Gauss' Theorem/Stokes' theorem

In summary, Gauss' Theorem and Stokes' Theorem are two fundamental mathematical theorems that relate surface and volume integrals of vector fields. While Gauss' Theorem deals with three-dimensional volume integrals, Stokes' Theorem deals with two-dimensional surface integrals. Both theorems have various applications in physics, engineering, and other fields, and are closely related to each other, with Stokes' Theorem being a generalization of Gauss' Theorem.
  • #1
superpig10000
8
0
Hi guys,

I am having trouble with this "simple" problem involving these two theorems:

Find the value of the integral (A dot da) over the surface s, where A = xi - yj + zk and S is the closed surface defined by the cylinder c^2 = x^2 + y^2. The top and bottom of the cylinder are z= 0 and z=d.

From common sense, integrating circular layers from z=0 to z=d should give the volume of a cylinder. The book doesn't have any sample problem so I don't know which theorem to apply, and how.

Here's a more complicated question:

Find the value of the integral (curl A da) over the surface s, where A = yi + zj + xk and S is the closed surface defined by the paraboloid z=1-x^2-y^2 where z >=0

I appreciate any help.
 
Physics news on Phys.org
  • #2
"Find the value of the integral (A dot da) over the surface s, where A = xi - yj + zk and S is the closed surface defined by the cylinder c^2 = x^2 + y^2. The top and bottom of the cylinder are z= 0 and z=d."
Find div A. (It is a constant.) Then just multiply by the volume of a cylinder.
 
  • #3
"Find the value of the integral (curl A da) over the surface s, where A = yi + zj + xk and S is the closed surface defined by the paraboloid z=1-x^2-y^2 where z >=0"
By either the div theorem or Stokes' theorem, the integral of curl over a closed surface=0. Prove it.
 

1. What is Gauss' Theorem?

Gauss' Theorem, also known as the Divergence Theorem, is a mathematical theorem that relates the surface integral of a vector field to the volume integral of the divergence of that vector field. It states that the total outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field over the enclosed volume.

2. What is Stokes' Theorem?

Stokes' Theorem is a mathematical theorem that relates the line integral of a vector field around a closed curve to the surface integral of the curl of that vector field over any surface bounded by that curve. It states that the line integral of a vector field around a closed curve is equal to the surface integral of the curl of that vector field over any surface bounded by that curve.

3. What is the difference between Gauss' Theorem and Stokes' Theorem?

The main difference between Gauss' Theorem and Stokes' Theorem is that Gauss' Theorem relates the surface integral of a vector field to the volume integral of the divergence of that vector field, while Stokes' Theorem relates the line integral of a vector field around a closed curve to the surface integral of the curl of that vector field. In other words, Gauss' Theorem deals with three-dimensional volume integrals, while Stokes' Theorem deals with two-dimensional surface integrals.

4. What are the applications of Gauss' Theorem and Stokes' Theorem?

Gauss' Theorem and Stokes' Theorem have many applications in physics, engineering, and other fields. Some common applications include fluid mechanics, electromagnetism, and heat transfer. These theorems are also used in vector calculus to solve various problems involving vector fields.

5. Are Gauss' Theorem and Stokes' Theorem related to each other?

Yes, Gauss' Theorem and Stokes' Theorem are closely related to each other. In fact, Stokes' Theorem can be seen as a generalization of Gauss' Theorem. This is because Gauss' Theorem can be derived from Stokes' Theorem by considering a closed surface as a special case of a closed curve in three-dimensional space. Both these theorems are fundamental concepts in vector calculus and have many important applications in mathematics and other fields of science.

Similar threads

Replies
3
Views
1K
  • Advanced Physics Homework Help
Replies
19
Views
701
  • Calculus and Beyond Homework Help
Replies
2
Views
987
  • Advanced Physics Homework Help
Replies
4
Views
3K
Replies
30
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
561
  • Advanced Physics Homework Help
Replies
1
Views
206
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
826
  • Advanced Physics Homework Help
Replies
17
Views
2K
Back
Top