Physics
Differential Calculus
Differential calculus is a branch of mathematics that deals with the study of rates at which quantities change. It involves the concept of derivatives, which represent the rate of change of a function with respect to its independent variable. In physics, differential calculus is used to analyze motion, rates of change, and gradients of physical quantities.
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11 Key excerpts on "Differential Calculus"
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
3 Differential Calculus This and the next chapter are concerned with the formalism of probably the most widely used mathematical technique in the physical sciences, namely the calculus. The current chapter deals with the process of differentiation whilst Chapter 4 is concerned with its inverse process, integration. The topics covered are essential for the remainder of the book; once studied, the contents of the two chapters serve as reference material, should that be needed. Readers who have had previous experience of differentiation and integration should ensure full familiarity by looking at the worked examples in the main text and by attempting the problems at the ends of the two chapters. Also included in this chapter is a section on curve sketching. Most of the mathematics needed as background to this important skill for applied physical scientists was covered in the first two chapters, but delaying our main discussion of it until the end of this chapter allows the location and characterisation of turning points to be included amongst the techniques available. 3.1 Differentiation • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Differentiation is the process of determining how quickly or slowly a function varies, as the quantity on which it depends, its argument , is changed. More specifically, it is the procedure for obtaining an expression (numerical or algebraic) for the rate of change of the function with respect to its argument. Familiar examples of rates of change include acceleration (the rate of change of velocity) and chemical reaction rate (the rate of change of chemical composition). Both acceleration and reaction rate give a measure of the change of a quantity with respect to time. However, differentiation may also be applied to changes with respect to other quantities, for example the change in pressure with respect to a change in temperature.
eBook - PDFA Course of Mathematical Analysis
International Series of Monographs on Pure and Applied Mathematics
- A. F. Bermant, I. N. Sneddon, S. Ulam, M. Stark(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
C H A P T E R III DERIVATIVES AND DIFFERENTIALS. THE Differential Calculus 1. The Concept of Derivative. Rate of Change of a Function 42. Some physical concepts. Let us consider the following simple physical phenomena: (1) rectilinear motion; (2) linear expansion of a mass; (3) heating of a body. To characterize these phenomena, the respective concepts have been introduced: (1) velocity, (2) den-sity, (3) specific heat; as it happens, all these represent particular aspects of the same concept from the mathematical point of view. We shall show this by analyzing each phenomenon separately. I. VELOCITY OF RECTILINEAR MOTION. Suppose that a body performs a rectilinear motion, and that we know the distance s tra-versed by the body after any given time t, i.e. we know s as a function of t: s = F(t) The equation s = F(t) is called the equation of motion, whilst the curve defined by it in the system of axes Ots is the graph of the motion. Let us consider the motion of the body in the course of an inter-val of time Δ t from the instant t to the instant t + Δ t. The body has traversed a distance s = F(t) after time t f and s + Δs = F(t + Δ t) after time t + Δ t. Thus in Δ t units of time it travels As = F(t +Δή - F(t). If the motion is uniform, s is a linear function of t: s = v 0 t + s 0 (Sec. 17). In this case Δβ = ν 0 Δί, and the ratio Δδ/Δί (= v 0 ) shows how many units of path s are traversed in unit time t ; the ratio remains constant here, i.e. depends neither on the instant t chosen, nor on 130 DERIVATIVES AND DIFFERENTIALS 131 As the increment of time A t. This constant ratio —r— is called the veloc-ity of the uniform motion*. If the motion is non-uniform, the ratio A sjA t of path to time depends both on t and on A t. It is called the average velocity of the motion in the time interval from t to t + A t and is written as v av : v av = As/At.
eBook - PDF- Jerry B. Marion(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
C H A P T E R 2 Vector Calculus 2.1 Introduction The application of vector methods to physical problems most frequently takes the form of differential operations. The rate of change of a vector function with respect to the spatial coordinates or with respect to the time are of particular importance. Such operations allow us, for example, to define the velocity vector of the motion of a particle or to describe the flow properties of a fluid. In this chapter we begin by defining the elemen-tary differential operations which immediately allow us to calculate the velocity and acceleration vectors in the commonly used coordinate systems. Angular velocity is considered next and this leads to a discussion of infinitesimal rotations. Treated next is the important differential operator, the gradient. The fact that the gradient operator may act on vector functions in different ways, leads finally to the divergence and the curl. The chapter concludes with a brief discussion of the simple integral concepts that are necessary in mechanics. 32 2.2 DIFFERENTIATION OF A VECTOR WITH RESPECT TO A SCALAR 33 2.2 Differentiation of a Vector with Respect to a Scalar If a scalar function φ = cp(s) is differentiated with respect to the scalar variable s, then since neither part of the derivative can change under a coordinate transformation, the derivative itself cannot change and must therefore be a scalar. That is, in the x f and x coordinate systems, φ = φ' and s — s', so that άφ = άφ' and ds = ds'. Hence, dcp/ds = d(p'/ds' = (άφ/ds)' (2.1) In a similar manner, we may formally define the differentiation of a vector A with respect to a scalar s.
eBook - PDF- G. M. Fikhtengol'ts, I. N. Sneddon(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER 5 DIFFERENTIATION OF FUNCTIONS OF ONE VARIABLE § 1. Derivative of a function and its computation 76. Problem of calculating the velocity of a moving point. Before proceeding to treat the foundations of the differential and integral calculus we draw the reader's attention to the fact that the ideas of calculus were originated as early as the seventeenth century, i.e. much earlier than the theories investigated in the preceding chapters. In the last chapter of this volume we shall survey the more important facts of the history of mathematical analysis and describe the merits of the two great mathematicians Newton and Leibniz, who completed the works of their predecessors by creating a really new calculus. In our discussion here we shall follow the modern demands of rigour, and not the history of the problem. As an introduction to the Differential Calculus we shall examine in this subsection the problem of velocity, and in the next subsection the problem of finding a tangent to a curve; both problems are historically connected with the formation of the basic concept of the Differential Calculus, which was later called the derivative. We begin by a simple example, namely we consider the free fall (in vacuum, when we can disregard the resistance of the air) of a heavy particle. If the time t (seconds) is measured from the beginning of the fall, the distance covered s (metres) is given by the well-known formula where g = 9.81 m/sec 2 . From these facts it is required to determine the velocity v of motion of the point at a given instant of time t, when the point is located at M (Fig. 31). [140] § 1. DERIVATIVE OF A FUNCTION 141 Introduce an increment At of the variable t and consider the instant t + At when the point is located at M ± . The increment MM X of the distance covered in the interval of time At we denote by As. Substituting into (1) t + At instead of t we obtain for the new value of distance the expression s+As = ^(t + Ai) whence As -^-(2t-At + At 2 ).
eBook - PDFPhysical Oceanography
A Mathematical Introduction with MATLAB
- Reza Malek-Madani(Author)
- 2012(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 3 Differential and Integral Calculus In this chapter we develop the essential concepts from differential and integral calculus and discuss the role they play in this text in the context of geophysical fluid dynamics. We will also use this opportunity to hint at the issues we will face when we need to approximate the typical rates of change that appear in the governing equations of motion. 3.1 Derivative The standard definition of the derivative of f , a function of one vari-able, at the point x = a is f prime ( a ) = lim h → 0 f ( a + h ) -f ( a ) h , (3.1) when that limit exists. Alternative ways of defining the same quantity are f prime ( a ) = lim h → 0 f ( a ) -f ( a -h ) h , (3.2) f prime ( a ) = lim h → 0 f ( a + h ) -f ( a -h ) 2 h , (3.3) or f prime ( a ) = lim h → 0 f ( a + 2 h ) + f ( a + h ) -2 f ( a ) 3 h , (3.4) which constitute just a few formulas, out of infinitely many such for-mulas, that lead to determining f prime ( a ). We use the concept of derivative primarily to relate the rates of growth of various variables in a physi-cal process. In this context it is not significant which of the definitions in (3.1)–(3.4) we use to develop our arguments. This choice becomes quite significant, however, in our second application of the definition of derivative, namely when we need to approximate f prime ( a ) by one of the many “rise-over-run” ratios on the right side of (3.1)–(3.4). In the con-text of solving differential equations, a subject we will take up in the 97 98 Physical Oceanography: A Mathematical Introduction next chapter, which one of the representations of f prime ( a ) in (3.1)–(3.4) is selected could have a significant impact on the accuracy of the numerical schemes one develops. Higher order derivatives of f are defined analogously, by applying the formulas in (3.1)–(3.4) to lower order derivatives. For example, f primeprime ( a ) is determined as f primeprime ( a ) = lim h → 0 f prime ( a + h ) -f prime ( a ) h .
eBook - PDF- David H. Eberly(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
C h a p t e r 9 Calculus T his appendix provides a brief summary of topics in calculus that you should be familiar with in order to fully understand how to model a physical system and implement the physical simulation on a computer. Calculus occurs in two flavors, Differential Calculus and integral calculus. Both disciplines are founded on the concepts of infinitesimal quantities and a limit, the measurement of what happens to a quantity as one or more parameters are varied. Calculus involves processing functions, the topic further subdivided based on the number of independent and dependent variables. Univariate calculus studies func- tions y = f (x ), where x is an independent variable and y is the dependent variable. Formally, the function is written as f : D → R, where D ⊂ IR is the domain of the function and R ⊂ IR is the range of the function. To be somewhat loose with the notation, an emphasis will be placed on the sets containing the domain and range by writing f : IR → IR. The domain and range are most likely proper subsets of IR, but those will be known within the context of the problem at hand. Multivariate calculus studies functions y = f (x 1 , ... , x n ), where x 1 through x n are n independent variables and y is a single dependent variable. The function may be written as f : IR n → IR, where IR n denotes the set of n-tuples of real numbers. As indi- cated in the last paragraph, the domain of f may be a proper subset of IR n and the range of f may be a proper subset of IR. The next natural extension is to study a collection of functions y i = f i (x 1 , ... , x n ) for 1 ≤ i ≤ m. We now have n independent variables and m dependent variables. Using vector notation, let Y = (y 1 , ... , y m ), X = (x 1 , ... , x n ), and F = ( f 1 , ... , f m ). The function may be written as Y = F(X), or F : IR n → IR m . This coordinate-free represen- tation looks just like the univariate case where n = m = 1.
eBook - PDFA Mathematical Bridge
An Intuitive Journey in Higher Mathematics
- Stephen Hewson(Author)
- 2009(Publication Date)
- WSPC(Publisher)
Chapter 5 Calculus and Differential Equations In 1687 Newton published his monumental work ‘The Mathematical Princi-ples of Natural Philosophy’. In the opening of this great book is a statement of Newton’s second law of motion, which may be paraphrased as follows: force equals mass times acceleration. It is remarkable that a physical law so simple to express would give rise to a branch of mathematics which would prove to be so essential to the further development of most areas of mathe-matics and the natural sciences. This mathematics is the theory of calculus and differential equations. In addition to their interest from the point of view of beautiful mathematics, differential equations are essential tools in the study of disciplines as diverse as economics and the biological sciences. We begin this chapter by motivating the concepts of differentiation and dif-ferential equations through a discussion of Newton’s second law of motion, rediscovering and extending the ideas of calculus discussed in the study of analysis. 5.1 The Why and How of Calculus Newton’s law of motion has a virtually limitless number of useful applica-tions, from ballistics to planetary dynamics. To obtain an equation for the position of a body in space at each point in time from Newton’s second law we need to probe the relationship between acceleration and position. 5.1.1 Acceleration, velocity and position What is acceleration? If a car travelling at a velocity of v ( t 0 ) miles per hour at a time t 0 smoothly increases its velocity to v ( t 1 ) miles per hour at a time t 1 then the magnitude of its acceleration a over this time period 289 290 A Mathematical Bridge is given by the change in velocity δv = v ( t 1 ) − v ( t 0 ) divided by the time δt = t 1 − t 0 taken to make this change: a = δv δt This is an exact formula because the car was accelerating uniformly over the time between t 0 and t 1 .
eBook - PDF- Tom M. Apostol(Author)
- 2019(Publication Date)
- Wiley(Publisher)
We begin with a function f defined at least on some open interval (a, b) on the x-axis. Then we choose a fixed point x in this interval and introduce the difference quotient f (x + h) − f (x) h , where the number h, which may be positive or negative (but not zero), is such that x + h also lies in (a, b). The numerator of this quotient measures the change in the function when x changes 160 Differential Calculus from x to x + h. The quotient itself is referred to as the average rate of change of f in the interval joining x to x + h. Now we let h approach zero and see what happens to this quotient. If the quotient approaches some definite value as a limit (which implies that the limit is the same whether h approaches zero through positive values or through negative values), then this limit is called the derivative of f at x and is denoted by the symbol f ′ (x) (read as “f prime of x”). Thus, the formal definition of f ′ (x) may be stated as follows: definition of derivative. The derivative f ′ (x) is defined by the equation f ′ (x) = lim h→0 f (x + h) − f (x) h , (4.4) provided the limit exists. The number f ′ (x) is also called the rate of change of f at x. By comparing (4.4) with (4.3), we see that the concept of instantaneous velocity is merely an example of the concept of derivative. The velocity v(t) is equal to the derivative f ′ (t), where f is the function which measures position. This is often described by saying that velocity is the rate of change of position with respect to time. In the example worked out in Section 4.2, the position function f is described by the equation f (t) = 144t − 16t 2 , and its derivative f ′ is a new function (velocity) given by f ′ (t) = 144 − 32t. In general, the limit process which produces f ′ (x) from f (x) gives us a way of obtaining a new function f ′ from a given function f. The process is called differentiation, and f ′ is called the first derivative of f.
eBook - PDFQuick Calculus
A Self-Teaching Guide
- Daniel Kleppner, Peter Dourmashkin, Norman Ramsey(Authors)
- 2022(Publication Date)
- Jossey-Bass(Publisher)
CHAPTER TWO Differential Calculus In this chapter you will learn about • The concept of the limit of a function; • What is meant by the derivative of a function; • Interpreting derivatives graphically; • Shortcuts for finding derivatives; • How to recognize derivatives of some common functions; • Finding the maximum or minimum values of functions; • Applying Differential Calculus to a variety of problems. 2.1 The Limit of a Function 97 Before diving into Differential Calculus, it is essential to understand the concept of the limit of a function. The idea of a limit may be new to you, but it is at the heart of calculus, and it is essential to understand the material in this section before going on. Once you understand the concept of limits, you should be able to grasp the ideas of Differential Calculus quite readily. Limits are so important in calculus that we will discuss them from two different points of view. First, we will discuss limits from an intuitive point of view. Then, we will give a precise mathematical definition. Go to 98. 57 58 Differential Calculus Chap. 2 98 Here is a little bit of mathematical shorthand, which will be useful in this section. Suppose a variable x has values lying in an interval with the following properties: 1. The interval surrounds some number a. 2. The difference between x and a is less than another number B, where B is any number that you choose. 3. x does not take the particular value a. (We will see later why this point is excluded.) The above three statements can be summarized by the following: |x − a| > 0 (This statement means x cannot have the value a.) |x − a| < B (The magnitude of the difference between x and a is less than the arbitrary number B.) These relations can be combined in the single statement: 0 < |x − a| < B. (If you need to review the symbols used here, see frame 20.) The values of x which satisfy 0 < | x − a | < B are indicated by the interval along the x-axis shown in the figure.
- Will H. Moore, David A. Siegel(Authors)
- 2013(Publication Date)
- Princeton University Press(Publisher)
Part II Calculus in One Dimension Chapter Five Introduction to Calculus and the Derivative In our experience, calculus and all things calculus-related prove the most stress- ful of the topics in this book for those students who have not had prior calculus coursework. We conjecture that this is due to the foreignness of the subject. While probability and linear algebra certainly have some complex concepts one must internalize, much of the routine manipulations students perform in ap- plying these concepts use operations they are used to: addition, multiplication, etc. In contrast, calculus introduces two entirely new operators, the derivative and the integral, each with its own set of rules. Further, these operators are often taught as a lengthy set of rules, leading to stressful rote memorization and little true understanding of what are relatively straightforward concepts, at least as used in most of political science. 1 To try to avoid this, we’re going to take a little more time with the topic. In this chapter we will cover the basics of Calculus and the derivative in what we hope is an intuitive manner, saving the rules of its use for the next chapter. If you are working through this chap- ter as part of a course and are not sure of something, this is the time to ask questions—before you end up trying to take derivatives without having a clear understanding what they are. The first section below provides a brief overview of calculus. The second section introduces the derivative informally, and the third provides a formal definition and shows how it works with a few functions. 5.1 A BRIEF INTRODUCTION TO CALCULUS For our purposes, the primary use of calculus is that it allows us to deal with continuity in a consistent and productive manner. This is likely a useless claim at this point, so let us explain. As we discussed in Chapter 4, a continuous function is one that we can draw without lifting pencil from paper.
eBook - PDFA Course of Higher Mathematics
Adiwes International Series in Mathematics, Volume 1
- V. I. Smirnov, A. J. Lohwater(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER II DIFFERENTIATION: THEORY AND APPLICATIONS § 3. Derivatives and differentials of the first order 45. The concept of derivative. We consider a point moving in a straight line. The path s traversed by the point, measured from some definite point of the line, is evidently a function of time t: A corresponding value of s is defined for every definite value of t. If t receives an increment At, the path s -f-As will then correspond to the new instant t + At, where As is the path traversed in the interval At. In the case of uniform motion, the increment of path is proportional to the increment of time, and the ratio As/At repre-sents the constant velocity of the motion. This ratio is in general dependent both on the choice of the instant t and on the increment At, and represents the average velocity of the motion during the interval from t to t + At. This average velocity is the velocity of an imaginary point which moves uniformly and traverses path As in time At. For example, we have in the case of uniformly accelerated motion: s = — gt 2 + v 0 t and As -^-g(t + At)* + v 0 (t + At)--Lgt*-v 0 t -ÂT= : Ät = i * + *o + -2-fl^. The smaller the interval of time t, the more we are justified in taking the motion of the point in question as uniform in this interval, and the limit of the ratio As/At, with At tending to zero, defines the velocity v at the given instant t : T As v = lim — rr · 101
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