# How to program Mathematica to do the differential operator calculation

I want to create a surface Laplacian under the spheroidal coordinates, Now, I already have the surface gradient defined, but I don't know how to find the Laplacian.

Since the surface gradient is too long, and it's a vector operator, I use this one as an example:

op[t_]:={Cos[t] D[#, t], Sin[t] D[#, t]}&;


Apply this on some u(t), I got:

In:= op[t][u[t]]

Out= {Cos[t] u'[t], Sin[t] u'[t]}


which is what I want.

However, I can't do divergence of this gradient. I've tried Composition, but it gives me a matrix instead of a scalar:

In:= Composition[op[t], op[t]][u[t]]

Out= {{Cos[t] (-Sin[t] u'[t] + Cos[t] u"[t]), Cos[t] (Cos[t] u'[t] + Sin[t] u"[t])},
{Sin[t] (-Sin[t] u'[t] + Cos[t] u"[t]), Sin[t] (Cos[t] u'[t] + Sin[t] u"[t])}}


which is obviously wrong.

• I use the double quote to make the codes look nice, the actual output has /[prime] within. – Ying Zhang Apr 18 '18 at 13:57
• I think I just found a similar question with the dot product of operators: mathematica.stackexchange.com/questions/75486/… – Ying Zhang Apr 18 '18 at 14:04
• I guess I didn't state the problem clearly. My problem is to do the dot product of two vector operators. That's why I create this simple vector operator as a representation of my surface gradient operator. – Ying Zhang Apr 18 '18 at 14:29

I did not exactly get what you are doing there but the following should work in a general setting.

We start with a simple surface parameterization f (I simply use the sphere and compute the Riemannian metric g and the volume density vol induced on the parameterization domain by f. Then we apply a well known formula for the Laplace-Beltrami operator in terms of g and vol.

f = X \[Function] {Cos[X[]] Cos[X[]], Cos[X[]] Sin[X[]], Sin[X[]]};

LaplaceBeltrami = u \[Function]
Quiet[Block[{X, XX = Table[X[[i]], {i, 1, 2}], Df, g, vol},
Df = X \[Function] Evaluate[D[f[XX], {XX, 1}]];
g = X \[Function] Evaluate[Df[XX]\[Transpose].Df[XX]];
vol = X \[Function] Evaluate[Sqrt[Det[g[XX]]]];

Evaluate[X] \[Function] Evaluate[
Simplify[
Tr[D[D[u[XX], {XX, 1}].Inverse[g[XX]] vol[XX], {XX, 1}]]/ vol[XX]
]
]
]
];

LaplaceBeltrami[u][{ϕ, θ}]

• Thanks for the reply. I guess I didn't state the problem clearly. My problem is to do the dot product of two vector operators. That's why I create this simple vector operator as a representation of my surface gradient operator. But it's also great to see your example! Thanks! – Ying Zhang Apr 18 '18 at 14:29