Calculating the Surface Integral Using Stokes' Theorem

In summary, the task at hand is to find the double integral at S of the curl of F.n. To do this, we first need to calculate the normal vector to the surface S, which is (2/7, 6/7, -3/7). Then, we compute the curl of F, which is -2y. Taking the dot product of the two, we get 6y/7. Next, we will perform the surface integral of this dot product over the intersection of the plane and the cylinder, which is an elliptical disk. This can then be converted to a double integral with x and y in the circular disk x^2+y^2=1. The key is to convert the area
  • #1
Hiche
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Homework Statement


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Homework Equations



The double integral at S of the (curl of)F.n.

The Attempt at a Solution



We find the the normal vector to the surface S is (2/7, 6/7, -3/7), right or not? We compute the curl of F which is -2y, right? Then, we calculate the dot product and we get 6y/7. Am I doing it right? If so, can someone tell me what to do next, and if not, can someone tell me how to properly solve it. I'm fairly new to the concept so please be patient with me.
 
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  • #2
Then you'll perform the SURFACE integral of the dot product 6y/7 over the intersection of the plane and the cylinder, which is an elliptical disk. The integral can be converted to a DOUBLE integral with x and y in the circular disk x^2+y^2=1. The trick is to convert the area element dS of the elliptical to the area element dxdy in the circular disk. Intuitively dS*cos(theta)=dxdy, where theta is the angle of the intersection plane to the x-y plane.
 

FAQ: Calculating the Surface Integral Using Stokes' Theorem

What is Stokes' Theorem and what does it state?

Stokes' Theorem is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a surface to the line integral of the same vector field along the boundary of the surface. It states that the surface integral of the curl of a vector field over a closed surface is equal to the line integral of the vector field along the boundary of the surface.

What is the significance of Stokes' Theorem in mathematics and physics?

Stokes' Theorem is significant because it allows us to calculate the surface integral of a vector field without having to directly evaluate the surface integral, which can be a difficult task. This theorem is also a key tool in understanding and applying concepts in fluid mechanics, electromagnetism, and other areas of physics.

How can Stokes' Theorem be applied in real-world problems?

Stokes' Theorem can be applied in various real-world problems, such as calculating the flow of a fluid through a surface, determining the circulation of a magnetic field around an object, and analyzing the motion of a solid body in a moving fluid. It is also useful in understanding the behavior of electric and magnetic fields in electromagnetic induction.

What are the conditions for Stokes' Theorem to be applicable?

The conditions for Stokes' Theorem to be applicable are that the surface must be a smooth, closed surface with a well-defined boundary, and the vector field must be continuously differentiable on the surface and its boundary. Additionally, the surface must be oriented in a consistent direction and the boundary must be traversed in the same direction as the surface normal.

Can you provide an example of a problem that can be solved using Stokes' Theorem?

One example of a problem that can be solved using Stokes' Theorem is calculating the work done by a force field on an object moving along a closed path. By applying Stokes' Theorem, the line integral of the force field can be transformed into a surface integral, making the calculation easier to solve. Another example is calculating the flux of a vector field through a closed surface, which can also be solved using Stokes' Theorem.

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