Solving the Elliptical Integral Using Stoke's Theorem

In summary, to solve this problem using Stoke's theorem, we first calculate the curl of the given vector field, which is a constant vector. Then, we use the formula \iint_s curl \overrightarrow{F}\cdot \overrightarrow{n} \cdot dS, where \overrightarrow{n} is a unit perpendicular vector to the elliptical surface and dS is the area of the ellipse. The bounds of integration can be determined by finding the semi-axes of the ellipse, which are a and a/2. By multiplying the curl with the area of the ellipse, we can find the desired integral without any further integration.
  • #1
daftjaxx1
5
0
I'm trying to solve this problem:

Compute [tex]\oint_c(y+z)dx + (z-x)dy + (x-y)dz [/tex] using Stoke's theorem, where c is the ellipse [tex]x(t) = asin^2t, \ y(t) = 2asintcost, z(t) = acos^2t, 0\leq t \leq \pi [/tex]


The version of stoke's theorem I learned is:
[tex]
\int_c \overrightarrow{F} \cdot d\overrightarrow{r}
= \int_s curl \overrightarrow{F} \cdot d\overrightarrow{S}
=\iint_s curl \overrightarrow{F}\cdot \overrightarrow{n} \cdot dS
[/tex]

where S is the elliptical surface bounded by the curve c, F is a vector field and n is the unit vector pointing out at that point.

In this case, [tex]F = <y+z, z-x, x-y>[/tex], and I calculated curl F to be [tex]<-2, 0, -2>[/tex].

So we have to find

[tex]\iint_s <-2, 0, -2> \cdot \overrightarrow{n} \cdot dS [/tex]

How would I find [tex]\overrightarrow {n} [/tex] and dS, and also the bounds of integration for the double integral?
 
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  • #2
Fortunately, since the curl is a constant vector, you don't have to do any integration! Just find [tex]curl \overrightarrow{F}\cdot \overrightarrow{n}[/tex] which will be a constant, of course, and multiply by the area of the ellipse.

I don't know that this is the easiest way but here's what I did off the top of my head: When t= 0, a point on the ellipse is (0, 0, a). When t= [tex]\frac{\pi}{4}[/tex], a point on the ellipse is (a/2, a, a/2). When t= [tex]\frac{\pi}{2}[/tex], a point on the ellipse is (a, 0, 0). Since the interior of the ellipse is a plane, the vectors from one of those points to the other 2 lie in that plane and so perpendicular to any normal to the plane: Take the cross product of the two vectors to find a perpendicular (and then, of course, divide by its length to get a unit perpendicular).
Take the dot product of that unit perpendicular with the curl you have already calculated and "integrate" that over the ellipse. Since, as I said before, the integrand is a constant, that is just the constant times the area of the ellipse. It should be easy to see, by looking at the points for t= 0, t= [tex]\frac{\pi}{4}[/tex], t= [tex]\frac{\pi}{2}[/tex] , and t= [tex]\frac{3\pi}{4}[/tex] that the ellipse has semi-axes of length a and a/2. You can calculate the area of the ellipse from that.
 
  • #3


To solve this problem using Stoke's theorem, we first need to find the normal vector n and the surface element dS for the elliptical surface S. The normal vector n can be found by taking the cross product of the two tangent vectors to the ellipse, which can be calculated using the parametric equations given for x(t), y(t), and z(t). The surface element dS can be calculated using the formula dS = ||\overrightarrow{T_1} \times \overrightarrow{T_2}|| dt, where \overrightarrow{T_1} and \overrightarrow{T_2} are the tangent vectors at a given point on the ellipse and dt is the infinitesimal element of arc length along the ellipse.

Once we have found n and dS, we can then evaluate the double integral using the bounds of integration 0 \leq t \leq \pi. This will give us the value of the surface integral, which is equal to the line integral along the curve c by Stoke's theorem. We can then solve for the line integral by setting up the parametric equations for x(t), y(t), and z(t) and evaluating the integral over the given range of t. This will give us the final solution to the problem.

In summary, to solve this problem using Stoke's theorem, we need to first find the normal vector and surface element for the elliptical surface, then evaluate the double integral over the given bounds to find the value of the surface integral, and finally use this value to evaluate the line integral along the curve c. This approach may be more efficient and simpler than directly evaluating the line integral, especially for more complex curves and surfaces.
 

1. What is an Elliptical Integral?

An elliptical integral is a type of mathematical function that involves finding the area under an ellipse curve. It is a key concept in calculus and is used to solve various problems in physics and engineering.

2. What is Stoke's Theorem?

Stoke's Theorem is a mathematical theorem that relates the surface integral of a vector field to the line integral of its curl. It is often used to simplify and solve complex integrals, such as the elliptical integral.

3. How does Stoke's Theorem help in solving the Elliptical Integral?

Stoke's Theorem allows us to transform a difficult surface integral into a simpler line integral, making it easier to solve. It essentially reduces the number of variables and dimensions in the integral.

4. What are the steps involved in solving the Elliptical Integral using Stoke's Theorem?

The steps involved in solving the Elliptical Integral using Stoke's Theorem are as follows:
1. Identify the vector field and the surface over which the integral is taken.
2. Calculate the curl of the vector field.
3. Apply Stoke's Theorem to transform the surface integral into a line integral.
4. Evaluate the line integral using appropriate integration techniques.

5. What are the applications of solving the Elliptical Integral using Stoke's Theorem?

The applications of solving the Elliptical Integral using Stoke's Theorem are found in various fields of science and engineering, such as electromagnetism, fluid dynamics, and heat transfer. It is also used in the calculation of physical quantities, such as the moment of inertia and the center of mass of an object.

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