More on the Fundamental Theorem of Calculus: Standard vs BodyandSoul

In recent posts we have compared the Standard Calculus approach with the BodyandSoul approach to the Fundamental Theorem as expressed in the corresponding proofs of the theorem, which concerns the relation between primitive function/integral $x(t)$, derivative $Dx = \frac{dx}{dt}$ and integrand $v(t)$ connected by the equations:

1. $Dx(t) = v(t)$
2. $x(t)=\int_0^t v(s)\, ds$.

We have noticed that in Standard Calculus as presented in e.g. the standard text book Calculus: A Complete Course by Adams and Essex, the integral $x(t)$ as an area under the graph $t\rightarrow v(t)$ is the primary given object (possibly somehow constructed as a limit of piecewise rectangular approximating areas), and the proof of the Fundamental Theorem consists of showing that $x(t)$ satisfies the differential equation $Dx = v$. 

In the BodyandSoul approach the primary given object is the differential equation $Dx=v$ with $v$ given data and $x$ unknown solution to determine, and the proof of the Fundamental theorem consists of showing that this equation can be solved by constructing approximate solutions by time-stepping

according to 

• $x(t+\Delta t)=x(t)+v(t)\Delta t$ with $\Delta t$ a time step tending to zero. 

In this approach $x(t)$ satisfies the differential equation $Dx(t)=v(t)$ by construction, because 

• $Dx(t)\approx \frac{x(t+\Delta t) - x(t)}{\Delta t}=v(t)$.

In the Standard approach the construction is hidden and the proof thus consists in verifying that the integral $x(t)$ satisfies the equation $Dx(t)=v(t)$. Now, this is cumbersome because computation of a derivative in principle is an ill-posed unstable process, which has to be regularized to be computationally meaningful in finite precision with the presence of perturbations of $x(t)$. In the Standard approach this is circumvented by performing symbolic exact differentiation. 

For example, it is verified by symbolic differentiation in infinite precision that e.g. $D t^3 = 3t^2$, which allows computation of the area under the function $3t^2$ from $t=0$ to $t=1$, as a form of magic seemingly without summation:

• $\int_0^13t^2 = 1^3 - 0^3 = 1$.   

The full picture is hidden to the student of Standard Calculus, presenting primarily the non-constructive magic of exact symbolic differentiation and less the constructive non-magic of time-stepping. BodyandSoul presents both aspects and opens to a deeper and more useful constructive understanding.

PS1 To define as in Standard Calculus the integral as an area follows the same non-constructive geometric approach as defining trigonometric functions through the lengths of the sides of rectilinear triangles (used in e.g. Adams). In BodyandSoul the integral and elementary functions such as trigonometric functions are constructed as solutions to elementary differential equations. Properties of elementary functions then come out as consequences of defining elementary differential equations and not from possibly far-fetched geometry.

The step from the (difficult) symbolic geometry of Euclide to the (easy) constructive analytic geometry of Descartes, marked the beginning of the scientific and industrial revolution and was thus not a small step for humanity. A modern Calculus course should reflect this step.

PS2 In BodyandSoul the trigonometric functions $x(t)=\sin(t)$ and $y(t)=\cos(t)$ are defined as solutions of the system describing a harmonic oscillator:

• $Dx(t) = y(t)  and Dy(t) = - x(t)$ for $t > 0$ with initial values $x(0)=0$ and $y(0)=1$,

and solutions are computed by time-stepping. 

The familiar relations $D\sin(t) = \cos(t)$ and $D\cos(t)= - \sin(t)$, thus result from the constructive time-stepping of the system where these relations are encoded. This is readily understood by the general student.

In Standard Calculus, $\sin(t)$ and $\cos(t)$ are defined geometrically and the relations $D\sin(t) = \cos(t)$ and $D\cos(t)= - \sin(t)$ have to be discovered and proved by tricky trigonometry, which the teacher may love, but the general student finds difficult.

Again, presenting this argument to a Standard Calculus teacher will lead nowhere.