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Greetings, and great to be back after all this time. I am having trouble with a mathematics question where we are asked to use a triple integral to find the volume of the ellipsoid, where x²/a²+ y²/b²+ z²/c² = 1 .
I have looked at all the ways to do it, and am still confused, having been working on this for days. If I integrate, where does the pi term come from to form the volume which I do know to be pi times abc ? Chris the Russian Christopher Lilly 07:45, 26 July 2011 (UTC)
Use the change of variables u=x/a, v=y/b, w=z/c. The Jacobian is abc, so the volume is abc times the volume of the unit sphere: . Sławomir Biały ( talk) 21:41, 26 July 2011 (UTC)
Thanks all so much, and he, that is I, am very impressed by all of this, which should be of much use indeed. Chris the Russian Christopher Lilly 01:20, 27 July 2011 (UTC)
( edit conflict) My solution would be to make a volume presrving, linear change or coordinates. The equation (x/a)2 + (y/b)2 + (z/c)2 = 1 gives an ellipsoid that cuts the x–axis at x = ±a, it cuts the y–axis at y = ±b and it cuts the z–axis at z = ±c. We can make a linear transformation of xyz–space that preserves the x–, y–, and z–axes; which also preserves volume. Such a transformation is given by (x,y,z) → (αx,βy,γz) where α, β and γ are real numbers such that the product αβγ = 1. We want the linear transforation to carry the ellipsoid to a sphere. A sphere has an equation of the form x2 + y2 + z2 = r2 for some real number r > 0. We must choose α, β and γ so that the equation (x/a)2 + (y/b)2 + (z/c) = 1 is transformed into the equation x2 + y2 + z2 = r2. If we make the linear transformation (x,y,z) → (rx/a,ry/b,rz/c) then the image of the set of points given by the equation (x/a)2 + (y/b)2 + (z/c) = 1 will be transformed into the set of points given by the equation x2 + y2 + z2 = r2 or, if you prefer (x/r)2 + (y/r)2 + (z/r)2 = 1. We still don't know what r is. We can determine r from the fact that we want (x,y,z) → (rx/a,ry/b,rz/c) to be volume preserving. The determinant of (x,y,z) → (rx/a,ry/b,rz/c) is given by r3/abc and r3/abc = 1 if and only if r = (abc)1/3. Thus, your problem reduces to evaluating the volume of the sphere x2 + y2 + z2 = (abc)2/3. To solve this you can use the formula for the volume of a solid of revolution. The formula (which can easily be proven if necessary) is as follows:
gives the volume of the solid given by rotating the graph y = ƒ(x), for , given in the xy–plane by a full 2π radians about out the x–axis in xyz–space. As a result, the volume of the sphere (and so the volume of the ellipsoid) is given by calculating
where k = (abc)1/3. It follows that V = 4⁄3πabc. I know that it might seem long winded; but I wanted to explain everything properly. The main details are regarding the linear transformation, which is the easiest bit once you understand it. But it's not always so clear for the the first time. Anyway, I hope this helps. — Fly by Night ( talk) 01:46, 27 July 2011 (UTC)
Yes, thank You. I see the key is in trying to simplify the ellipsoid equation into that of a sphere, but being in mind of the differences between the two. To be sure, as seems obvious, an ellipsoid is just a skewed kind of sphere, where in the end it makes sense that their volumes should be of a similar nature. I have spent most of this week on this problem, which it appears is one given to unsuspecting maths students all over the world, designed to get us to think. From this I can begin to understand how the triple integral works, and this shall assist me a lot in giving a proper answer, thank You. Chris the Russian Christopher Lilly 23:36, 27 July 2011 (UTC)
All eigenvalues of real skew-symmetric matrices are imaginary or zero. Does this work in the other direction, i.e. are all real matrices whose eigenvalues are imaginary or 0 similar to a real skew-symmetric matrix? I.e. if all eigenvalues of some real matrix A are imaginary, does this imply the existence of a transformation P such that P^(-1)AP is skew-symmetric? 83.134.166.168 ( talk) 16:56, 26 July 2011 (UTC)
Mathematics desk | ||
---|---|---|
< July 25 | << Jun | July | Aug >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Greetings, and great to be back after all this time. I am having trouble with a mathematics question where we are asked to use a triple integral to find the volume of the ellipsoid, where x²/a²+ y²/b²+ z²/c² = 1 .
I have looked at all the ways to do it, and am still confused, having been working on this for days. If I integrate, where does the pi term come from to form the volume which I do know to be pi times abc ? Chris the Russian Christopher Lilly 07:45, 26 July 2011 (UTC)
Use the change of variables u=x/a, v=y/b, w=z/c. The Jacobian is abc, so the volume is abc times the volume of the unit sphere: . Sławomir Biały ( talk) 21:41, 26 July 2011 (UTC)
Thanks all so much, and he, that is I, am very impressed by all of this, which should be of much use indeed. Chris the Russian Christopher Lilly 01:20, 27 July 2011 (UTC)
( edit conflict) My solution would be to make a volume presrving, linear change or coordinates. The equation (x/a)2 + (y/b)2 + (z/c)2 = 1 gives an ellipsoid that cuts the x–axis at x = ±a, it cuts the y–axis at y = ±b and it cuts the z–axis at z = ±c. We can make a linear transformation of xyz–space that preserves the x–, y–, and z–axes; which also preserves volume. Such a transformation is given by (x,y,z) → (αx,βy,γz) where α, β and γ are real numbers such that the product αβγ = 1. We want the linear transforation to carry the ellipsoid to a sphere. A sphere has an equation of the form x2 + y2 + z2 = r2 for some real number r > 0. We must choose α, β and γ so that the equation (x/a)2 + (y/b)2 + (z/c) = 1 is transformed into the equation x2 + y2 + z2 = r2. If we make the linear transformation (x,y,z) → (rx/a,ry/b,rz/c) then the image of the set of points given by the equation (x/a)2 + (y/b)2 + (z/c) = 1 will be transformed into the set of points given by the equation x2 + y2 + z2 = r2 or, if you prefer (x/r)2 + (y/r)2 + (z/r)2 = 1. We still don't know what r is. We can determine r from the fact that we want (x,y,z) → (rx/a,ry/b,rz/c) to be volume preserving. The determinant of (x,y,z) → (rx/a,ry/b,rz/c) is given by r3/abc and r3/abc = 1 if and only if r = (abc)1/3. Thus, your problem reduces to evaluating the volume of the sphere x2 + y2 + z2 = (abc)2/3. To solve this you can use the formula for the volume of a solid of revolution. The formula (which can easily be proven if necessary) is as follows:
gives the volume of the solid given by rotating the graph y = ƒ(x), for , given in the xy–plane by a full 2π radians about out the x–axis in xyz–space. As a result, the volume of the sphere (and so the volume of the ellipsoid) is given by calculating
where k = (abc)1/3. It follows that V = 4⁄3πabc. I know that it might seem long winded; but I wanted to explain everything properly. The main details are regarding the linear transformation, which is the easiest bit once you understand it. But it's not always so clear for the the first time. Anyway, I hope this helps. — Fly by Night ( talk) 01:46, 27 July 2011 (UTC)
Yes, thank You. I see the key is in trying to simplify the ellipsoid equation into that of a sphere, but being in mind of the differences between the two. To be sure, as seems obvious, an ellipsoid is just a skewed kind of sphere, where in the end it makes sense that their volumes should be of a similar nature. I have spent most of this week on this problem, which it appears is one given to unsuspecting maths students all over the world, designed to get us to think. From this I can begin to understand how the triple integral works, and this shall assist me a lot in giving a proper answer, thank You. Chris the Russian Christopher Lilly 23:36, 27 July 2011 (UTC)
All eigenvalues of real skew-symmetric matrices are imaginary or zero. Does this work in the other direction, i.e. are all real matrices whose eigenvalues are imaginary or 0 similar to a real skew-symmetric matrix? I.e. if all eigenvalues of some real matrix A are imaginary, does this imply the existence of a transformation P such that P^(-1)AP is skew-symmetric? 83.134.166.168 ( talk) 16:56, 26 July 2011 (UTC)