@acer Can you explain the "true" results that I got?

That's very weird. Try doing a restart. You do not need to with anything to do basic arithmetic.

@tuvok1153 

First_Principle, as it's written, requires its argument to be a procedure (or appliable as a procedure); it will not (correctly) accept expressions. So, you need to use it like

First_Principle(x-> tan(x));  First_Principle(x-> sin(x)^2);

or

First_Principle(tan);  First_Principle(sin^2);

If you wish for it to work on expressions, it's a little more complicated to write:

First_Principle:= proc(f::algebraic, x::name)
local h, A:= Limit(simplify((eval(f, x= x+h)- f)/h), h= 0);
     A = value(A)
end proc:

First_Principle(tan(x), x);

Note that this requires you to pass, as the second argument, the variable with respect to which the derivative is being taken.

If you want it to work in either case, you could do this

First_Principle:= proc(f::{appliable,algebraic}, x::name)
local
     _X,
     h,
     A:= Limit(simplify(`if`(nargs=2, eval(f, x= x+h)-f, f(_X+h)-f(_X))/h), h= 0)
;
     A = value(A)
end proc:

First_Principle(tan);

First_Principle(tan(x), x);

@MikeS Your code does not work because you have MATRIX. It should be Matrix.l

Please provide an example of what you mean.

@Kitonum There's a difference between matrices being identical and them having equal entries. If you test is(A=A1), the result is false. Equal only checks that the entries are equal.

@Erfan You have misspelled the command name. It's KroneckerProduct, not KroneckeProduct.

@Hulalarip To get more information about that, you'll need to post your system of equations.

You intend to use the C variables as symbolic names, right? rather than assigning them values?

If m and m' both range from 1 to N, then m'' = m-m' will be zero or negative for many pairs (m,m'). Yet m'' is one of your function indices. How do you handle that?

You wrote, "Sketch, on the same plane, the curve...." But the curve does not lie in any plane. The curve is a helix on an elliptic cylinder around the x-axis.

Update: Perhaps "same plane" is meant as "same graph" because any 3-D graph can be considered to be lying in the plane of the paper or screen on which it is drawn.

You have a typo. The equation of the circular cylinder should be x^2 + y^2 = a^2. You have x^2+y^3^=a^2.

@Aakanksha 

What's the point of summing 1000 terms when you don't have numeric values for Xi and gamma[o]? You'll just get an expression with 1000 terms.

What is the parameter Xi? What is gamma[o]?

@acer 

Better yet,

plots:-display(plot([[0,0]], color= white), plots:-pointplot( DataTable3 ));