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Calculus K M Golden First Lecture What is calculus about and why should I study it The story of calculus began in the late 1600 s with the revolutionary results of Isaac New ton 1642 1727 and Gottfried Leibniz 1646 1716 in understanding motion and rates of change Some of the underlying concepts however such as the infinite and the in ni tesimal were thought about and articulated by the ancient Greeks Zeno and Archimedes well over 2000 years ago The word calculus itself comes from Latin and means a small stone or pebble used in gaming voting or reckoning It is hoped that after you have used this book you will realize that this very long story is still being written and is even more vibrant and essential today than it was over 300 years ago With historical hindsight it can be said that the development of calculus is certainly one of the greatest intellectual achievements of the past two millenia Without calculus most of the incredible advances in science and engineering which occurred in the twentieth century and have become part of everyday life such as air and space travel television computers weather prediction med ical imaging nuclear bombs wireless phones the internet microwave ovens etc could not have happened Calculus provides the language and basic concepts used to formulate most of the fundamental laws and principles of the various disciplines throughout the physical mathematical biological economic and social sciences as well as electrical mechanical computer bio civil and materials engineering Of course within mathematics calculus serves as the inescapable gateway to all higher level courses Moreover historically it is the seed which gave rise to many of the principal branches of mathematics which themselves often have deep links to important areas of application throughout the sciences and engi neering Calculus is also an essential tool used widely thoughout business and industry such as in the nancial insurance transportation manufacturing and pharmaceutical in dustries and in the development of computer communications and medical technologies The role of mathematics and calculus in particular in serving as the operating system77 of science and engineering cannot be underestimated It is for these reasons that students majoring in elds throughout the sciences engineering medicine and business as well as in mathematics are required to take calculus Clearly the better you know the language of a country you may wish to visit the more likely you ll be successful in meeting com municating with and learning from the people there Similarly the better you know the operating system of the eld in which you intend to work the better positioned you will be to succeed be it in doing well in subsequent classes spoken in the language of calculus or in formulating and solving problems in a job situation The main objects of study in calculus are functions such as x m2 gz sinz and 2x One of the central questions which calculus addresses is to understand what we mean by the rate of change of a function f locally at a point m In other words how much does 1 change for small changes or differences in m and how can this be measured precisely For functions whose graphs are straight lines such as 2x the answer is provided by the slope of the line which is 2 for Mm and is the same rate of change for all m For example if we start at z 1 or any m and increase the value of x by 110 to z 11 then the value of k correspondingly increases by twice as much or 15 For functions whose graphs are not straight lines such as x 2 and gz sinz the calculus It is fascinating that the global question about the area under x on an interval is intimately connected to the seemingly unrelated question about the local rate of change of 1 described above This deep and perhaps unexpected connection between the two great problems of calculus is one of its most important results called the fundamental theorem of calculus Generalizing the notion of the integral as well as the fundamental theorem to more than one variable again leads to many new and important ideas used throughout higher mathematics the sciences and engineering As you can see calculus provides us with a framework to analyze the most basic and essential properties of functions The reason calculus has such a monopoly in describing our world is that almost any quantitative model of a physical chemical biological engineering industrial or financial system involves the use of functions Moreover almost any type of analysis of these functions used for understanding or predicting the behavior of the system particularly how it evolves in time will invariably involve calculus This is not surprising however given the basic and fundamental nature of the tools developed in calculus For example in physics and engineering one regularly encounters functions describing the posi tion of a particle or object as a function of time t such as t for motion in one dimension The function Mt might represent the height of a falling rock or the position of a mass hanging by a spring which is stretched and then let go As we will see very soon the derivative z t is what we normally refer to as the velocity vt z t of the object or the rate of change of the position Particular attention will be paid in this book to motion examples not only due to their fundamental importance but because most everyone has a lot of experience with motion and velocity in everyday life and hopefully a good intuition about it In fact in this context it is natural to consider the rate of change or derivative 1t of the velocity vt which we know as the acceleration at 1t from common experience in cars airplanes and amusement park rides such as the thrill of going from 0 to 60 mph in less than 5 seconds in a sports car or of being steam catapulted off the deck of an aircraft carrier In terms of the original function zt the derivative is taken twice so that the acceleration at 1t m t is the second derivative of the position Other examples of functions encountered in science and engineering include the current 25 in an electrical circuit perhaps being used in a computer or in a radar or MRI system the average temperature Tz in a region of the ocean as a function of depth 2 the size Nt of a rapidly growing viral population the electric field strength Et received by a wireless phone during a call or by the cornea during laser eye surgery the sound pressure Pt received by your ear during a favorite song the uid velocity 12z y in a cross section of a pipe artery or tornado or on the surface of an airplane wing or submarine hull the gene activity function Ag along a DNA strand measuring how many times a gene 9 is transcribed or called during a cellular process a company s stock price St or total revenues Rt the density of water pT as a function of its temperature T particularly near where water boils at 100 C the surface air pressure pmyt over the United States as a function of time or the thickness Hzyt of the Antarctic ice sheet with time the concentration Ct of salt during a chemical reaction the electrical action potential Vt of a neuron the amount of strain ez y 2 produced in a material under stress and the wave function m y z describing the quantum state of an electron in an atom As mentioned above calculus provides the language and concepts used to formulate most of the fundamental laws and principles of science and engineering This is because these laws and principles are usually and most naturally expressed locally in terms of rates of change or dervatives or globally in terms of integrals or in some cases in terms of derivatives and integrals If a law is expressed locally which is most common then it typically has the form of an equation involving the derivatives of a function as well as perhaps the function itself These equations are called differential equations For example nding a function yt that satis es the differential equation y t yt or nding a function t that satis es m t 4mt O are interesting and important problems Just as with calculus itself it is difficult to underestimate the importance of differential equations They play the central role in expressing the fundamental principles and laws governing our world as well as in understanding and predicting the behavior of much of what is in it be it naturally occurring or man made A principal reason for the importance of differential equations in the description of our world is that Newton s famous law F ma or Force mass x acceleration which underlies so much of our understanding of motion and dynamics is a differential equation The acceleration at 1t m t is the second derivative of the position Such differential equations involving the second derivative but no higher derivatives and perhaps the rst derivative and the function itself are called second order differential equations like m t 4mt O or m t 732 Can you solve either of these equations Of course equations involving only the rst derivative and the function itself like y t yt are called rst order differential equations Because so much of the description of our world rests on Newton s laws as well as on other laws related to or depending in some way on Newton s framework second order differential equations occupy a special place throughout mathematics the sciences and engineering The pervasiveness of this second order structure has dictated to some extent which types of differential equations have received the most attention from theoretical computational and experimental researchers over the past couple hundred years Another striking example of how mathematics and multivariable calculus and differen tial equations in particular facilitates the quantitative description of our world is Maxwell s equations In 1865 James Clerk Maxwell saw a fundamental inconsistency in a set of dif ferential equations describing the basic phenomena of electromagnetism This set of second order differential equations expressed mathematically the known experimental facts at the time By adding a missing term to the equations and transforming them into a consistent set he was able to make the startling prediction that light is an electromagnetic wave and that electromagnetic waves of all frequencies could be produced This discovery spurred a tremendous amount of experimental and theoretical research on electromagnetism in the late 1800 s and laid the groundwork for much of what we take for granted in the 21st cen tury lndeed Maxwell s equations provide the framework in which all electromagnetic wave phenomena are analyzed such as the propagation re ection and scattering of light radar microwaves x rays and lasers and they govern the propagation of electrical nerve impulses and electrical activity in the heart and brain as well It is amazing that this beautiful set of equations can be viewed as a statement appropriate to the electromagnetic eld of the fundamental theorem of calculus and related results about functions of several variables Further generalizations and applications of these ideas to studying the geometry of higher dimensional surfaces called manifolds eventually came to scientific fruition when they served as the mathematical foundation of Einstein s theories of relativity These startling developments in the understanding of our world and in creating the mathematical framework for analyzing it laid the groundwork for more surprising advances Ernest Rutheford used basic ideas in calculus to investigate the internal structure of the atom which culminated in the discovery of the atomic nucleus He was awarded the Nobel Prize for Chemistry in 1908 for this work Further investigations of atomic phenomena revealed that in many experiments matter such as electrons exhibit wavelike behavior which led to the development of the theory of quantum mechanics Since it s generally more difficult to specifically localize a wave such a realization shifted the point of View away from solving Newton s deterministic differential equations for the particle position zt often referred to as classical mechanics at least for atomic scale phenomena A new function Izt called the wave function became the focus of interest For an electron in one dimension it describes among other things the probability of nding the electron in an interval ab at time t In fact this probability is found with an integral In 1926 Erwin Schrodinger developed his famous wave equation for m t for which he later won the Nobel Prize for Physics in 1933 This second order differenential equation has connections to Newton s law yet is also similar to the classic equation of heat conduction except it involves the number i H which stands for imaginary not impossiblel lts solutions can then exhibit wave behavior The development of quantum theory laid the foundation for the discovery of DNA the transistor semiconductors the materials used to make the brains of current computers and mobile phones nuclear weapons MRI imaging and radiation treatment for cancer There are myriad other examples where mathematics and calculus and differential equa tions in particular has enabled a major scientific or technical advance One of the inter esting ways in which this happens is through the transference of ideas and techniques that are developed in one field of study and are then subsequently applied to another The reason this is possible is because the language and methods of mathematics and calculus in particular are universal If they work on one function of a certain type or which satis es a certain type of equation then similar methods will usually work on another function of that type even if it comes up in a completely different eld For example mathematical techniques used to study air ow over an airplane or insect wing can be used to study the ow of blood in the heart as the underlying second order differential equations are the same In chemistry one often needs to solve Schrodinger s equation for the wave function describing systems with many interacting particles as well as with only a few The ideas behind techniques developed to exactly calculate or more realistically approximate the so lutions are often related to techniques developed to solve similar problems throughout the sciences Such problems include studying the propagation of light microwaves or sound through media with many interacting scatterers like the atmosphere ocean or the human body and the behavior of Brownian motion and diffusion processes like the motion of a pollen grain on a droplet of water If a mathematical theory based heavily on calculus was developed originally to understand the ow of electrons through semiconductors the basic building blocks of silicon based computers yet happens to also shed light on almost the same functions describing uid ow through sea ice arising in the study of bacteria and algae living inside the salty liquid inclusions lacing the ice which covers the polar oceans of earth so much the better Similarly harmonic spectral or Fourier analysis a branch of mathematics which arose out of calculus originally in connection with solving the second order differential equation for heat conduction perhaps gained most prominence in the late 1800 s by providing a mathematical foundation for the frequency or spectral analysis of light sound and music Fourier analysis and related areas of mathematics now lie at the heart of many types of medical imaging digital sound and video recordings oil and mineral recovery climate analy sis electrical brain wave analysis image compression for transmission across the internet design of components in telecommunications networks remote sensing of the earth from satellites as well as a broad range of techniques used in scientific computing and algorithm design It is interesting to note that the heat equation has also provided the foundation for a theory of pricing options on a stock which are called derivative securities or financial derivatives77 since their prices depend on the underlying stock price and how it uctuates sometimes seemingly randomly over small time scales as well as perhaps makes larger moves over longer time scales Fischer Black and Myron Scholes found through mathematical ana lyis that option prices typically obey a type of heat or diffusion equation now called the Black Scholes model for which the Nobel Prize in Economics was awarded in 1997 The operating system for analyzing functions provided by calculus is what allows a particular set of ideas such as Fourier analysis and the heat equation to achieve such wide in uence From experience with personal computers you know that any application which works on one computer running Windows or MAC OS or Linux as its operating system will in principle work on any other computer running the same operating system even if that computer is on the other side of the world down the hall or on a trip to Jupiter It is this type of standardization as well as standardization of protocols for how computers communicate with one another which led to the explosion of personal computers and the internet in the late 1900 s Similarly calculus standardized how functions in any field are quantitatively analyzed in terms of how to calculate rates of change and integrals Calculus provided a universal platform upon which fundamental laws can be formulated and analyzed as well as a language through which scientists in different elds studying similar types of functions could communicate and transfer methods of analysis and solution The development of calculus gave rise to an explosion of inquiry and ndings throughout the sciences and engineering These advances most certainly enabled the Industrial Revolution of the early 1900 s as calculus nally provided a systematic way to study the dynamic and not just the static the dawn of the Nuclear Age in the 1940 s as well as the rise of the Information Age in the late 1900s As a vivid illustration of the impact of these advances suppose that Newton or his estate could have received licensing fees or royalties from the sale of all the products and services whose creation used the derivative the integral or his related laws of motion in any way After all Microsoft receives fees for the use of its operating system Windows For example as we have pointed out all electromagnetic wave phenomena are analyzed and exploited commercially within the framework of Maxwell s di erentz39al equations which rest upon Newton s framework The entire telecommunications and internet industries owe their very existence to the development of calculus as well as many subsequent breakthroughs such as Maxwell s equations More generally Newton effectively provided us with a platform upon which the bulk of the technical infrastructure of our modern world has been built Newton s fortune would then dwarf that of Bill Gates co founder of Microsoft many times over even if Newton s cut were 01 lndeed part of Microsoft s revenues would have to be paid in fees to Newton s estatel From the above discussions you now have some idea what calculus is about and why it s very important to study and understand it well Hopefully you have glimpsed a bit of the depth and gravitas of calculus although at this point you are only seeing the tip of this iceberg As we progress I will try to show you what lies beneath the topics we re studying how they have laid the groundwork for major advances throughout science and engineering and where they might lead to in higher mathematics Hopefully you will come to realize that the light of calculus penetrates even into the most remote reaches of our world illuminating understanding and providing a framework in which one may ask and answer quantitative questions about how and why things work the way they do Per ardua ad astra