I am interested in the following: Let $f(x,y,z)$ be a given (known) function in Cartesian coordinates, and let $B_d(p_0) = \{ y: ||y-p_0|| < d \}$ (i.e., a sphere centered at p_0). I want to numerically compute in MATLAB
$$\int \int \int_{B_d(p_0)} f(x,y,z) dx dy dz$$
Now, I remember that if $p_0 = 0$, this isn't too bad: I just change to spherical coordinates and compute
$$\int_0^{2 \pi} \int_0^{\pi} \int_0^d f(r,\theta,\phi) r^2 \sin(\theta) dr d\theta d \phi$$
where $\theta$ is my polar angle (from the positive z-axis down) and $\phi$ is my azimuthal angle (from the positive x-axis in the x-y plane). In this form, this is just an integral over a cube , and I can just call integral3 on this, with my integrand being $f(r,\theta,\phi) r^2 \sin(\theta)$. But, if I have a circle centered elsewhere, it seems a bit trickier.
I thought I could shift variables as such: Let $p_0 = (x_0,y_0,z_0)$. Then, set $x' = x-x_0, y' = y-y_0, z' = z-z_0$. I believe that this linear shift won't affect the integration, so now I have
$$\int \int \int_{B_d(0)} f(x',y',z') dx' dy' dz'$$
This I can now switch to polar coordinates as
$$\int_0^{2\pi} \int_0^{\pi} \int_0^d f(r',\theta',\phi') r'^2 \sin(\theta') dr d\theta' d\phi'$$.
with $r' = x'^2 + y'^2 + z'^2$, $\theta' = arccos(\frac{z'}{r'})$, $\phi' = arctan\frac{y'}{x'})$.
I guess I'm worried about how to actually input this; there has to be some additional shifting or changing going on, because if I just tell MATLAB to integrate $f(r',\theta',\phi')$, over a sphere of radius 0, it looks to me like it would be no different than integrating over $(r,\theta,\phi)$ without the shift. Is there some change to the Jacobian or something I need to do to reflect this shift of variables to this shifted polar coordinate system? Would I first define a function $f(x,y,z)$, then define a new shifted function $g(x',y',z') = f(x,y,z)$, then spherical coordinate $g$?
For example, let's say we want to integrate $f(x,y,z) = \sin(x) \sin(y) \sin(z)$ over a ball of radius 1 centered at (2,3,4). First, I can define $x' = x-2, y' = y-3, z' = z-4$, and I have $f(x',y',z') = \sin(x'+2) \sin(y'+3) \sin(z'+4)$. Then, I can define my spherical coordinate system, and I replace $x' = r' \sin(\theta') \cos(\phi'), y' = r' \sin(\theta') \sin(\phi')$ and $z' = r' \cos(\theta')$. This leads to the final integrand being (including the Jacboian from the spherical coordinate change)
$$F(r',\theta',\phi') = \sin(r'\sin(\theta')\cos(\phi')+2) \sin(r' \sin(\theta') \sin(\phi')+3) \sin(r' \cos(\theta')+4) r'^2 \sin(\theta')$$
This function i can then apply a Matlab routine like integral3 over $0\leq r' \leq 1$, $0 \leq \theta' \leq \pi$ and $0 \leq \phi' \leq 2\pi$.
Does this sound correct? Thanks!
Best Answer
Remember that Jacobian only includes derivatives of coordinate change, so any translation will have no effect on it.
Another way of thinking of this is just making normal variable substitution in integral and finding new $dx^\prime, dy^\prime, dz^\prime$. As you do this, you'll see that integral didn't change except having new variables. This problem now becomes completely self-contained, and you can safely switch to spherical coordinates from your new integral.
So yes, your reasoning is correct.