Topic 1-5: Poor physics student's calculus guide
Calculus, in the context of differentiation and integration, is an essential means of physics, and classical mechanics is not an exception. However, the portion of calculus needed in physics is not the high-powered abstract stuff. Thus we do not need to worry about d-e description, the uniqueness proof of integration, the proof of Simpson's method and the proof of Sterling's formula. This guide is attempted to give a quick boost to students who have not learned calculus but have been pushed into calculus-based physics so that they can also tackle many standard problems included in this book. This guide will also serve as a tool for those who want a quick review of calculus. The prerequisite to this guide is a sound knowledge of high school algebra, especially the concept of functions, and some basic trigonometry.
1. Continuous Functions
Let us review functions first. Instead of reciting the formal mathematical definition
of a function, we begin with some actual examples of functions.
f(x) = 2x + 1
is a function. It means that
if x = 1, then f(1) = 2 x 1 + 1 = 3,
if x = 2, then f(2) = 2 x 2 + 1 = 5,
if x = -3.24, then f(-3.24) = 2 x (-3.24) + 1 = -5.48,
and so on. In other words if a value is assigned to x, f(x) can always be computed from the given formula of the function.
Now suppose x approaches a value, say 2. We consider two different ways for x to approach 2. One is for x to approach 2 from somewhat larger values than 2; for example, x = 2.1, x = 2.09, x = 2.08, ... toward x = 2. In this case we say that x approaches 2 from above and use a notation x → 2+ to indicate this approach. Obviously the other approach is for x to approach 2 from below, that is, from the values persistently smaller than 2. We denote the second approach as x → 2-. As x approaches 2 from above, f(x) also approaches some value. We indicate this value to which f(x) approaches as
Similarly when x approaches 2 from below, f(x) approaches a value that is denoted as
In this function it is obvious that
When the condition in the above equation is satisfied, we say that f(x) is continuous at x = 2. In general for any point x0, if
we say that f(x) is continuous at x0. If f(x) is continuous at any point x0 that satisfies the condition a ≤ x0 ≤ b, we say that f(x) is continuous in the range
is continuous everywhere except at the point x = 0. When x approaches 0, the function f(x) behaves as
Thus the function is not continuous at x = 0. The approximation of the graph representd by the function is again plotted in the figure at the right. This graph is called a hyperbola.
is not continuous at x = 0 since
The graph of this function is presented in the figure at the right.
This can also be written as
There are several simple rules for the limiting process. Let f(x) and g(x) be
continuous at point x = c. Then
2. DifferentiationConsider a function γ(x,Δx) defined as
The following figure illustrates the meaning of γ(x,Δx) .
From the left-hand picture it is clear that
As Δx→0, point B approaches point A, and the line AB approaches a line that is denoted as AP in the right-hand picture of the above figure; the line AP contacts the curve f(x) at point A, or, in a simplified way of saying, at point x. Point C also approaches point A along a horizontal line denoted as AQ. As Δx→0, tan(θ) approaches the slope of line AP. Since γ(x,Δx) always equals tan(θ) during this limiting process, it also approaches the slope of the contact line AP. We denote the limit of tan(θ) when Δx→0 as f '(x) and call it the differentiation of function f(x) at point x, just the differentiation of f(x), or the derivative of f(x). We also use the notations
interchangeably with f'(x). In summary we have
3. Examples of Differentiation
In physics the usage of calculus is not so much proofs of theorems as the repeated practice of a limited number of prederived formulas. Let us consider such frequently used equations here.
(1) f(x)=c, where c is a constant. Then
Since f(x)=c is just a horizontal line intersecting the vertical axis of f(x) at the value of c, all its contact lines are horizontal lines and thus have slope 0 as the above equation indicates.
(3) f(x)=x2 .
We list some general formulas for differentiation without derivation.
. . . . .
The following relation in differentiation is also very useful:
Let us illustrate the usage of this relation.
The following group of relations is also very useful, and is listed here without derivation.
We illustrate an example of the last relation:Since
4. Integration with One Variable
Integration is in essence a means to calculate the area under a curve. Consider the following picture.In the left-hand picture we want to calculate the shaded area. We divide the interval from a to b into N narrow strips each of which has a width Δx defined asThe summation over the area of each narrow rectangle as shown in the right-hand picture will give us an approximation to the area under the curve. Apparently this approximation becomes better and better as N→∞ or Δx→0. Let us denote the area that we want to calculate as I. Then using the summation notation Σ, we can writewhere each term inside the summation symbol represent the area of a narrow rectangle. This limiting process has a special symbol called the definite integral of function f(x) from a to b, and is written asWhen this definition of integral is combined with the concept of differentiation, the integral becomes calculable. Suppose we haveAccording to the definition of differentiation, we haveWhen Δx is very small, it is a good approximation to just writeDo the same for f(a+Δx), and we will haveThusContinuing for f(a+2Δx), f(a+3Δx) …, we will haveSincewe can write as an approximationThis approximation also becomes better and better as Δx→0. Substituting this approximation into Eq. (1), we haveIn summary we get
Now let us work on some simple examples to show how Eq. (2) works.
(a) (b) (c) (d) (e) (f) (g) (h)The variable of integration, x, in Eq. (1) can be changed to another variable, say z. Supposethen Sometimes the boundaries of integration is not needed, and we can just writeIt is called the indefinite integral.
5. Partial Differentiation and Multivariable IntegrationWhen a function has more than one independent variables, say f(x, y), the differentiation can be performed on one of the variable with the other variable just regarded as a constant.In the case of f(x, y) the differentiation can be performed either with regard to x or y. Such differentiation is called "partial differentiation" and carries the notation of Thus if Multiple integration is the natural extension of the one-variable integration. For example, the integration is clearly a function of x; let us call it g(x). Thus Then If the integration boundaries, a, b, c and d are just constants, the order of integration are interchangeable, so If then
6. Differential EquationsIn physics the most important equations of motion take the form of differential equations, that is, equations involving differentials. The methods to solve differential equation form an important branch of applied mathematics. Here we will not go into details of solving differential equations, but only consider some simple examples.
The equationis the simplest first order linear differential equation; first order means that it only contains one order of d/dx. If with c as an arbitrary constant is the general solution of the differential equation. To determine the arbitrary constant c we need some initial conditions; for example, if f(0)=3, then c=3, so f(x)=3 is the completely determined solution of the differential equation.
The second example isThe general solution iswhere c is the arbitrary constant needed to be determined by an initial condition.
The third example isObserve that if then Substituting this into the equation yields Thus f(x)=Aekx is the general solution with A as the arbitrary constant needed to be determined by an initial condition. In essence even if a differential equation is given, we need some initial conditions in order to determine the solution completely. This also means that even if two motions have exactly the same equation of motion, the solutions can be quite different if their initial conditions differ. More complicated differential equation will be encountered in physics but will be dealt with when they emerge.
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