Surface Integrals: Why Dot ds with Normal?

Click For Summary
SUMMARY

The discussion clarifies the necessity of using the dot product in surface integrals, specifically in the context of calculating the flux of a vector field A through a surface. The integral expression \oint A \cdot n \, ds is highlighted, where ds represents the area element and n is the normal vector to the surface. The dot product is essential as it captures only the component of the vector field A that is perpendicular to the surface, which is critical for accurately determining flux. This understanding resolves the initial confusion regarding the use of the dot product in surface integrals.

PREREQUISITES
  • Understanding of vector fields and their properties
  • Familiarity with surface integrals in multivariable calculus
  • Knowledge of the concept of flux in physics
  • Basic proficiency in vector calculus operations, including dot products
NEXT STEPS
  • Study the concept of flux in vector fields and its applications
  • Learn about different types of surface integrals beyond flux calculations
  • Explore the mathematical derivation of surface integrals in multivariable calculus
  • Investigate the role of normal vectors in various integral theorems, such as Gauss's theorem
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are working with surface integrals and vector fields, particularly those seeking to deepen their understanding of flux calculations.

likephysics
Messages
638
Reaction score
4

Homework Statement


This is not a HW prob. Just a question.
When doing surface integrals, why should the area element ds be dotted with the normal. I don't get it .
[tex]\oint[/tex]A.n ds

Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
likephysics said:

Homework Statement


This is not a HW prob. Just a question.
When doing surface integrals, why should the area element ds be dotted with the normal. I don't get it .
[tex]\oint[/tex]A.n ds

It doesn't have to be, in general. There are other types of surface integrals as well. The type you've listed above is designed to determine the flux of the vector field A through the surface you are integrating over. The dot product is there because, by definition of flux, you are only concerned with the component of A that is perpendicular to the surface at each point on the surface.
 
Ok. Thanks. I can live with it now.
 

Similar threads

Calculate surface integral on sphere
  • · Replies 7 ·
Replies
7
Views
3K
Unable to simplify dS (Stokes' theorem)
  • · Replies 1 ·
Replies
1
Views
1K
Divergence Theorem Verification: Surface Integral
  • · Replies 3 ·
Replies
3
Views
2K
Flux/Surface Integral across a Plane
  • · Replies 5 ·
Replies
5
Views
2K
Stokes' Theorem for a Circle in a Plane
  • · Replies 7 ·
Replies
7
Views
3K
Surface Integrals: Evaluate A.n dS on 2x+y=6 Plane in 1st Octant
  • · Replies 10 ·
Replies
10
Views
3K
If you do not answer the above questions, you will not have a correct answer.
  • · Replies 6 ·
Replies
6
Views
2K
Find all points where surface normal is perpendicular to plane
  • · Replies 4 ·
Replies
4
Views
2K
Rates of change: surface area and volume of a sphere
  • · Replies 3 ·
Replies
3
Views
5K
Stokes' Theorem 'corollary' integral in cylindrical polar coordinates
  • · Replies 6 ·
Replies
6
Views
2K