What is Calculus?

Aug 24, 2017

Here is my stab at an answer. Please feel free to add or amend. It would probably benefit by the inclusion of the limit definition of the derivation along with some notation. I will expand later on.

I would start by categorizing the two main fields of calculus, that of differentiation and integration.

Differential Calculus, or Differentiation

If we have a function of one variable, ie of the form  y=f(x), then in its most basic form differentiation is the study of how a small change in one variable $x$ affects the other variable $y$.

As an real life example, consider the average speed of a moving car:

$\text{average speed" = "distance travelled" / "time taken}$

Obviously this is be its very definition, an average, but suppose there was a formal mathematical relationship between distance and time, then could we form a function that would give us the instantaneous velocity at any given point in time? The study of differential calculus provides techniques for us to take the ratio of an small change in distance, compared to a small change in time, then as make as we make the small change infinitesimal small we get the actual instantaneous speed.

Similarly if we wanted to find the gradient of the tangent to a curve at some particular point $A$ we would estimate the gradient by using a chord to a nearby point $B$. As we move this nearby point $B$ closer to the tangent point $A$ the slope of the chord approaches the slope of the tangent with more and more accuracy. Again differential calculus provides techniques for us to make the point $B$ infinitesimally close to the point $A$ so that we can calculate the actual gradient of the tangent

Integral Calculus, or Integration

Suppose we wanted to calculate the area under a curve, $y = f \left(x\right)$, bounded the $x$=axis, and two points $a$ and $b$. We could start by splitting the interval $\left[a , b\right]$ into $n$ regular strips, and estimating the area under the curve using trapezia (this is the essence of the trapezium rule which provides an estimate of such an area). If we increase $n$ then generally we would hope for a better approximation.

The study of integration provides techniques for us to take an infinitely large number of infinitesimally small strips to gain an exact solution.

The Fundamental Theorem of Calculus

Given the above two concepts at first it would seem unlikely that there is any relationship between the two., However the
Fundamental Theorem of Calculus is a theorem that relates the rate of change of the area function (that provides the area under a curve), and the function itself. In words this means the derivative of the area function is the function itself.

Aug 26, 2017

The list of sciences that would suffer from the absence of calculus is a good measure of its importance.

Explanation:

Physics, Chemistry, all engineering sciences, statistics, economics, finance, biology, computer science, linguistics, to name but a few, are all areas that would be a desert without the use of calculus.

Leibnitz and Newton worked to define the velocity of a planet moving on a curved trajectory. That was not possible without calculus, and both had to invent differential calculus. Differential calculus allows to compare quantities along a curve, and thus their time rate of change.

All of classical physics can be summarized in this operation. Given second derivative (which is Force/mass), find the position as a function of time. This process is called integration. Half of calculus is made with integration, the other half with derivation. All of classical physics rests on these two parts of the calculus.

Quantum mechanics, quantum field theory, electromagnetism, fluid mechanics all use integration and derivation and much more. I rest my case. I hope this helps you gauge the place that calculus occupies in science.