Stokes' Theorem parameterization

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Start date Apr 23, 2016
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Stokes Theorem

In summary: The unit normal vector is easy to find. You can just use the parametrization of the circle to find it.

Apr 25, 2016

LCKurtz

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reminiscent said:

So would it be <0,0,sqrt(2)>?
Do I have to use polar coordinates?

This is getting a bit silly. Do you really have to ask whether <0,0,sqrt(2)> is a unit vector?

And before you talk about polar coordinates or rectangular coordinates, let's see what your integrand is. In post #15 we called the ##\vec V## the curl of ##\vec F## and the integrand is ##\vec V\cdot d\vec S##. What does that work out to?

<h2> What is Stokes' Theorem parameterization?</h2><p>Stokes' Theorem parameterization is a mathematical tool used to evaluate line integrals over a closed curve in a three-dimensional space. It relates a surface integral to a line integral by using a vector field and a differential form.</p><h2> How is Stokes' Theorem parameterization used in scientific research?</h2><p>Stokes' Theorem parameterization is commonly used in physics, engineering, and other scientific fields to calculate the circulation of a vector field around a closed curve. It is also used to evaluate flux integrals over a surface and to solve problems involving conservative vector fields.</p><h2> What are the conditions for applying Stokes' Theorem parameterization?</h2><p>In order to apply Stokes' Theorem parameterization, the surface must be smooth and orientable, and the vector field must be continuous and differentiable on the surface. The curve must also be a closed curve that lies on the boundary of the surface.</p><h2> How is Stokes' Theorem parameterization related to Green's Theorem?</h2><p>Stokes' Theorem parameterization is a higher-dimensional version of Green's Theorem, which relates a line integral to a double integral over a region in the plane. Green's Theorem can be seen as a special case of Stokes' Theorem parameterization when the surface is a flat plane.</p><h2> What are the practical applications of Stokes' Theorem parameterization?</h2><p>Stokes' Theorem parameterization has many practical applications, including calculating the work done by a force field on a particle moving along a closed path, determining the flow of a fluid around a closed loop, and finding the total electric charge enclosed by a closed surface. It is also used in the study of fluid dynamics, electromagnetism, and other areas of physics and engineering.</p>

FAQ: Stokes' Theorem parameterization

What is Stokes' Theorem parameterization?

Stokes' Theorem parameterization is a mathematical tool used to evaluate line integrals over a closed curve in a three-dimensional space. It relates a surface integral to a line integral by using a vector field and a differential form.

How is Stokes' Theorem parameterization used in scientific research?

Stokes' Theorem parameterization is commonly used in physics, engineering, and other scientific fields to calculate the circulation of a vector field around a closed curve. It is also used to evaluate flux integrals over a surface and to solve problems involving conservative vector fields.

What are the conditions for applying Stokes' Theorem parameterization?

In order to apply Stokes' Theorem parameterization, the surface must be smooth and orientable, and the vector field must be continuous and differentiable on the surface. The curve must also be a closed curve that lies on the boundary of the surface.

How is Stokes' Theorem parameterization related to Green's Theorem?

Stokes' Theorem parameterization is a higher-dimensional version of Green's Theorem, which relates a line integral to a double integral over a region in the plane. Green's Theorem can be seen as a special case of Stokes' Theorem parameterization when the surface is a flat plane.

What are the practical applications of Stokes' Theorem parameterization?

Stokes' Theorem parameterization has many practical applications, including calculating the work done by a force field on a particle moving along a closed path, determining the flow of a fluid around a closed loop, and finding the total electric charge enclosed by a closed surface. It is also used in the study of fluid dynamics, electromagnetism, and other areas of physics and engineering.