Surface integral without using Gauss' theorem

In summary, the problem is to calculate the surface integral of A.n over the region in the first octant bounded by (y^2)+(z^2) = 9 and x = 2. The solution involves finding the normal vector n and then projecting the surface onto the xy-plane. The limits of integration are then set up and the integral is evaluated over all three faces of the closed surface. The final result should be 180, but the original attempt only yielded 108. After correcting for the missing faces, the correct result was obtained.
  • #1
maupassant
10
0

Homework Statement



Calculate §§ A.n dS if
A= 2y(x^2)i-(y^2)j + 4xzk
over the region in the first octant bounded by (y^2)+(z^2) = 9 and x = 2

Homework Equations





The Attempt at a Solution



Let n = (yj + zk) / 3

then A.n = [-(y^3) +4xz^3] / 3

Since we 'll project the surface onto the xy-plane:
|n.k| = z/3 and z = SQRT(9-y^2)

Putting all together I obtain
= §§R (4xz^3 - (y^3))/z dx dy



Now making the appropriate changes and setting up the limits of integration:


§y=30 §x=20 4x(9-y^2) - (y^3)/sqrt(9-y^2) dx dy



However I always obtain 108 as a result and not 180 as my book suggested me (and after verification by Gauss' divergence theorem.

Is there a problem with the limits of integration? Wrong projection? I really have no clue ...
Thanks for the help!
 
Physics news on Phys.org
  • #2
You've only calculated the integral over one side/face of the surface...there are three more faces that make up the closed surface bounding the given region...you need to calculate the surface integral over all 3 of those as well.
 
  • #3
Thanks a lot!
I finally got it (at least I hope so ;-) !
 

1. What is a surface integral?

A surface integral is a mathematical concept used to calculate the flux or flow of a vector field across a surface. It involves integrating a vector field over a given surface to determine the total amount of flow passing through the surface.

2. How is a surface integral without using Gauss' theorem different from a regular surface integral?

A surface integral without using Gauss' theorem is calculated by directly integrating the vector field over the given surface, without using the divergence theorem to convert it to a volume integral. This method is often used when the surface is not a closed surface or when the vector field is not defined over a region in space.

3. What are some applications of surface integrals without using Gauss' theorem?

Surface integrals without using Gauss' theorem are commonly used in physics and engineering to calculate the flux of electric and magnetic fields, the flow of fluids, and the surface area of objects. They are also used in computer graphics to render 3D objects and in computer vision to analyze 3D images.

4. What are the limitations of using surface integrals without using Gauss' theorem?

One limitation of surface integrals without using Gauss' theorem is that they can only be used for surfaces that can be parameterized by a single variable. Additionally, they only apply to vector fields that are continuous and differentiable on the surface being integrated over.

5. How do you calculate a surface integral without using Gauss' theorem?

To calculate a surface integral without using Gauss' theorem, you first need to parameterize the given surface and express the vector field in terms of the parameters. Then, use the appropriate integration method, such as double or triple integration, to integrate the components of the vector field over the surface. Finally, evaluate the resulting expression to find the total flux or flow across the surface.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
282
  • Calculus and Beyond Homework Help
Replies
8
Views
826
  • Calculus and Beyond Homework Help
Replies
12
Views
934
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
982
  • Calculus and Beyond Homework Help
Replies
2
Views
987
  • Calculus and Beyond Homework Help
Replies
2
Views
261
  • Calculus and Beyond Homework Help
Replies
4
Views
954
  • Calculus and Beyond Homework Help
Replies
3
Views
487
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Back
Top