' - - File: Identities2.txt
'    File to test operations by using identities.

a = 12.3E-2                                        ' Any constant; + if Sqrt[ or logs are used.

G{t} = (t - a^.5)^2 + Log[t + a^3]                ' Any function
F{t}  = G{t + 2} + Log[Abs[t+1.4E1]]         ' Any function that references another function.

A<t> = <t, t^3, Sqrt[t] >                              ' Any vector
B<t> = <G, Sinh[t^2], F >                          ' Any vector that references functions.


' - - - - - - - - - - - -Check this ID: derivative (A x B) = A x Derivative(B) + Derivative(A) x B

G<t> = vCross[A,B]                                    ' A x B
Gd<t> = vDiff[G]                                         ' d (A x B) / dt

'Do the right-hand side above: - - - - - - -
Fb<t> = vCross[A,vDiff[B]]                         '  A x B'     where B' is derivative of B.
Fa<t > = vCross[vDiff[A],B]                        '  A' x B     where A' is derivative of A.
S<t> = vAdd[Fb,Fa]                                    '  A x B'   +  A' x B

'  Does Gd<t> = S<t>?                               ' Gd<t> - S<t> = 0?
Df<t> = vSub[Gd,S]                                     ' Subtract the two vectors.

' ANS: we get <-6.25E-2, 0, 0> with obvious round off errors since the derivatives
'   of functions within functions followed by cross products and subtraction of these
'   vectors would magnify any small differences.

' Even though numerical examples does not prove in an absolute sense, the validity
'    of a program, subtraction of vectors Gd & S derived by different means is close to 
'    zero. A small procedure error, whether from Diff, vector manipulation, or just doing
'    numerical computations would magnify and be off from zero difference. Different
'   parameter substitution of random values yields the same results.

<t = 3.456>

'  If you use <t = 0.1247>, we get <0,0,0> for the difference.


