How can I make Mathematica compute an operator?

Question

I would like Mathematica to write down the Laplace-Beltrami operator in $R^N$ $$ \Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{N-1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \delta f, $$ $$ \delta = \sum\limits_{j = 1}^{N - 1} {\frac{1} {{q_j \left( {\sin \theta _j } \right)^{N - j - 1} }}\frac{\partial } {{\partial \theta _j }}\left( {\left( {\sin \theta _j } \right)^{N - j - 1} \frac{\partial } {{\partial \theta _j }}} \right)} , $$ $$ \begin{array}{*{20}c} {q_1 = 1,} & {q_j = \left( {\sin \theta _1 \sin \theta _2 \cdots \sin \theta _{j - 1} } \right)^2 ,} & {j = 2, \ldots ,N - 1} \\ \end{array} $$ as a function of $N$, i.e. I want to specify $N$ and compute the symbolic sum of $\delta$. I tried with

q[1] := 1

q[j_] := (Product[Sin[Subscript[θ, i]], {i, 1, j - 1}])^2

δ[N_] := 
 Sum[1/(q[j] (Sin[Subscript[θ, j]])^(N - j - 1)) \!\(
\*SubscriptBox[\(∂\), 
SubscriptBox[\(θ\), \(j\)]]\((
\*SuperscriptBox[\((Sin[
\*SubscriptBox[\(θ\), \(j\)]])\), \(N - j - 1\)]\ 
\*SubscriptBox[\(∂\), 
SubscriptBox[\(θ\), \(j\)]])\)\), {j, 1, N - 1} ]

but it does not work:

Syntax::sntxi: Incomplete expression; more input is needed.

Anyone has any idea?

Thanks

Answer 1

The partial derivatives are causing the problems in your above code. If you realy just want to write it down here is a version that just prints the definition:

\[Delta][N_] :=
Sum[Row[{1/(q[j] Sin[Subscript[\[Theta], j]]^(N - j - 1)), (
"\[PartialD]")/Subscript["\[PartialD]\[Theta]", j], "(", 
Sin[Subscript[\[Theta], j]]^(N - j - 1), ("\[PartialD]")/
Subscript["\[PartialD]\[Theta]", j], ")"}], {j, 1, N - 1} ]

This will create a sum of terms which can not be manipulated algebraically by mathematica since they are row expressions.