#### 7.1 back substitution (2.7.96)

I wonder if someone has a good strategy for backsubstituting?

For instance, deﬁne `f(x,y,z) = sqrt(x**2+y**2+z**2)`

and then take a derivative of a
function such as `diff(f(x,y,z),x)`

. At this point I would like to back substitute for
f, i.e., remove `sqrt(x**2+y**2+z**2)`

where ever it occurs and replace it with
f.

I ﬁnd that I need contortions involving `subs(%1=f,...);`

and that this trial and error
procedure is not pedagogically reasonable.

Is there a scheme for doing this kind of back substitition so that physically meaningful
variables can be employed consistently in the manner which pencil and paper work is done in
physics and chemistry?

I think this is a very general problem in Maple. You may try using the alias function, but
that is certainly not a ﬁnal solution:

alias( a= 1/sqrt(x^2+y^2) );
diff( sqrt(x^2+y^2) , x );
a*x

Maple automatically substitutes the subexpression `1/sqrt(...)`

by a. Note that
unfortunately `alias( a=sqrt(...) )`

does not work because of the internal structure of
Maple expressions. But subs does not work also:

1/sqrt(...);
subs( sqrt(...)=b );
1/sqrt(...)

I and certainly many Maple users have the problem of really "intelligent" backsubstitution in
a way like "Find the most simple expressions using these substitutions or functions
...". E.g. if should be possible to ﬁnd out that `x^4+y^2`

should be expressed by
`f(x^2,y)`

if `f(x,y)=x^2+y^2`

has been previously deﬁned or given as a possible
substitution.

comment and a caveat. ASSUMING the range/characteristics of a variable does not make it
so in the ’eyes’ of "whattype". This is, I believe, an error on Maple’s part. It screws up the
performance of such programs as `BesselI(n,x)`

when n is only ASSUMEd integer, not
TYPEd integer.