Curl On Bottom of Sphere: Determine F(x,y,z)

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Also, the limits of integration should be from 0 to 2pi as t goes from 0 to 2pi.In summary, the problem is to find the flux of curl(F) through the surface described by the equation 625=z^2+x^2+y^2 where z<=20, using Stokes' theorem to convert the double integral into a single circulation integral around the top of the bottom section of the sphere. The approach of using the parameter t and substituting x,y, and z into the integral before computing it is correct. The limits of integration should be from 0 to 2pi.
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bfr
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Homework Statement



Determine the curl on teh surface of the bounded region consisting of the bottom part of the sphere with equation 625=z^2+x^2+y^2 where z<=20, in the force field F(x,y,z)=<x^2 * y,x*y^2 * z,2x>

Homework Equations



http://img187.imageshack.us/img187/291/1fdf437d8e18a23191b63dfnj8.png

The Attempt at a Solution



I used Stokes' theorem to change the double integral for curl into a single circulation circle around the top of the bottom section of the sphere:

625=z^2+x^2+y^2; z=20; 225=r^2

Let x=15cos t; y=15sin t; z=20 (I'm still writing x,y, and z instead of their substituted values in the following integral though): integral (<x^2*y,xy^2z,2z> dot <-15sin(t),15cos(t)>) dt from t=0 to t=2pi ("dot" represents a dot product). Is this right?
 
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bfr said:

Homework Statement



Determine the curl on teh surface of the bounded ...

That doesn't make much sense to me. The curl of a vector function is simply the curl of the vector function. It has different values at different point on the surface, so to determine "the curl of a vector on a surface" doesn't make a whole lot of sense.

Surely the question is supposed to be to "find the flux of curl(F) through the surface"?

The Attempt at a Solution



I used Stokes' theorem to change the double integral for curl into a single circulation circle around the top of the bottom section of the sphere:

625=z^2+x^2+y^2; z=20; 225=r^2

Let x=15cos t; y=15sin t; z=20 (I'm still writing x,y, and z instead of their substituted values in the following integral though): integral (<x^2*y,xy^2z,2z> dot <-15sin(t),15cos(t)>) dt from t=0 to t=2pi ("dot" represents a dot product). Is this right?

Assuming that you are indeed trying to find the flux of curl(F) through the surface (as opposed to trying to find the value of curl(F) at every point on the surface); then this approach looks correct.

However, since x and y depend on the parameter t, you need to make sure you substitute x=15cos t, y=15sin t and z=20 into the integral before computing it.
 

Related to Curl On Bottom of Sphere: Determine F(x,y,z)

1. What is the curl on the bottom of a sphere?

The curl on the bottom of a sphere refers to the direction and magnitude of the rotational force at any point on the bottom surface of the sphere.

2. How do you determine the curl on the bottom of a sphere?

The curl on the bottom of a sphere can be determined by calculating the partial derivatives of the vector function F(x,y,z) with respect to x, y, and z and then taking the cross product of the resulting vectors.

3. What does the curl on the bottom of a sphere tell us about the flow of a vector field?

The curl on the bottom of a sphere indicates the presence of rotational motion in the vector field. It tells us about the direction and strength of the rotational component of the flow at any point on the bottom surface of the sphere.

4. What are some real-world applications of determining the curl on the bottom of a sphere?

One application is in fluid dynamics, where the curl can help analyze the circulation and vorticity of a fluid. It is also useful in electromagnetism, as the curl can help determine the direction and strength of magnetic fields.

5. Are there any limitations to using the curl to determine the flow of a vector field on the bottom of a sphere?

The curl is only a measure of rotational motion and does not provide information about the overall flow of the vector field. Additionally, it may not be applicable in non-conservative vector fields where the curl may not exist.

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