Why I get the "Set::write: "Tag Times in is Protected." error?

Question

Why I get the error below with this code:

Needs["VectorAnalysis`"]

ρ = 1000;
g = 9.80665;
μ = 0.001;
p[x_, y_, z_, t_] = {x^2, Sqrt[y], -z, t^3};
f[x_, y_, z_, t_] = {-x^2, Sqrt[y]/2, z, t^2};

ρ Dt[v, t] = -Gradient[p] + μ Laplacian[v] + f

---

Error:

Set::write: "Tag Times in 1000\ Dt[v,t] is Protected."

Answer 1

Any time you try to execute a command of the form a*b=c, you'll generate this error:

Of course, that's exactly what you've done in your last line:

ρ Dt[v, t] = -Gradient[p] + μ Laplacian[v] + f

We can see the issue more clearly if we examine the left side of the equation in FullForm

FullForm[a*b]
(* Out: Times[a,b] *)

Furthermore, Times is protected.

Attributes[Times]
(* Out: {Flat, Listable, NumericFunction, OneIdentity, Orderless, Protected} *)

Thus, you're trying to set the value of a protected symbol. As rm-rf points out, it's not just about removing the ρ, as otherwise you're trying to set the value of Dt.

Perhaps more to the point - if you want to represent an equation that you want to solve, you should type:

ρ Dt[v, t] == -Gradient[p] + μ Laplacian[v] + f

Note the double equals sign. This tests for equality, rather than setting a value - which is what you generally want when solving an equation.