Parametrize a surface and calculating a surface integral

In summary, LKurtz gives the parametrization of the intersection of a cylinder with a plane of constant z. He suggests that x= cos(\theta), y= (1/\sqrt{2})sin(\theta), z= z so the two-dimensional surface is in terms of the two parameters \theta and z. The derivatives with respect to the two parameters are vectors in the tangent plane at each point and their cross product is the "fundamental vector product" which is the surface area.
  • #1
physmatics
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Homework Statement


Calculate the surface integral I = [tex]\int\int[/tex] f dS of the function f(x,y,z) = [tex]\sqrt{1/2 + y^{2}}[/tex] over the surface S given by [tex]x^{2} + 2*y^{2} = 1[/tex], [tex]0 \leq z \leq x^{2} + y^{2}[/tex]. (Clue: parametrize the surface.)

Homework Equations


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The Attempt at a Solution


The surface is, as far as I can tell, the elliptic cylinder [tex]x^{2} + 2*y^{2} = 1[/tex], from z = 0 to z = 1.
Now, I have trouble parametrizing the surface. Can I just parametrize it as an ellips in [tex]R^{2}[/tex]? The equation of that ellips would be [tex]x = \sqrt{1 - 2*y^{2}}[/tex]. Then, how do I parametrize the ellips given the equation? And also, why is 'parametrizing the surface' a clue? I really don't get it...
Sorry for clumsy use of LaTeX, I'm not very familiar with it.

Thank you very much!
 
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  • #2
You try to parameterize a surface in a way to express it in a nice form that hopefully makes the integral easy. This problem suggests cylindrical-like coordinates. Here's what I would try. Write your cylinder like this:

[tex]x^2 + \frac{y^2}{\left(\frac 1 {\sqrt{2}}\right)^2} = 1[/tex]

This suggests

[tex]x = \cos(\theta),\ y = \frac 1{\sqrt{2}}\sin(\theta)[/tex]

for the ellipse so you could try this parameterization for your surface:

[tex]\vec R(\theta,z) =\langle \cos(\theta),\frac 1{\sqrt{2}}\sin(\theta),z\rangle[/tex]

Try that. It might get a little messy but I think you will find it works. Be brave.:smile:
 
  • #3
Strictly speaking, what LKurtz gives is the parametrization of the intersection of that cylinder with a plane of constant z. For the surface itself you will need to add z= z.

You then have the surface parameterized by [itex]x= cos(\theta)[/itex], [itex]y= (1/\sqrt{2})sin(\theta)[/itex], [itex]z= z[/itex] so the two-dimensional surface is in terms of the two parameters [itex]\theta[/itex] and [itex]z[/itex].

We can then write
[tex]\vec{R}(\theta, z)= \langle cos(\theta), (1/sqrt{2})sin(\theta), z\rangle[/tex]

The derivatives with respect to the two parameters,
[tex]\vec{R}(\theta, z)_\theta= \langle -sin(\theta), (1/\sqrt{2})cos(\theta), 0\rangle[/tex]
[tex]\vec{R}(\theta, z)_z= \langle 0, 0 , 1 \rangle[/tex]
are vectors in the tangent plane at each point and their cross product (the "fundamental vector product" the surface), a vector perpendicular to the tangent plane at each point, with "[itex]d\theta dz[/itex]" is the vector differential of surface area and its magnitude is the differential of surface area.
 
  • #4
HallsofIvy said:
Strictly speaking, what LKurtz gives is the parametrization of the intersection of that cylinder with a plane of constant z.

Huh? Maybe your scroll-down bar isn't working??
 

1. What does it mean to parametrize a surface?

Parametrizing a surface means representing a 2-dimensional surface in terms of two independent variables, usually denoted as u and v. This allows us to describe points on the surface using a set of parameters instead of just x, y, and z coordinates.

2. How do you parametrize a surface?

To parametrize a surface, we can use a parametric equation in terms of u and v, such as x = f(u,v), y = g(u,v), z = h(u,v). This equation should represent all possible points on the surface in terms of the parameters u and v.

3. What is a surface integral?

A surface integral is a mathematical tool used to calculate the flux, or flow, of a vector field across a surface. It is similar to a double integral, but instead of integrating over a region in the xy-plane, we integrate over a surface in 3-dimensional space.

4. How do you calculate a surface integral?

To calculate a surface integral, we first need to parametrize the surface and then set up the integral using the parametric equations and a given function or vector field. The limits of integration will depend on the shape and orientation of the surface, and the integral can be evaluated using various techniques such as the divergence theorem or Green's theorem.

5. What are some applications of calculating surface integrals?

Surface integrals have many applications in physics and engineering, such as calculating the electric flux through a closed surface, finding the mass or center of mass of a curved object, and determining the surface area of a 3-dimensional object. They are also used in fluid mechanics to analyze the flow of fluids over surfaces.

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