Solving Stokes' Theorem: Find \int_{\partial S} F \cdot ds

In summary, the problem involves finding the line integral of the vector field F over the ellipsoid S and using Stokes' theorem to relate it to the boundary of S. In this case, the boundary is 0 since the surface has no boundary. This can be shown by dividing the surface into smaller flat surfaces and noticing that the line integrals around them cancel out, leaving only the boundary. The answer becomes 0 when the surface has no boundary.
  • #1
Here is the problem:

S is the ellipsoid [tex]x^2+y^2+2z^2=10[/tex]

and F is a vector field [tex]F=(sin(xy),e^x,-yz)[/tex]

Find: [tex]\int \int_S ( \nabla \mbox {x} F) \cdot dS[/tex]

So, I know that Stokes' Theorem states that:
[tex]\int \int_S ( \nabla \mbox {x} F) \cdot dS = \int_{\partial S} F \cdot ds[/tex]
where [tex]\partial S[/tex] equals the boundary of the ellipsoid. How do you find [tex]\partial S[/tex]? My professor just told me that any closed surface has no boundary and therefore the answer is 0, but would someone show me how I can show this? And can someone tell me under what conditions does the answer become 0 using Stokes' Theorem? Thanks a lot.
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  • #2
Stokes' theorem is derived by dividing a bounded surface up into a bunch of small flat surfaces and then noticing that the line integrals around all these surfaces cancel each other out except at the boundary of the big surface. Think about that and what it means if the surface you're splitting up has no boundary.
  • #3

To solve this problem, we can start by finding the boundary of the ellipsoid \partial S. The boundary of a surface is defined as the set of points where the surface ends or meets another surface. In this case, the ellipsoid is a closed surface, meaning it has no boundaries. This can be seen by visualizing the ellipsoid and noticing that it has no edges or boundaries where it meets another surface. Therefore, \partial S = \emptyset and the integral becomes:

\int \int_S ( \nabla \mbox {x} F) \cdot dS = \int_{\emptyset} F \cdot ds = 0

This shows that the answer is indeed 0, as your professor stated.

Stokes' Theorem is applicable when the surface is closed, meaning it has no boundaries or edges. In this case, the surface is a closed ellipsoid and therefore the theorem can be used. In general, Stokes' Theorem can be used for any closed surface, whether it is a sphere, cylinder, or any other shape with no boundaries.

In summary, to solve this problem we first determine that the boundary of the ellipsoid is \emptyset. Then, using Stokes' Theorem, we can rewrite the integral as \int_{\emptyset} F \cdot ds = 0. This shows that the answer is 0 and also shows the conditions under which Stokes' Theorem can be used - for closed surfaces with no boundaries.

1. What is Stokes' Theorem?

Stokes' Theorem is a mathematical tool that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field along the boundary of the surface.

2. How is Stokes' Theorem used to solve for \int_{\partial S} F \cdot ds?

Stokes' Theorem states that \int_{\partial S} F \cdot ds = \iint_S (\nabla \times F) \cdot dS, so to solve for \int_{\partial S} F \cdot ds, we need to calculate the curl of the vector field F and then evaluate the surface integral over the region S.

3. What is the difference between a surface integral and a line integral?

A surface integral is an integral over a two-dimensional surface, while a line integral is an integral along a one-dimensional curve. In terms of Stokes' Theorem, a surface integral is related to a line integral through the curl of the vector field.

4. What are the prerequisites for understanding Stokes' Theorem?

To understand and use Stokes' Theorem, one should have a solid understanding of vector calculus, particularly the concepts of vector fields, line integrals, and surface integrals. It is also helpful to have some familiarity with differential forms and the concept of a boundary.

5. Can Stokes' Theorem be used to solve any type of problem?

No, Stokes' Theorem is specifically used to relate surface integrals and line integrals for vector fields. It cannot be used to solve problems in other areas of mathematics or science.

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