# How to change coordinates of integration?

Is there some built-in routine, some easier method to change variables of integration, without/before solving the integral? Say I have

Integrate[expression1[x, y, z], x, y, z]


how to change it to

Integrate[expression2[r, theta, phi], r, theta, phi]

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For the specific example of Cartesian -> Spherical transformation, you could use

TransformedField["Cartesian" -> "Spherical", expression1[x, y, z], {x,y,z} -> {r, theta, phi}] *
CoordinateTransformData["Spherical" -> "Cartesian", "MappingJacobianDeterminant"][{r, theta, phi}]


to get expression2[r, theta, phi]

Alternatively, you could use CoordinateChartData["Spherical", "VolumeFactor"][{r, theta, phi}] in the integral.

This functionality is new in version 9. In version 8, you'd have to do the calculations by hand:

mapping = {x, y, z} -> {r Cos[phi] Sin[theta], r Sin[phi] Sin[theta], r Cos[theta]}
jd = Simplify@Det@D[mapping[[2]], {{r, theta, phi}}]

expression1[x, y, z] jd /. Thread[mapping]

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This looks nice, but anything for ver. 8? – Vladimir Mar 12 '13 at 17:48
Ok, I can get around with JacobianDeterminant for now. I was just hoping there are some bulit-in routines, and it's true for ver. 9. Thanks – Vladimir Mar 12 '13 at 17:52
@Vladimir I don't think there are builtins for v8, see my update – Szabolcs Mar 12 '13 at 17:55
@Szabolcs I see you were already talking about JacobianDeterminant, but I don't see it spelled out anywhere. So I added an answer containing that approach and a variation thereof. – Jens Mar 12 '13 at 18:06

For completeness, in version 8 (and version 9 too) you can load the VectorAnalysis package and do one of the following:

Needs["VectorAnalysis"]

JacobianDeterminant[Spherical[r, θ, ϕ]]
(* ==> r^2 Sin[θ] *)

Times @@ ScaleFactors[Spherical[r, θ, ϕ]]
(* ==> r^2 Sin[θ] *)


So instead of invoking JacobianDeterminant, you can get the scale factors individually. The coordinates ($r$, $\theta$, $\phi$) in each case can be named as I did above - if you leave that out, it defaults to built-in capitalized standard names.

Putting this into an integral

To show that this leads to a syntax which is significantly more concise than the new version 9 syntax, see this specific integral. I define a function of Cartesian variables, then do the variable transformation in the integral, as was asked for in the question:

f[x_, y_, z_] := x + 2 y + z

Integrate[
f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[
Spherical @@ #] &@{r, θ, ϕ},
r, θ, ϕ]

(*
==>
-(1/8) r^4 (ϕ Cos[θ]^2 + θ (2 Cos[ϕ] -
Sin[ϕ]) +
Cos[θ] Sin[θ] (-2 Cos[ϕ] + Sin[ϕ]))
*)
`

I honestly think this is much prettier than what you now have to write in version 9. But I also like the completely manual approach in Szabolcs' answer.

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However, methods from "VectorAnalysis" are limited to 3D. This is why for ver. 8. I would even do explicit calculation, like Szabolcs. – Vladimir Mar 12 '13 at 18:24
Sure, but your example is also 3D. And the price you pay in version 9 is a much more lengthy syntax. I'll show the much more concise integration syntax for my approach in an edited answer. – Jens Mar 12 '13 at 18:31
I haven't used this package before. You're right that it is simpler than what v9 offers. Maybe new Mma functions are getting a bit more wordy than they should be ... – Szabolcs Mar 12 '13 at 18:48
@Jens Wow, these pure functions are powerful :) Yes, I see what you mean! – Vladimir Mar 12 '13 at 18:55