# Dealing with Norm, Complex situation not desired

I am working on a notebook for my calculus students and am dealing frequently with the norm of vector valued functions. I always seem to run into this kind of situation.

Clear[a, s, r];
r[s_] = {a Cos[s/a], a Sin[s/a]};
T[s_] = r'[s];
κ[s_] = Norm[T'[s]]


Where the output is:

(*Sqrt[Abs[Cos[s/a]/a]^2 + Abs[Sin[s/a]/a]^2]*)


Then I have to bring on the Simplify command to fix the problem. In this case, for example, I do:

κ[s_] = Simplify[Norm[T'[s]], s > 0 && a > 0]


Which provides the same answer students achieve when doing the problem by hand, namely, 1/a.

Am I dealing with the Norm command properly, or would folks like to give better advice?

Update:

Here is an idea that is extremely helpful if you are going to use these assumptions frequently in a large notebook:

Clear[a, s, r];
$Assumptions = a > 0 && s ∈ Reals; r[s_] = {a Cos[s/a], a Sin[s/a]}; T[s_] = r'[s]; κ[s_] = Norm[T'[s]]  The output is: (*Sqrt[Abs[Cos[s/a]/a]^2 + Abs[Sin[s/a]/a]^2]*)  But now I can quickly simplify. Clear[a, s, r];$Assumptions = a > 0 && s ∈ Reals;
r[s_] = {a Cos[s/a], a Sin[s/a]};
T[s_] = r'[s];
κ[s_] = Norm[T'[s]]//Simplify


The output is:

(*1/a*)

• What you're doing seems to be the most reasonable answer in this case. If you want a simple result, you have to simplify. This is the only way to make use of all the assumptions.
– Jens
Jun 14 '15 at 3:53
• Since all values are real, you could implement the Norm directly as Sqrt[T'[s].T'[s]]. This doesn't obviate the need for simplifying, although you have one fewer condition to specify. Rather, it merely gets rid of the Abs. Jun 14 '15 at 4:09
• \[Kappa][s_] = Norm[T'[s]] //Simplify[#, {a > 0, Element[s, Reals]}] & Jun 14 '15 at 12:42

I frequently do what march advises in a comment, especially when I do not want to Simplify but I do want a differentiable norm:

norm = Sqrt[#.#] &;


If you were going to put this in a package for students to use, then you might prefer

norm[v_?VectorQ] := Sqrt[v.v];


Of course, it's only good for real-component vectors.

• Yes, I've been seeing this type of method as being most desirable. I think I'll use the Sqrt when teach multivariable calculus next semester. Jul 16 '15 at 2:01

I agree with Jens that Simplify seems like the right approach.
Perhaps you would find value in making its use more terse.

Define:

simp[cond_: {}][expr_] :=
Simplify[
expr,
Union @@ (Flatten[{$Assumptions, cond}] /. ({All -> #} & /@ Variables @ Level[expr, {-1}])) ]  Then you can do: Norm[T'[s]] // simp[All > 0]  1/a  Or: $Assumptions = {All > 0};

Norm[T'[s]] // simp[]

1/a