How Do You Apply Stoke's Theorem to a Rectangle in the x-y Plane?

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In summary, Stoke's 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. The simple case of Stoke's Theorem refers to the two-dimensional version, which relates a line integral along the boundary of a two-dimensional surface to a double integral over the surface itself. In physics, Stoke's Theorem is used to calculate the work done by a force field on a moving object and to calculate the flow of a fluid through a surface. In order for Stoke's Theorem to be applicable, the surface must be smooth and orientable, the boundary of the surface must be a closed curve, and the vector field must
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element1945
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The simple case of Stoke's Theorem for a rectangle in the x-y plane. I use green's theorem because n = k.

The question is... what should i choose as a vector function F? I just want to prove stoke's theorem for this case. Do i need a vector function?
 
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I have no idea what you mean by this. What are "n" and "k"? And, of course, the function F depends on the particular problem. If you are trying to prove Stoke's theorem, you must let F be any differentiable vector function.
 

FAQ: How Do You Apply Stoke's Theorem to a Rectangle in the x-y Plane?

What is Stoke's Theorem?

Stoke's 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.

What is the simple case of Stoke's Theorem?

The simple case of Stoke's Theorem refers to the two-dimensional version, which relates a line integral along the boundary of a two-dimensional surface to a double integral over the surface itself.

How is Stoke's Theorem used in physics?

In physics, Stoke's Theorem is used to calculate the work done by a force field on a moving object, as well as to calculate the flow of a fluid through a surface.

What are the conditions that must be met for Stoke's Theorem to be applicable?

In order for Stoke's Theorem to be applicable, the surface must be smooth and orientable, the boundary of the surface must be a closed curve, and the vector field must be continuous on the surface and its boundary.

Is Stoke's Theorem related to other fundamental theorems in calculus?

Yes, Stoke's Theorem is closely related to other fundamental theorems in calculus, such as Green's Theorem and the Divergence Theorem. All of these theorems are special cases of the more general Stokes' Theorem, which is used in higher dimensions.

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