A cornerstone of school/university mathematics is Calculus in the form of Symbolic Calculus which is basically a catalog/table of derivatives of elementary functions compiled using the following set of rules for symbolic computations for functions F(t) and G(t) depending on a number variable t with derivatives DF(t) and DG(t) with D=d/dt, as shown in Symbolic Differentiation:
- D(F+G) = DF + DG (or d(F+G)/dt = dF/dt + dG/dt)
- D(F*G) = DF*G + F*DG (or d(F*G)/dt = dF/dt*G + F*dG/dt)
- D(F/G) = (DF*G – F*DG)/(G*G)
- D(F(G(t)) = DF(G(t))*DG(t) (chain rule) (or dF(G(t))/dt = dF/dG * dG/dt)
The idea is that given an initial value problem to solve: Du = f(t) for t > 0, with u(0) = u0, that is the integral ∫ f(t)*dt to compute, search in the catalog for a function F(t) with DF = f , which if successful will produce the solution u(t) as u(t) = F(t) + C with the constant C determined so that F(0) + C = u0.
Compare with Digital vs Symbolic Calculus.
To prove 4. consider dF(G(t)) =F(G(t+dt))-F(G(t))=D(F(G(t))*dt and dG(t) = (G(t+dt) – G(t)=DG(t)*dt to give D(F(G(t))*dt = dF(G(t)) = F(G(t)+dG(t)) – F(G(t)) = DF(G(t))*dG(t) = DF(G(t)*DG(t)*dt. Here F(G(t)) is a composite function that is a “function of a function” and the rule states the derivative of a composite function is equal to the product of the individual derivates of the two functions forming the composite.
With Digital Calculus the initial value problem Du = f(t) is solved by time stepping for any f(t) and the use of Symbolic Calculus can be directed to illumination of certain basic connections rather than lengthy possibly fruitful search for actual evaluations in applications. Symbolic Calculus is valuable for thought processes but limited as concerns actual evaluation of intergrals. Before the computer Symbolic Calculus was the magic making computation of integrals possible without massive hand calculation.
Recall that essence of the symbols sin(t) and cos(t) is embodied in the code
- du = v*dt , dv =-u*dt for t>0 with u(0)=0 and v(0)=1,
which computes u(t) = sin(t), v(t) = cos(t) including the information that Dsin(t) = cos(t) and Dcos(t) = -sin(t).