# Keeping equations in terms of specific variable

I'm new to Mathematica and really struggling with something seemingly simple. I'm trying to do an integration and keep the results in terms of the symbol $$T_1$$ and $$A_i$$:

$$T_1=\frac{\text{pi} A_i}{2 J_a}$$

I want to integrate $$j_1$$:

$$j_1=J_a \sin \left(\frac{\text{pi} t}{T_1}\right)$$

The result I get is:

$$-\frac{1}{2} A_i \cos \left(\frac{2 t J_a}{A_i}\right)$$

I'm not sure how to get the results in terms of the desired variables. I understand that Mathematica evaluates symbols as early as possible, and so I have tried playing with Hold but it didn't help. I've also tried to recast the problem as simultaneous equations and then using Solve or Eliminate as mentioned here:

Rewriting expression in terms of factor

Here's an example of something I've tried:

$$j_1=J_a \sin \left(\frac{\text{pi} t}{\text{Hold}\left[T_1\right]}\right)$$

$$a_1 = Integrate[\%, t]$$

$$\text{ReleaseHold}\left[\text{Solve}\left[\left\{\text{expr}=a_1,T_j=T_1\right\},\text{expr},\left\{J_a\right\}\right]\right]$$

I tried introducing $$T_j$$ since I figured $$T_1=T_1$$ probably wouldn't make sense.

The results are still in terms of $$J_a$$ though:

$$\left\{\left\{\text{expr}\to -\frac{\text{pi} A_i^2 \cos \left(\frac{2 t J_a}{A_i}\right)}{4 J_a T_j}\right\}\right\}$$

Any advice greatly appreciated!

• It's also not a good idea to use anything with subscripts if you're, as you say, "new to Mathematica", since those are troublesome to use with anything except for pretty formatting. Apr 19, 2020 at 12:00
• Thanks, I didn't know. If I want to have constants t_1 through t_10 that are defined through other constants, and I want to be able to programmatically iterate through them, what is the suggested approach? An array format instead? t[[1]]? Apr 19, 2020 at 12:34
• Yes, something like t[1] (that is, single brackets and not double ones) should work. Apr 19, 2020 at 12:35
• And what about for replacing letter subscripts, like A_j? Are dollar signed (A\$j) used or what? Seems so ugly, guess I need to read about Symbolize and the Notation package.... Apr 19, 2020 at 12:37

Format[Ai] := Subscript["A", "i"]
Format[T1] := Subscript["T", 1]

j1 = Ja Sin[π t/T1];

expr = Integrate[j1, t] /. Solve[T1 == π Ai/(2 Ja), Ja][[1]]


• Is there a way to do it where T1 is still defined? Because I need it (and T2, T3, ...) in other places. Didn't know about "Format", thanks! I could imagine using a Module or Block, but it seems so ugly... Apr 19, 2020 at 13:32
• Define the replacement rule t1r = T1 -> π Ai/(2 Ja); Then whenever you want to replace T1 use expression /. t1r Apr 19, 2020 at 14:05
• Perhaps I'm misunderstanding, but if I only have the replacement rule then there would be no way for the system to know there is a relationship between T1 and Ja and so I'd either have answers without T1 (if I used the replacement rules) or I'd have answers with a mix of T1 and Ai/Ja, but without the simplifications that can result by knowing how they inter-relate. Apr 19, 2020 at 14:24
• If you Set the value of T1, i.e. T1 = ... then T1 can never appear since it will always be replaced by its definition. A rule enables you to either not use the rule and let T1 appear or use the rule and thereby substitute for T1. Alternatively, you can define the equation T1 == ... and use the equation in a system of equations (Solve, Eliminate, Reduce, FindInstance, ...) or as an assumption in Simplify or similar. Apr 19, 2020 at 14:37
• Ok, that makes sense, thank you! If I want to do the system of equations route, do I need to do something like eq1 = T1 == ..., eq2 = T2 == ..., or is there some way to reference the T1 == and T2 == without assigning the equality to another symbol? Apr 19, 2020 at 19:32