Question

Change of variables: Triple integration problem...

Answer 1

Perform the integration by transforming the ellipsoidal region of integration into a spherical region of integration and then evaluating the transformed integral in spherical coordinates.

\[\int\limits_{}^{}\int\limits_{}^{}\int\limits_{}^{}(y ^{2}+z ^{2}) dV\]

Where G is the region enclosed by the ellipsoid:\[\frac{x ^{2} }{ a ^{2} }+\frac{ y ^{2} }{ b ^{2} }+\frac{ z ^{2} }{ c ^{2} } = 1\]

Firstly I started by going off an example from the class which was that:
\[x = au\]\[y=bv\]\[z=cw\]
and also:
\[u^{2}+v^{2}+w^{2} = 1\]

The question I have is about the Jacobian how would I go about finding it; the example I was given of a problem similar to this without just 1dV rather than: \[(y ^{2}+z ^{2})\]
Can anyone help me to understand how to work out the jacobian for a problem like this, the example i'm given just has 1 dV so the change of variables just puts an abc outside.

Answer 2

@oldrin.bataku @dumbcow

Answer 3

shoot i dont have a lot of practice with spherical coordinates or the Jacobian Matrix

Answer 4

@UnkleRhaukus , @.Sam.

Answer 5

@genius12

Answer 6

Neither, I made sense of the last problem because u and v was given it was just a region

Answer 7

so far so good! those substitutions to indeed give you a 'spherical' region in \(uvw\)

Answer 8

now for the Jacobian, we're interested in a *locally* linear map between the \(uvw\) space to \(xyz\)... and what you gave is exactly that:$$x=au\\y=bv\\z=cw$$writing in vector form we get:$$\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}\begin{bmatrix}u\\v\\w\end{bmatrix}$$since this map is linear we're good. now compute its determinant to figure out how it *scales* volumes from \(uvw\) to \(xyz\); matching up with intuition, we get \(abc\). intuitively, the unit cube \([0,0,1]\times[0,1,0]\times[1,0,0]\) in \(uvw\) maps to the prism \([0,0,c]\times[0,b,0]\times[a,0,0]\) yielding a volume of \(L\times W\times H=abc\)

Answer 9

so our Jacobian was just:$$\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}$$ and its determinant is merely \(abc\)

Answer 10

That's exactly like the example, so I would have that outside the integral like this:

\[abcint_{}^{}\int\limits_{}^{}\int\limits_{}^{}\int\limits_{}^{}(y^{2}+z ^{2}) dV\]

Answer 11

bah not int of course just abc

Answer 12

well the change of variables formula is:$$\iiint f(\mathbf{x})\,dV=\iiint f(\phi(\mathbf{x}))|\det \phi^{-1}|\,dV$$where \(\phi\) maps to new coordinates... in this case \(|\det\phi^{-1}|=abc\) and so it's a constant meaning you can just pull it out:$$\iiint f(\mathbf{x})\,abc\,dV=abc\iiint f(\mathbf{x})\,dV$$

Answer 13

so f(x) wouldn't be changed then in this case \[(y ^{2}+z ^{2})\] or would it be?

Answer 14

sure but you need to replace \(y=bv,z=cw\)

Answer 15

yea of course so instead it would be like this:
\[((bv)^{2}+(cw)^{2})\]

Answer 16

@oldrin.bataku Now im stuck at what to do next, I thought the only problem i'd have is this, but now the converting part of this problem is throwing me off so we have:

\[abc \int\limits_{}^{}\int\limits_{}^{}\int\limits_{}^{}((bv)^{2}+(cw)^{2}) dV\]

How can I tackle that in spherical coordinates now?

Answer 17

well we're integrating a spherical region in \(uvw\) so use spherical coordinates:$$x=\rho\sin\phi\cos\theta\\y=\rho\sin\phi\sin\theta\\z=\rho\cos\phi$$and then use the Jacobian \(\rho^2\sin\phi\)

Answer 18

err:$$u=\rho\sin\phi\cos\theta\\v=\rho\sin\phi\sin\theta\\w=\rho\cos\phi$$

Answer 19

$$abc\iiint\limits_R (b^2v^2+c^2w^2)\,dV\\\ \ \ =abc\int_0^{2\pi}\int_0^{\pi}\int_0^1 (b^2\rho^2\sin^2\phi\sin^2\theta+c^2\rho^2\cos^2\phi)\rho^2\sin\phi\,\,d\rho\,d\phi\,d\theta\\\ \ \ =abc\int_0^{2\pi}\sin\theta\int_0^{\pi}(b^2\sin^2\phi\sin^2\theta+c^2\cos^2\phi)\int_0^1\rho^4\ d\rho\ d\phi\,d\theta$$

Answer 20

ahh ok!

Answer 21

thank you very much!

Answer 22

So, the Jacobian from u,v,w, coordinates to rho, phi, theta coordinates is identical to the jacobian from x,y,z, coordinates to rho, phi, theta coordinates? Is this always true?

E.g. if I have a starting coordinate system A and make a transformation to a second system, C, have another starting system B and make a transformation to system B, both will be identical?