How do you verify Stokes Theorem?

Verifying Stokes’ Theorem for a Specific Case Verify that Stokes’ theorem is true for vector field F ( x , y , z ) = 〈 y , 2 z , x 2 〉 F ( x , y , z ) = 〈 y , 2 z , x 2 〉 and surface S, where S is the paraboloid z = 4 – x 2 – y 2 z = 4 – x 2 – y 2 .

How is direction determined 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 are the conditions for Stokes Theorem?

Stokes’ Theorem is about tiny spirals of circulation that occurs within a vector field (F). The vector field is on a surface (S) that is piecewise-smooth. Additionally, the surface is bounded by a curve (C). The curve must be simple, closed, and also piecewise-smooth.

How do you select a surface for Stokes Theorem?

For Stokes’ theorem, use the surface in that plane. For our example, the natural choice for S is the surface whose x and z components are inside the above rectangle and whose y component is 1.

What is positive orientation for Stokes theorem?

If you look at your right hand from the side of your thumb, your fingers curl in the counterclockwise direction. Think of your thumb as the normal vector n of a surface. If your thumb points to the positive side of the surface, your fingers indicate the circulation corresponding to curlF⋅n.

Which operation is used in Stokes theorem?

the 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.

Does Stokes theorem require a closed surface?

closed surfaces A surface S⊂R3 is said to be closed if it has no (Stokes) boundary. An example of such a surface is the unit sphere S={(x,y,z)∈R3:x2+y2+z2=1},n oriented outwards.

Why is Stokes theorem used?

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.

What are applications of Stokes theorem?

Stokes’ Theorem is applied to derive the retarded vector potential of loop antennas for the radiation of electric field and magnetic field. Simulations of the ideal and the actual magnetic dipole antenna suggest a satisfactory agreement of the two antennas.

What does it mean when Stokes theorem is 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.

How do you verify Stokes’ theorem?

Verify that Stokes’ theorem is true for vector field and surface S, where S is the paravbolid . Verifying Stokes’ theorem for a hemisphere in a vector field. As a line integral, you can parameterize C by . By (Figure), Therefore, we have verified Stokes’ theorem for this example.

How to apply Stokes’theorem to the hemisphere?

The hemisphere does not naturally have an orientation. But to use Stokes’ theorem, we must apply one. All we need here is to check whether the orientation we chose for the line integral is the same as that for the surface integral — use the right hand rule.

How to verify Stokes’theorem for vector field?

Verify Stokes’ Theorem for the given vector field f(x, y, z) and surface Σ . f(x, y, z) = 2yi − xj + zk; Σ: x2 + y2 + z2 = 1, z ≥ 0 This was the solution given. I understand the line integral part, but not the surface integral.

What is the parameterization of s in Stokes’ theorem?

Therefore, a parameterization of S is The curl of F is and Stokes’ theorem and (Figure) give Use Stokes’ theorem to calculate line integral where and C is oriented clockwise and is the boundary of a triangle with vertices and