Newton's Discovery of the Refraction of Light

Episode Eleven – Newton’s Achievements

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Integral calculus also has many applications. One of them is finding areas or volumes. For instance, if you want to find the volume of an irregular shape such as a Coke bottle, you can use integration to do so.

Many properties in physics are calculated as integrals. For instance, finding the coordinates of the centre of mass of an object, studying magnetic flux and so forth.

And then there is the fact that since integration is the reverse of differentiation, you can use it to derive many equations, just as you can using differential calculus.

The applications of calculus are far too numerous to mention all but a few of the most general applications. Suffice to say that every branch of higher mathematics uses it and calculus is applicable to almost any field of knowledge known to man.

Which is true of mathematics in general, right? Mathematics is all about describing and deriving the various relationships between entities and their properties.

Well, calculus is a fundamental technique used to find many of these. Without which much of higher mathematics would not be possible.

Pythagoras

Pythagoras was among the first to catch on to the general practicability of math. But, he took it a bit too far …

Newton-Raphson Method

This is a method of numerical analysis which allows one to find approximate solutions for real-valued functions. That is, mathematical functions with solutions that are real numbers.

Yes, it has been generalized to complex values, but we will not go into that here.

Let us suppose that you have the equation x3 + 5x – 3 = zero. What value of x satisfies this equation? The Newton-Raphson method allows an approximate value to be found.

It is an iterative process, requiring repeated cycles through the Newton-Raphson formula in order to get increasingly accurate results.

This is not the only such method used to solve functions. But, it is a commonly used one and some computer algorithms use it to solve functions.

But why should we care? Well, many equations where you need to solve for x can be solved with the Newton-Raphson method. In fact, many equations where we have no known method for finding exact solutions can have this method applied. This allows approximate values to be found.

You can also use it for optimization problems, similar to the optimization described when we discussed calculus. In fact, the Newton-Raphson method itself uses calculus, differentiation in particular.

Its applications are many and varied. It is frequently used in the analysis of flow in large networks. Such as water distribution networks or electrical flow through electrical grids.

It also has various uses in numerical analysis and other areas of mathematics, but we will not go into that. Suffice to say that it is an extremely important tool in solving and analysing a great many equations.

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