Surface integral of scalar function

In summary, the task is to find the mass of a spherical surface S of radius R where the mass density at each point (x, y, z) is equal to the distance from that point to a fixed point (x_0, y_0, z_0) on S. This can be solved by using spherical coordinates and choosing (x_0, y_0, z_0) to be the origin, simplifying the integral to 8πR^3.
  • #1
malicx
52
0

Homework Statement


Find the mass of a spherical surface S of radius R such that at each point (x, y, z) in S the mass density is equal to the distance of (x, y, z) to some fixed point (x_0, y_0, z_0) in S.


Homework Equations


Integral of a scalar function over a surface.

The Attempt at a Solution


I was thinking about converting this into spherical coordinates, but I see no way of doing that nicely since the distance formula would get very messy. I am also assuming they are using the euclidean distance, since this is an intro multivariable course.

I don't need help evaluating, just with getting it set up.

This is from Vector Calculus, 5e. by Marsden and Tromba, 7.5 #9.
 
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  • #2
It doesn't matter what the fixed point [tex](x_0, y_0, z_0)[/tex] is, so you can make a choice that makes spherical coordinates less horrible.
 
  • #3
ystael said:
It doesn't matter what the fixed point [tex](x_0, y_0, z_0)[/tex] is, so you can make a choice that makes spherical coordinates less horrible.

So basically, choose (x_0, y_0, z_0) to be 0, and using spherical coordinates, the distance is
[tex]\sqrt{(2Rsin(\phi)cos(\theta))^2 + (2Rsin(\phi)sin(\theta))^2} + (2Rcos(\phi))^2}[/tex]
= 2R.

So, [tex]\int_0^{2\pi} \int _0^\pi 2R*R^2sin(\phi) \, d\phi d\theta ?[/tex]

[tex] = 8\pi*R^3 [/tex]

Edit: The answer is supposed to be (16/3)Pi*R^3, so I lost a factor of 2/3 somewhere...
 
Last edited:
  • #4
Well, you probably shouldn't choose [tex](x_0, y_0, z_0)[/tex] to be the origin if your sphere [tex]S[/tex] is centered at the origin. Read the question again; [tex](x_0, y_0, z_0)[/tex] is supposed to lie on [tex]S[/tex].
 
  • #5
Make life easy for yourself and take [itex]\theta_0=\phi_0=0[/itex].
 
  • #6
Cyosis said:
Make life easy for yourself and take [itex]\theta_0=\phi_0=0[/itex].

Wow, I made that way harder than it had to be. Thank you both, that was driving me nuts!
 

1. What is a surface integral of a scalar function?

A surface integral of a scalar function is a mathematical concept used in vector calculus to measure the flux or flow of a scalar field over a two-dimensional surface. It involves calculating the integral of a scalar function over a given surface, taking into account both the magnitude and direction of the function at each point on the surface.

2. How is a surface integral of a scalar function different from a regular integral?

A surface integral of a scalar function is different from a regular integral in that it involves integrating over a two-dimensional surface rather than a one-dimensional interval. This means that the limits of integration are defined by the boundaries of the surface rather than fixed values.

3. What is the significance of the surface normal in a surface integral of a scalar function?

The surface normal is an important factor in calculating a surface integral of a scalar function. It represents the direction perpendicular to the surface at each point and is used to determine the orientation of the surface and the direction of the flux or flow of the scalar field.

4. Can a surface integral of a scalar function have a negative value?

Yes, a surface integral of a scalar function can have a negative value. This can occur when the scalar function has different values on different parts of the surface, resulting in opposing directions of flux or flow. The overall value of the integral will depend on the magnitude and direction of the function at each point on the surface.

5. What are some real-world applications of surface integrals of scalar functions?

Surface integrals of scalar functions have many practical applications in fields such as physics, engineering, and fluid dynamics. They are used to calculate the amount of fluid passing through a given surface, the flux of electric fields, and the work done by a force applied over a surface, among other things.

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