How to find the limits of a volume integral?

  • #1
Hawkingo
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Homework Statement


If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##.

The Attempt at a Solution


From gauss divergence theorem we know
##\int \int _ { S } \vec { F } \cdot \hat { n } d s = \int \int \int _ { V } \vec { \nabla } \cdot \vec { F } d v##
So ##\int \int _ { S } \vec { F } \cdot \hat { n } d s = \int \int \int _ { V } [ \frac { \partial } { \partial x } ( x ) + \frac { \partial } { \partial y } ( y ) + \frac { \partial } { \partial z } ( z ) ] d x d y d z##
=##3 \int \int \int _ { V } d x d y d z##

Here if I'll calculate the answer using gauss divergence theorem, then I have to find the volume integral of the function, but how will I take the limits of the integration. Can someone help please?
Thank you.
 
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Answers and Replies

  • #2
Orodruin
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The volume integral you have left is just the integral of the volume element. This gives the volume enclosed by the surface, ie, the volume of a sphere in this case.
 
  • #3
Hawkingo
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The volume integral you have left is just the integral of the volume element. This gives the volume enclosed by the surface, ie, the volume of a sphere in this case.
Thanks, I was confused because the book solved in a very long and difficult manner and the limits used were seemed to be inappropriate, but by your way I arrived at the answer in just one step. Thanks a lot
 
  • #4
Hawkingo
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The volume integral you have left is just the integral of the volume element. This gives the volume enclosed by the surface, ie, the volume of a sphere in this case.
It was solved because the equation was of a sphere, but if it was something unknown can you suggest me a way how to derive the limits from the equation without knowing the figure?
 
  • #5
LCKurtz
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It was solved because the equation was of a sphere, but if it was something unknown can you suggest me a way how to derive the limits from the equation without knowing the figure?
Using the figure is the way to determine the limits because the coordinates are intimately related to the geometry.
 
  • #6
vela
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It was solved because the equation was of a sphere, but if it was something unknown can you suggest me a way how to derive the limits from the equation without knowing the figure?
Presumably you've already covered this in your class. There is no cookbook way of coming up with the limits. Each problem is different.

You should be able to do this problem by evaluating the three-dimensional integral. Maybe you should start there, since you know what the answer is supposed to be.
 
  • #7
Ray Vickson
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Homework Statement


If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##.

The Attempt at a Solution


From gauss divergence theorem we know
##\int \int _ { S } \vec { F } \cdot \hat { n } d s = \int \int \int _ { V } \vec { \nabla } \cdot \vec { F } d v##
So ##\int \int _ { S } \vec { F } \cdot \hat { n } d s = \int \int \int _ { V } [ \frac { \partial } { \partial x } ( x ) + \frac { \partial } { \partial y } ( y ) + \frac { \partial } { \partial z } ( z ) ] d x d y d z##
=##3 \int \int \int _ { V } d x d y d z##

Here if I'll calculate the answer using gauss divergence theorem, then I have to find the volume integral of the function, but how will I take the limits of the integration. Can someone help please?
Thank you.

Another way is to note that ##\vec{F} \cdot \vec{n} = r## = radius of the sphere, so your integral is ##\int \!\! \int_S r \, dS = r \int \! \! \int_S dS.## Do you recognize that last integral?
 
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