**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.

__Example 1__

f(*x*) = 2*x* + 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

In this function it is obvious that

When the condition in the above equation is satisfied, we say that f(

we say that f(

a ≤

Sometimes we write

and plot the function as a graph as shown in the figure at the right. In principle a value is given to

The function

The function

is continuous everywhere except at the point

Thus the function is not continuous at

The function

is not continuous at

The graph of this function is presented in the figure at the right.

In general we can rephrase the continuity condition of a function as follows:

Let

When a function is continuous at some point

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. Differentiation**

The following figure illustrates the meaning of *γ(x,Δx)* .

From the left-hand picture it is clear that

Thus

As

interchangeably with

.

**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.

Examples:

(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.

(2) *f(x)=x*.

.

(3) *f(x)=x ^{2}* .

.

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

we have

4. Integration with One VariableIntegration 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 froma to binto N narrow strips each of which has a width Δxdefined 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 asN→∞ or Δx→0. Let us denote the area that we want to calculate asI. 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 functionf(x)fromatob, 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 haveWhenΔxis 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 asΔx→0. Substituting this approximation into Eq. (1), we haveIn summary we getNow 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, sayz. 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, sayf(x, y), the differentiation can be performed on one of the variable with the other variable just regarded as a constant.In the case off(x, y)the differentiation can be performed either with regard toxory. 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 ofx; let us call itg(x). Thus Then If the integration boundaries,a, b, canddare 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 equation

is the simplest first order linear differential equation; first order means that it only contains one order ofd/dx. If withcas an arbitrary constant is the general solution of the differential equation. To determine the arbitrary constantcwe need some initial conditions; for example, iff(0)=3, thenc=3, sof(x)=3is the completely determined solution of the differential equation.The second example is

The general solution iswherecis the arbitrary constant needed to be determined by an initial condition.The third example is

Observe that if then Substituting this into the equation yields Thusf(x)=Aeis the general solution with^{kx}Aas 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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