# Triple product in spherical coordinates

Given six real numbers $$a,b,c,d,e,f$$ (say between $$0$$ and $$\pi$$) I would like to express the following determinant in a compact and "reasonable" way: $$\det \begin{bmatrix} \sin a \cos b & \sin c \cos d & \sin e \cos f\\ \sin a \sin b & \sin c \sin d & \sin e \sin f\\ \cos a & \cos c & \cos e \end{bmatrix}.$$

Can someone help me? I do not have at my disposal any version of Mathematica and Wolfram|Alpha does not help.

• Sin[e] (Cos[c] Sin[a] Sin[b - f] - Cos[a] Sin[c] Sin[d - f]) - Cos[e] Sin[a] Sin[c] Sin[b - d] works for all complex values of the variables – Coolwater Feb 13 '19 at 16:07
• @Coolwater Thanks a lot! Anything better than that :-)? I was hoping for some cancellation or factorization... Indeed I need to take the Fourier series of that expression (Fourier series in a,b,c,d,e,f). – Romeo Feb 13 '19 at 16:10
• You can use a free, somewhat slower version of Mathematica via the Wolfram Open Cloud – b3m2a1 Feb 13 '19 at 16:52
• This is an option for you to do this yourself: sandbox.open.wolframcloud.com/app/view/… – b3m2a1 Feb 13 '19 at 21:30

{{Sin[a] Cos[b], Sin[c] Cos[d], Sin[e] Cos[f]},
{Sin[a] Sin[b], Sin[c] Sin[d], Sin[e] Sin[f]},
{Cos[a], Cos[c], Cos[e]}} // Det // FullSimplify


Cos[f] (Cos[c] Sin[a] Sin[b] - Cos[a] Sin[c] Sin[d]) Sin[e] + Cos[d] Sin[c] (-Cos[e] Sin[a] Sin[b] + Cos[a] Sin[e] Sin[f]) + Cos[b] Sin[a] (Cos[e] Sin[c] Sin[d] - Cos[c] Sin[e] Sin[f])

As @coolwater says, this is equivalent to Sin[e] (Cos[c] Sin[a] Sin[b - f] - Cos[a] Sin[c] Sin[d - f]) - Cos[e] Sin[a] Sin[c] Sin[b - d].

You get the Fourier expansion directly with

{{Sin[a] Cos[b], Sin[c] Cos[d], Sin[e] Cos[f]},
{Sin[a] Sin[b], Sin[c] Sin[d], Sin[e] Sin[f]},
{Cos[a], Cos[c], Cos[e]}} // Det // TrigToExp // Expand


-(1/16) I E^(-I a + I b - I c - I d - I e) + 1/16 I E^(I a + I b - I c - I d - I e) + 1/16 I E^(-I a + I b + I c - I d - I e) - 1/16 I E^(I a + I b + I c - I d - I e) + 1/16 I E^(-I a - I b - I c + I d - I e) - 1/16 I E^(I a - I b - I c + I d - I e) - 1/16 I E^(-I a - I b + I c + I d - I e) + 1/16 I E^(I a - I b + I c + I d - I e) - 1/16 I E^(-I a + I b - I c - I d + I e) + 1/16 I E^(I a + I b - I c - I d + I e) + 1/16 I E^(-I a + I b + I c - I d + I e) - 1/16 I E^(I a + I b + I c - I d + I e) + 1/16 I E^(-I a - I b - I c + I d + I e) - 1/16 I E^(I a - I b - I c + I d + I e) - 1/16 I E^(-I a - I b + I c + I d + I e) + 1/16 I E^(I a - I b + I c + I d + I e) + 1/16 I E^(-I a + I b - I c - I e - I f) - 1/16 I E^(I a + I b - I c - I e - I f) + 1/16 I E^(-I a + I b + I c - I e - I f) - 1/16 I E^(I a + I b + I c - I e - I f) - 1/16 I E^(-I a - I c + I d - I e - I f) - 1/16 I E^(I a - I c + I d - I e - I f) + 1/16 I E^(-I a + I c + I d - I e - I f) + 1/16 I E^(I a + I c + I d - I e - I f) - 1/16 I E^(-I a + I b - I c + I e - I f) + 1/16 I E^(I a + I b - I c + I e - I f) - 1/16 I E^(-I a + I b + I c + I e - I f) + 1/16 I E^(I a + I b + I c + I e - I f) + 1/16 I E^(-I a - I c + I d + I e - I f) + 1/16 I E^(I a - I c + I d + I e - I f) - 1/16 I E^(-I a + I c + I d + I e - I f) - 1/16 I E^(I a + I c + I d + I e - I f) - 1/16 I E^(-I a - I b - I c - I e + I f) + 1/16 I E^(I a - I b - I c - I e + I f) - 1/16 I E^(-I a - I b + I c - I e + I f) + 1/16 I E^(I a - I b + I c - I e + I f) + 1/16 I E^(-I a - I c - I d - I e + I f) + 1/16 I E^(I a - I c - I d - I e + I f) - 1/16 I E^(-I a + I c - I d - I e + I f) - 1/16 I E^(I a + I c - I d - I e + I f) + 1/16 I E^(-I a - I b - I c + I e + I f) - 1/16 I E^(I a - I b - I c + I e + I f) + 1/16 I E^(-I a - I b + I c + I e + I f) - 1/16 I E^(I a - I b + I c + I e + I f) - 1/16 I E^(-I a - I c - I d + I e + I f) - 1/16 I E^(I a - I c - I d + I e + I f) + 1/16 I E^(-I a + I c - I d + I e + I f) + 1/16 I E^(I a + I c - I d + I e + I f)

More legible in $$\LaTeX$$: $$-\frac{1}{16} i e^{-i a+i b-i c-i d-i e}+\frac{1}{16} i e^{i a+i b-i c-i d-i e}+\frac{1}{16} i e^{-i a+i b+i c-i d-i e}-\frac{1}{16} i e^{i a+i b+i c-i d-i e}+\frac{1}{16} i e^{-i a-i b-i c+i d-i e}-\frac{1}{16} i e^{i a-i b-i c+i d-i e}-\frac{1}{16} i e^{-i a-i b+i c+i d-i e}+\frac{1}{16} i e^{i a-i b+i c+i d-i e}-\frac{1}{16} i e^{-i a+i b-i c-i d+i e}+\frac{1}{16} i e^{i a+i b-i c-i d+i e}+\frac{1}{16} i e^{-i a+i b+i c-i d+i e}-\frac{1}{16} i e^{i a+i b+i c-i d+i e}+\frac{1}{16} i e^{-i a-i b-i c+i d+i e}-\frac{1}{16} i e^{i a-i b-i c+i d+i e}-\frac{1}{16} i e^{-i a-i b+i c+i d+i e}+\frac{1}{16} i e^{i a-i b+i c+i d+i e}+\frac{1}{16} i e^{-i a+i b-i c-i e-i f}-\frac{1}{16} i e^{i a+i b-i c-i e-i f}+\frac{1}{16} i e^{-i a+i b+i c-i e-i f}-\frac{1}{16} i e^{i a+i b+i c-i e-i f}-\frac{1}{16} i e^{-i a-i c+i d-i e-i f}-\frac{1}{16} i e^{i a-i c+i d-i e-i f}+\frac{1}{16} i e^{-i a+i c+i d-i e-i f}+\frac{1}{16} i e^{i a+i c+i d-i e-i f}-\frac{1}{16} i e^{-i a+i b-i c+i e-i f}+\frac{1}{16} i e^{i a+i b-i c+i e-i f}-\frac{1}{16} i e^{-i a+i b+i c+i e-i f}+\frac{1}{16} i e^{i a+i b+i c+i e-i f}+\frac{1}{16} i e^{-i a-i c+i d+i e-i f}+\frac{1}{16} i e^{i a-i c+i d+i e-i f}-\frac{1}{16} i e^{-i a+i c+i d+i e-i f}-\frac{1}{16} i e^{i a+i c+i d+i e-i f}-\frac{1}{16} i e^{-i a-i b-i c-i e+i f}+\frac{1}{16} i e^{i a-i b-i c-i e+i f}-\frac{1}{16} i e^{-i a-i b+i c-i e+i f}+\frac{1}{16} i e^{i a-i b+i c-i e+i f}+\frac{1}{16} i e^{-i a-i c-i d-i e+i f}+\frac{1}{16} i e^{i a-i c-i d-i e+i f}-\frac{1}{16} i e^{-i a+i c-i d-i e+i f}-\frac{1}{16} i e^{i a+i c-i d-i e+i f}+\frac{1}{16} i e^{-i a-i b-i c+i e+i f}-\frac{1}{16} i e^{i a-i b-i c+i e+i f}+\frac{1}{16} i e^{-i a-i b+i c+i e+i f}-\frac{1}{16} i e^{i a-i b+i c+i e+i f}-\frac{1}{16} i e^{-i a-i c-i d+i e+i f}-\frac{1}{16} i e^{i a-i c-i d+i e+i f}+\frac{1}{16} i e^{-i a+i c-i d+i e+i f}+\frac{1}{16} i e^{i a+i c-i d+i e+i f}$$

In terms of spherical harmonics:

A = {{Sin[a] Cos[b], Sin[c] Cos[d], Sin[e] Cos[f]},
{Sin[a] Sin[b], Sin[c] Sin[d], Sin[e] Sin[f]},
{Cos[a], Cos[c], Cos[e]}} // Det;
B = (8 I π^(3/2))/(3 Sqrt[3]) *
(Y[1, -1, e, f] Y[1, 0, c, d] Y[1, 1, a, b] -
Y[1, -1, c, d] Y[1, 0, e, f] Y[1, 1, a, b] -
Y[1, -1, e, f] Y[1, 0, a, b] Y[1, 1, c, d] +
Y[1, -1, a, b] Y[1, 0, e, f] Y[1, 1, c, d] +
Y[1, -1, c, d] Y[1, 0, a, b] Y[1, 1, e, f] -
Y[1, -1, a, b] Y[1, 0, c, d] Y[1, 1, e, f]) /. Y -> SphericalHarmonicY;
A == B // FullSimplify
(* True *)

• Thanks a lot... as I was asking above, is any simplification possible? I would like to take the Fourier series of this horrible function... – Romeo Feb 13 '19 at 16:11
• Maybe a spherical-harmonic decomposition instead of a Fourier transform? – Roman Feb 13 '19 at 16:30
• Thank you so much for your valuable help. I am sorry to bother you again. Could you please compute the Fourier series of the function abs(det ...) where ... is the horrible matrix above? And btw how does spherical harmonic decomposition works? Thank you again for your time. – Romeo Feb 13 '19 at 17:41
• @Romeo go here and do it yourself... – b3m2a1 Feb 13 '19 at 21:30