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Surface integral of sphere

  • Thread starter pizza_dude
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  • #1

Homework Statement


find the values of the integral

[itex] \int_{S} \vec A\cdot\ d\vec a [/itex]

where,

[itex] \vec A\ = (x^2+y^2+z^2)(x\hat e_{1}+y\hat e_{2}+z\hat e_{3}) [/itex]

and the surface S is defined by the sphere [itex] R^2=x^2+y^2+z^2 [/itex]

Homework Equations


first i must evaluate the integral directly, so i don't think there are any specific formulas other than ones you must derive from the geometry specific to the problem. i also have to calculate using gauss' theorem but for that there's a simple equation.

The Attempt at a Solution


really looking for an explanation on surface integrals. i know that [itex] d\vec a [/itex] is a small area on the surface of the sphere and equations must be derived from the geometry. im having a hard time visualizing this and how it's suppose to work. for now, i would appreciate a good explanation of surface integrals to help me visualize the problem.

thanks in advance.
 

Answers and Replies

  • #2
Orodruin
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##d\vec a## is the area element, which is an infinitesimal area multiplied by the unit normal of the surface. Try to answer the following: What is the unit normal of the sphere? What is the product of ##\vec A## with ##d\vec a##? How can you simplify the integrand using this?
 
  • #3
BvU
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Hello Pizza, welcome to PF :)

You familiar with spherical coordinates ? Reason I ask is because after all this is all happening on a sphere. Can you transform ##\vec A(x, y, z)## to ##\vec A(r,\theta, \phi)## ?

[edit] Well, I see you got some good help already. Bedtime for me!
 
  • #4
[itex] d\vec a [/itex] is perpendicular to the surface of the area (2 possible directions).
 
  • #5
Orodruin
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[itex] d\vec a [/itex] is perpendicular to the surface of the area (2 possible directions).
Yes, this is what I said, but which vector is perpendicular to the sphere surface at a given point on the sphere?
 
  • #6
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I think Orodruin is asking "which spherical coordinate system unit vector is perpendicular to the sphere surface at a given point on the sphere?"

What is your physical interpretation of the vector A at the surface of the sphere? In what direction is it pointing at each point over the surface?

Chet
 
  • #7
I think Orodruin is asking "which spherical coordinate system unit vector is perpendicular to the sphere surface at a given point on the sphere?"

What is your physical interpretation of the vector A at the surface of the sphere? In what direction is it pointing at each point over the surface?

Chet
[itex] \hat r [/itex] is the answer i believe you may be looking for. im not quite sure how to physically interpret [itex] \vec A [/itex]. i just know that it's some vector along the surface of the sphere. sorry if im making this difficult.
 
  • #8
Orodruin
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[itex] \hat r [/itex] is the answer i believe you may be looking for. im not quite sure how to physically interpret [itex] \vec A [/itex]. i just know that it's some vector along the surface of the sphere. sorry if im making this difficult.
Correct, so ##d\vec a = \hat r\, da## where da is the infinitesimal area. Now, can you write ##\vec A## in terms of the position vector ##\vec r##?
 
  • #9
[itex] \vec A = (x, y, z) \vec r [/itex] which i believe is equal to[itex] (x \hat e_{1} + y\hat e_{2} + z\hat e_{3}) [/itex] which is the second half of [itex] \vec A [/itex] mentioned in the problem
 
  • #10
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What is your understanding of what (x2+y2+z2) represents at the surface of a sphere of radius r?

Chet
 
  • #11
BvU
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[itex] \vec A = (x, y, z) \vec r [/itex] which i believe is equal to[itex] (x \hat e_{1} + y\hat e_{2} + z\hat e_{3}) [/itex] which is the second half of [itex] \vec A [/itex] mentioned in the problem
This [itex] \vec A = (x, y, z) \vec r [/itex] can't be right: what can possibly its components (if existent) ?

Dear dude, I am starting to worry if you understand how ## (x \hat e_{1} + y\hat e_{2} + z\hat e_{3} )## is to be interpreted. Can you express it in Cartesian coordinates ? And in Spherical coordinates ?
 

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