Stokes theorom question with a line

In summary, the conversation discusses using Stokes' theorem to solve a problem involving a line C intersecting a sphere and a plane. The suggested approach is to parameterize the line and use the projection of the elliptic cylinder onto the xz-plane to find a suitable parameterization for the problem. The limits for the parameter theta are 0 to 2pi, and the final solution involves finding a parameterization for the line and using it to solve the integral.
  • #1
Austturt
1
0

Homework Statement


F[/B]=(y + yz- z, 5x+zx, 2y+xy )

use stokes on the line C that intersects: x^2 + y^2 + z^2 = 1 and y=1-x

C is in the direction so that the positive direction in the point (1,0,0) is given by a vector (0,0,1)

2. The attempt at a solution
I was thinking that I could decide my surface Y to be where the plane is cutting the sphere but I'm not sure how to parametrize this or if it's the right way to do it?
 
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  • #2
You only have to parameterise the line don't you, not the surface?

A natural parameterisation might be to use as parameter the angle ##\theta## that the line from the centre of the circle C to ##\mathbf{x}(\theta)## makes with the x-y plane. It should be doable from there.

If the integration gets too messy you could try rotating the coordinate system by 45 degrees around the z axis so that the circle C has a constant x' coordinate. But I'd leave that as plan B for now.
 
  • #3
The limits for ##\theta## given by the curve ##C## would be ##0 \leq t \leq 2 \pi##, where the curve ##C## is the boundary curve of the intersection of ##y = 1 - x## and ##x^2 + y^2 + z^2 = 4##.

Find a parameterization ##\vec r(t)## such that:

$$\iint_S \text{curl}(\vec F) \cdot d \vec S = \oint_C \vec F \cdot d \vec r = \int_0^{2 \pi} \vec F( \vec r(t) ) \cdot \vec r'(t) \space dt$$

Hint: With ##y = 1 - x##, the sphere becomes an elliptic cylinder in the xz-plane: ##x^2 + (1 - x)^2 + z^2 = 4##. The projection of the elliptic cylinder onto the xz-plane produces an ellipse that is not centered at the origin.
 

What is Stokes' theorem?

Stokes' theorem is a mathematical theorem that relates the integral of a vector field over a surface to the line integral of the same vector field around the boundary of the surface.

What is the significance of Stokes' theorem?

Stokes' theorem is a fundamental tool in vector calculus and is used to simplify computations involving vector fields and their integrals. It is also closely related to other important theorems, such as Green's theorem and the divergence theorem.

How is Stokes' theorem applied in real-world situations?

Stokes' theorem has various applications in physics and engineering, including in fluid mechanics, electromagnetism, and differential geometry. It is commonly used to calculate the circulation of a fluid around a closed curve or to determine the flux of a vector field through a surface.

What is the relationship between Stokes' theorem and the fundamental theorem of calculus?

Stokes' theorem can be seen as a higher-dimensional analogue of the fundamental theorem of calculus. Both theorems relate an integral over a region to an integral over the boundary of that region, but Stokes' theorem applies to vector fields in multiple dimensions, while the fundamental theorem of calculus applies to scalar functions in one dimension.

Are there any limitations to Stokes' theorem?

Stokes' theorem is only applicable to smooth surfaces and vector fields. In addition, the surface must be oriented and the boundary curve must be oriented in the same direction for the theorem to hold. These limitations may require some modifications to be made in certain situations.

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