How to parameterize these surfaces?

• Sho Kano
In summary, the problem is to calculate the surface integral of y + z^2 over the upper hemisphere with radius a centered at the origin and x >= 0. The parameterizations for the surface are correct, and the surface integral can be simplified to pi*a^2. However, it is not clear how the value of ds will be expressed in the parameters.
Sho Kano

Homework Statement

Calculate ##\iint { y+{ z }^{ 2 }ds } ## where the surface is the upper part of a hemisphere with radius a centered at the origin with ##x\ge 0##

Homework Equations

Parameterizations:
##\sigma =\left< asin\phi cos\theta ,asin\phi sin\theta ,acos\phi \right> ,0\le \phi \le \frac { \pi }{ 2 } ,\frac { -\pi }{ 2 } \le \theta \le \frac { \pi }{ 2 } \\ N=(asin\phi )\sigma \\ \left| N \right| ={ a }^{ 2 }sin\phi \\ \\ \alpha =\left< rcos\theta ,rsin\theta ,0 \right> ,0\le r\le a,\frac { -\pi }{ 2 } \le \theta \le \frac { \pi }{ 2 } \\ N=-k\\ \left| N \right| =1##

The Attempt at a Solution

are these the right parameterizations?

Check for yourself: ##\ \sigma =\left< a\sin\phi \cos\theta ,a\sin\phi \sin\theta ,a\cos\phi \right> \ ##seems right to me. For ##\ \iint ds\ ## you would then get ##\ \pi a^2, \ ## right ?

It is not clear to me what you do to express ##\ ds \ ##. What is ##N## and what is the function of ##N## ?

BvU said:
Check for yourself: ##\ \sigma =\left< a\sin\phi \cos\theta ,a\sin\phi \sin\theta ,a\cos\phi \right> \ ##seems right to me. For ##\ \iint ds\ ## you would then get ##\ \pi a^2, \ ## right ?

It is not clear to me what you do to express ##\ ds \ ##. What is ##N## and what is the function of ##N## ?
Oh sorry, by the integral I mean a surface integral. N is the normal. Both parameterizations seem right to me...i originally had ##a## instead of ##r## for the second parameterization. But that would just give me a circle, not a disk (a surface)

Are we mixing up two threads with almost the same title ?
Not clear to me why you need ##N## in this thread. But you sure need ##ds## and I haven't seen how you are going to express that in the parameters

1. What is surface parameterization and why is it important?

Surface parameterization is the process of describing a surface in terms of a set of parameters. This allows for a surface to be represented mathematically and manipulated in various ways. Parameterization is important because it allows for easier analysis and visualization of a surface, as well as making it possible to perform calculations and simulations on the surface.

2. How do I determine the appropriate parameters for a given surface?

The appropriate parameters for a surface will depend on the specific characteristics and features of the surface. Generally, parameters should be chosen to represent the key variables that define the surface, such as its shape, size, and orientation. It may also be helpful to consider the intended use of the parameterization and any constraints that need to be taken into account.

3. Can a surface be parameterized in multiple ways?

Yes, a surface can often be parameterized in multiple ways. The choice of parameterization will depend on the specific needs and goals of the application. Some parameterizations may be more useful for certain types of analysis or manipulation, while others may be better suited for visualization or simulation purposes.

4. Are there any limitations to surface parameterization?

Yes, there are some limitations to surface parameterization. For example, some surfaces may be difficult to parameterize accurately or may require a large number of parameters to fully describe. Additionally, certain types of surfaces may not lend themselves well to certain types of parameterization methods. It is important to carefully consider the limitations of a chosen parameterization approach and to select the most appropriate method for the surface at hand.

5. How can I validate the accuracy of a surface parameterization?

The accuracy of a surface parameterization can be validated by comparing it to other known parameterizations or to the original surface data. This can be done through visual inspection or by using mathematical techniques such as error analysis. It is also important to consider the purpose of the parameterization and whether it is providing the desired results for the intended application.

• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
826
• Calculus and Beyond Homework Help
Replies
9
Views
467
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
14
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
3K
• Calculus and Beyond Homework Help
Replies
5
Views
895
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
3K