Surface Integrals: Understanding & Examples

In summary, the conversation discusses a general issue of understanding for evaluating surface integrals and the geometric significance of the dot product r.n. The conversation also includes attempts at finding the integrals for specific examples and clarifying the use of dS as a vector or scalar. The conversation ends with a clarification of the Cartesian form for the integral on the x=0 face and the question of whether order matters in the integral.
  • #1
alexvenk
11
0

Homework Statement


Its more of a general issue of understanding than a specific problem

I have to evaluate a few surface integrals and I am not sure about the geometric significance of what I am evaluating or even of what to evaluate. Examples.

If n is the unit normal to the surface S evaluate Integral of: r . n dS over a) a unit cube. b) a sphere

Not sure what I am evaluating or what the significance of it is. If someone could explain the general steps to take ill try and apply it to the specific problems

Homework Equations

The Attempt at a Solution

 
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  • #2
Let's consider r.n over the cube. In particular, at a face parallel to the YZ plane. Can you think of a geometric interpretation of the dot product r.n?
Regardless of any intepretation, are you able to do the integrals?
 
  • #3
haruspex said:
Let's consider r.n over the cube. In particular, at a face parallel to the YZ plane. Can you think of a geometric interpretation of the dot product r.n?
Regardless of any intepretation, are you able to do the integrals?
If i know what integral to do, i can do integration, its just a matter of knowing what I'm integrating. Is r a position vector? r.n gives us the projection of vector r onto the normal of the surface.
 
  • #4
alexvenk said:
If i know what integral to do, i can do integration, its just a matter of knowing what I'm integrating. Is r a position vector? r.n gives us the projection of vector r onto the normal of the surface.
Because you had written a dot between the r and n, and I know n is a vector even though you did not show it as one typographically, I assumed r was also a vector - presumably the position vector. And yes, their dot product will be the component of r in the normal direction. So for a face parallel to the YZ plane, the x component. Note that this is constant for that face.
dS is also a vector usually, though because of the n vector already there I'm not sure whether the question setter has already substituted n dS, dS being a scalar area element, for the vector dS. I'd need to see the typography of the original.
If dS is still a vector then clearly the integral produces a vector.
 
  • #5
yeah ok it makes sense that all of the sides of a cube will have a normal that is just in the directions of one of the axis

Here is the question:
http://imgur.com/jqJTF5h

im struggling with pretty much all the questions there, but we are specifically talking about 4 here.
 
  • #6
alexvenk said:
yeah ok it makes sense that all of the sides of a cube will have a normal that is just in the directions of one of the axis

Here is the question:
http://imgur.com/jqJTF5h

im struggling with pretty much all the questions there, but we are specifically talking about 4 here.
That helps. Seeing how questions 4, 5 and 6 are printed, I'm reasonably sure that the author is using ##\vec{dS}## and ##\vec n dS## interchangeably with the same meaning. Can you see how to write the integral in Cartesian form for the x=0 face?
 
  • #7
haruspex said:
That helps. Seeing how questions 4, 5 and 6 are printed, I'm reasonably sure that the author is using ##\vec{dS}## and ##\vec n dS## interchangeably with the same meaning. Can you see how to write the integral in Cartesian form for the x=0 face?

Im not entirely certain but ill give it a shot:

http://imgur.com/umVYFS3
 
  • #8
alexvenk said:
Im not entirely certain but ill give it a shot:

http://imgur.com/umVYFS3
Your Cartesian substitution for dS is wrong. This is not a volume integral.
 
  • #9
haruspex said:
Your Cartesian substitution for dS is wrong. This is not a volume integral.
ahh yeah i was wondering whether that would be right or not, itll be dydx then, which evaluates at 0 right? do i then sum all the sides?
 
  • #10
alexvenk said:
ahh yeah i was wondering whether that would be right or not, itll be dydx then, which evaluates at 0 right? do i then sum all the sides?
No, not dydx. This is orthogonal to the x axis.
 
  • #11
haruspex said:
No, not dydx. This is orthogonal to the x axis.
oh, i see, so dydz / dzdy then? does order matter?
Thanks a bunch for your help and patience!
edit:typo
 

1. What is a surface integral?

A surface integral is a mathematical tool used to calculate the total value of a function over a surface. It involves breaking the surface into small pieces and calculating the value of the function at each point on the surface, then adding up all these values to get the total value.

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

A regular integral calculates the area under a curve in two dimensions, while a surface integral calculates the value of a function over a surface in three dimensions. This means that instead of just considering the x and y coordinates, a surface integral takes into account the z coordinate as well.

3. What are some real-world applications of surface integrals?

Surface integrals have many applications in physics, engineering, and other sciences. For example, they can be used to calculate the flow of a fluid over a surface, the amount of heat transferred across a surface, or the mass of an object with a varying density over its surface.

4. How do you set up a surface integral?

To set up a surface integral, you first need to define the surface over which the integral will be evaluated. This can be done using a parametric equation or by defining the surface as a graph of a function. Then, you need to determine the bounds for the two variables (usually u and v) that define the surface. Finally, you need to determine the function that will be integrated over the surface.

5. What is the relationship between a surface integral and a line integral?

A line integral is similar to a surface integral, but it calculates the value of a function over a curve instead of a surface. In fact, surface integrals can be thought of as a generalization of line integrals to higher dimensions. Just as a regular integral is the limit of a Riemann sum, a surface integral is the limit of a double integral over smaller and smaller regions on the surface.

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