Infinite polygons have flatter outcome – For a circle where the number of polygons take the denominator, like 6/0.1=60 or 6/0.01 is 600 it would be indicative that smaller the number (polygons) bigger is the answer. As the denominator approaches zero the answer becomes bigger and bigger so that we cannot divide by zero any more to avoid a value going big.
Likewise, Zeno’s paradox for carries a similar concept. Every single step you need to talk would require a half of that step so that it can be expressed as : 1+0.1+0.01+0.001 +….=1.111 sec. This is equated with Achilles concept where it would take 10+t to catch-up a tortoise (t is time).
Zeno meets Planck’s length root h/C3 (10-35m).
If one takes a large possible distance – say diameter of the Universe - and
divide by smallest possible distance (h), the most you’d need to express becomes
obvious.
iPhone tracks the distance covered while walking using calculus – length of the stride multiplied by the height of the person is the distance travelled (in km). Similar idea is applied to calculate circumference of each step which is a fine straight line; pi=C/d.
Many circle-ellipse-parabola drawings and segments are found in the third chapter of the book. For instance, the Quadratic parabola = 1+1/4+1/16+1/32, illustration was intuitive. An animation for facial surgery and Avtar’s (movie) face carry great level of polygonal details, and this became extravagant when this was the first instance to use polygon for creating faces. Use of rotating screw for propelling goes back to Archimedes.
Laws of motion discussed: Knowledge of what comes from a moment into existence and then perishes – Plato on objective geometry. The apparent position of the stars -parallax - finger shifting sideways when you view with an eye closed – Galileo, Copernicus, Aristotle and Ptolemy get due credits for using calculus in letting us know where we are weighed up in the Universe.
Water clock invention was also based on calculus and the law of odd numbers gets a mention:1+3+5+7=16 which is 4 squares (42) likewise, 1+3+5=9 which is 3 squares (32). This concept is explained diagrammatically with good figures.
Swings take same amount of time and only the speed varies! This was first checked by using Chandeliers in a Cathedral. This was later confirmed in the case of stars and our own Earth which has an elliptical orbit thus getting to roll speeder at the Perigees! Kepler who had damaged vision and hands due to smallpox, did works for establishing laws of planetary motions. Over elliptical orbits equal areas cover equal times, so planets have to move faster at places thus do not have constant speeds. The closer they get to the sun the faster they move. T2/A3 is same for all planets – T is the time, and A is the average distance from the sun. Thus T2=A3 for Venus it is about T2 0.378 and A3=0.378.
Latitude can be ascertained by
looking at the Sun or Stars but longitudes have no counterpart. Galileo helped
solve it and found that fifteen (15) degree longitude translates to 1000 miles
at equator – Navigation by Huygens also was helpful in creating a position of
the pivotal point amidst a traverse. GPS
uses at least 4 of the 24 satellites (orbiting at 12000 miles over head) for
this purpose – Time received vs time transmitted in GPS would not be possible without
calculus. Wireless or GPS to atomic clocks that use Cesium atoms use calculus.
Dawn of differential calculus discusses the aspect of Algebra meeting geometry. For instance: An XY plane expresses relationship between two parameters and a simple example includes number of slices taken vs calories consumed with a bread; thus Y=200X. Through equations shapes are arrived at and Descartes and Fermat used this concept in studies leading to optimization of how to do things in the best possible way. This way designing of boxes to stuff as much gold as possible is described using a simple geometry in metric values: 10x10x25=45 (adding boxes) or 10x10x25 = 2500 (getting volume) but in the place of above if 15x15x15 size is used, it might give a volume of 3375 which is better room compared with 2500 (which means change is geometrical parameters for getting best volume).
Fermat helped FBI which had millions of records of fingerprints storing and retrieving them through calculus. Just like this JPG and MP3 formats retrieval uses
the algorithm. This chapter also includes the concept of bending of light that
helped Snell arrive at his famous formula for refractive index.
In Crossroads it was a pleasing aspect to see that a Function of function is power function. And thus, an Exponential function would help one arrive at how bacteria multiply. Likewise, Logarithm which is used for non-round numbers, and which enable us to write any positive number as powers of ten. Exponential growth and decay mechanism, similarly, use the same concept and is applied to reduce noise off a microphone when it picks up the sound of its own speaker. This is the same with nuclear chain reactions.
In a further explanation on the Y and X factors the
author clearly explains what dy/dx actually mean; for instance, it is (-velocity)
that is rate of speed. Dy/dt is acceleration on a graph of distances and against
time for a moving car the slope is indicative of a car’s speed – universal rate. Area under the curve is important factor. Linear function of the constant rates for
instance in the case of slice, 400 calories / 2slices is 200 calories / slice. Many
aren’t aware of using calculus dy\dx curves for velocity which is distance
covered.
Infinitesimal piece of curve as parabola helps us to understand a couple of caveats about derivatives. A curve with V shape is exampled with a sharp corner at one point and when we zoom into it, it will look like a corner. However, if it is a curve, it might not look like a corner. Derivative as rates of change of day length help us to know how much extra sunlight a year can be obtained. Usain bolt Running style and slowing down near the end point leading to losing the first place is an example of this derivative application.
Leibnitz introduced the word calculus which means small stone. Incidentally, Newton and Leibnitz had these calculi (stones) in bladder and kidney, and it is written that the former died of this stone in bladder. Integration is global operation instead of microscope we are using telescope and peering far off into the future (past?). The concept of pi gets a cue - 3.14 corresponds to 3+100+1(1/10)1 + 4(1/10)2.
Mathematician from Kerala solved power series related to cosine and sine. Jyestadeva and Nilantha attributed the set to Madhava (1350-1425) in what is Kerala school of Mathematics and Astronomy today. Newtown had mashed up work of Greeks, Islamic Algebra and French geometry for arriving at infinity principle.
Fictions of the mind: Infinitesimals supposed to be the tiniest number, smaller than everything but greater than nothing! For instance, 23 is 8, but (2.001)3? Algebra has answers for that; x+dx may be used to get at (x+dx)3 or using Pascal’s triangle it can be arrived at 8+12 dx+6 dx2 + dx3.
Differentials can also be used to describe the fundamentals of the calculus; Upshot of dA=ydx=f(x) dx. Altimeter for measuring the altitude lot of number fractions illustrated with a grand sketch. The author goes on to justify that finding HIV with assistance from calculus dV/V=-cdt was possible with greater accuracy based on the above principles. Similarly, estimation of virus mutation (at every genome) was also done using calculus.
The calculus comes to aid in describing the Universe. Planck, Einstein, Bohr, Dirac, and Heisenberg
used calculus in all their calculations. With statistics initiating the estimation
of gravity using F=ma, Moon’s acceleration was compared to the acceleration of
the falling bodies on Earth and with that of Galileo’s experiment. It was
differentiated by 602 according to the inverse law prediction. Moon
was 60 times farther from the centre of the Earth. Known as Two-body problem,
the irregularities in the orbit of Uranus suggested an unknown planet and using
calculus it was proved as ‘Neptune’.
Katherin Johnson of NASA used calculus to predict the position of moving space crafts and to bring back John Glenn, the first American to orbit the Earth. NASA has a building named after her!
A bowl of soup is a discrete collection of molecules all bouncing erratically. Then, if you think the soup as continuum using x,y,z for the given point at temp T, partial differentiation makes ordinary differentiation equation a child’s play. Pejorative is using art of differential equation that depend on just one variable.
The final chapter has talks about the flights inspired from kites and birds. Dreamline Boeing applied Mathematics and calculus to anticipate how the airplane’s wing would flex when working at 600 miles an hour - avoiding aeroelastic flutter a nastier version of venetian blinds. Marking waves, standing, sinusoidal, Fourier series and String theory (vibration burst out of some of sine waves), and overtones, all get through this application of calculus. Appearance of patient after facial surgery, flow of blood through artery, rumbling of ground after earthquake etc. all use the concept of calculus.
A derivative of sine wave is another sine wave shifted by a quarter cycle (for other waves it gets distorted), so Sine wave is always efficient. Chaldini Patterns in guitar gets a mention too.
Microwave ovens which used microwaves during world wars as Magnetron for detecting signals can be used to calculate the speed of light. ‘Frequency’ of vibrations (listed on the door frame of ovens) multiplied by ‘Wavelength’ can give that! (2.45 billion cy/sec into 12cm wavelength is close to the speed of light = 29.4 billion cm/sec). Percy Spencer had placed peanut cluster besides these Magnetrons, and he found later that it was like cooked one leading to the discovery of heating using MW. It was later tested with eggs. Calculus provides the analysis in CT and brain imaging.
The Future of calculus would be dealing with application to biology over writhing number of DNA. The diameter of typical nucleus is about 5 millionth of a meter and is 400000 times smaller than DNA! That has to fit inside this smaller!! Not only this the alignment or arrangement is so neat that the DNA copy (multiplication) is flawless!!!
Non-Linearity is a factor that is solved by Calculus. For
instance, if you listen to your favourite songs again and again, the same
pleasure is not doubled as cause and effect are proportional.
Chaos over contour maps is discussed in detail followed by
role of computers in getting Artificial Intelligence to the fore. But the
eye-catching postulate of this book is when the author mentions Quantum Electron
Dynamics. QED merges Maxwell’s theory of Electricity & Magnetism with Heisenberg’s
& Schrodinger’s Quantum Theory and Einstein’s Special relativity, which turns
out to be a pleasant aspect when you close the book and imagine how we all are
enjoying this merger!
