokes theorem?
What is the use of Stokes theorem?
Summary. 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 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 is Stoke’s theorem for magnetic field?
Stokes’ theorem relates a vector surface integral over surface S in space to a line integral around the boundary of S. We use Stokes’ theorem to derive Faraday’s law, an important result involving electric fields.
Which operation is used in Stokes theorem?
curl operation
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.
Which of the following is the mathematical expression of Stokes theorem?
With this definition in place, we can state 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.
What does Green’s theorem calculate?
In summary, we can use Green’s Theorem to calculate line integrals of an arbitrary curve by closing it off with a curve C0 and subtracting off the line integral over this added segment. Another application of Green’s Theorem is that is gives us one way to calculate areas of regions.
In which case stokes theorem is not applicable?
Stokes theorem does not always apply. The first condition is that the vector field, →A, appearing on the surface integral side must be able to be written as →∇×→F, where →F would either have to be found or may be given to you. If →F cannot be found, then Stokes theorem cannot be used.
Can you use Stokes theorem on a line?
Stokes’ theorem can be used to transform a difficult surface integral into an easier line integral, or a difficult line integral into an easier surface integral. Through Stokes’ theorem, line integrals can be evaluated using the simplest surface with boundary C.
Who invented Green’s theorem?
The same is true of Green’s Theorem and Green’s Function. The form of the theorem known as Green’s theorem was first presented by Cauchy [7] in 1846 and later proved by Riemann [8] in 1851.
What does Green’s theorem represent?
Green’s theorem gives a relationship between the line integral of a two-dimensional vector field over a closed path in the plane and the double integral over the region it encloses. The fact that the integral of a (two-dimensional) conservative field over a closed path is zero is a special case of Green’s theorem.
What is the physical meaning of Stokes theorem?
In other words, while the tendency to rotate will vary from point to point on the surface, Stokes’ Theorem says that the collective measure of this rotational tendency taken over the entire surface is equal to the tendency of a fluid to circulate around the boundary curve.
What does Stokes theorem find?
Stokes’ Theorem. The divergence theorem is used to find a surface integral over a closed surface and Green’s theorem is use to find a line integral that encloses a surface (region) in the xy-plane. The theorem of the day, Stokes’ theorem relates the surface integral to a line integral. Since we will be working in three dimensions,…
What is the intuition behind Stokes theorem?
Stokes’ Theorem: Physical intuition Stokes’ theorem is a more general form of Green’s theorem. Stokes’ theorem states that the total amount of twisting along a surface is equal to the amount of twisting on its boundary. Suppose we have a hemisphere and say that it is bounded by its lower circle. (picture)
What is stock theorem?
Stokes’ Theorem. It states that the circulation of a vector field, say A, around a closed path, say L, is equal to the surface integration of the Curl of A over the surface bounded by L.