How to Evaluate Surface Integrals Using Divergence Theorem?

In summary: Just try to find the area or volume. Once you have that, you can work out the equation for each face.
  • #1
Hoofbeat
48
0
Please help! I'm soo confused with surface integrals and have several to do by tues for my tutorial. I don't really understand how to approach surface integrals! :cry: Could someone give me an over-view and help me through the question below - hopefully then I can manage the rest myself :cool: (Btw $ = integral sign)

Q. If n is the unit normal to the surface S, evaluate $$ r.n dS over:
a) the unit cube bounded by the coordinate planes and the the planes x=1, y=1 & z=1;
b) the surface of a sphere of radius a centred on the origin.

I *think* that I have to start by finding an equation for the cube (how?!) and then using divergence theorem but I really have no idea what I'm doing (I couldn't follow the lectures and books confuse me with their notation! :rolleyes:

Thanks
 
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  • #2
I guess you needn't really know the "equations" for the surfaces of the cube and the sphere.Just some calculus,Gauss-Ostrogradski theorem & some basic space geometry...

How about posting your work...?

Daniel.
 
  • #3
dextercioby said:
How about posting your work...?

That's the thing, I haven't done any of it! I'm genuinely confused as to even start the problem. I'm thinking I have to consider each face of the cube, but I really don't work well in 3D and can't even think how to express each plane as an equation! Then how do I progress? Do I have to calculate the Div or do it a different way :rolleyes: I thought I was following the lectures ok as the first few were fine and I could do the respective problem sheets but with the last few lectures on surface integrals & Divergance I've been so confused! :cry:
 
  • #4
All you need is the Divergence (or Gauss's/Ostrogradksy's) theorem.
Try setting up the integral first. You don't have to parametrize the surface.
 

FAQ: How to Evaluate Surface Integrals Using Divergence Theorem?

1. What is a surface integral?

A surface integral is a mathematical concept that calculates the total value of a function over a specific surface in three-dimensional space. It involves breaking down the surface into small elements and summing up the contributions of each element.

2. How is a surface integral different from a regular integral?

A regular integral calculates the total value of a function over a one-dimensional interval, while a surface integral calculates the total value of a function over a two-dimensional surface. It involves integrating in two directions (u and v) instead of just one (x, y, or z).

3. What is the significance of vectors in surface integrals?

Vectors play a crucial role in surface integrals as they represent the direction and magnitude of the quantity being integrated. In surface integrals, vectors are used to define the orientation and area of the surface elements, which are essential for accurately calculating the integral.

4. How do you set up a surface integral?

To set up a surface integral, you need to define the surface over which the integral will be calculated, determine the orientation of the surface, and choose a parametric representation of the surface in terms of u and v. Then, the surface integral can be expressed as a double integral over the parametric domain.

5. What are some real-world applications of surface integrals and vectors?

Surface integrals and vectors have various applications in physics, engineering, and computer graphics. They are used to calculate flux through a surface, determine the mass and center of mass of a three-dimensional object, and solve problems related to fluid dynamics. They are also used in computer graphics to create realistic 3D images and animations.

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