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) = 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
a ≤ x ≤ b; we sometimes use the notation [a, b] to indicate the range a ≤ x0 ≤ b. For the function f(x) of this example, it is continuous in the range [-∞, +∞], i.e. it is continuous everywhere.

Sometimes we write

       y = f(x) = 2x + 1,

and plot the function as a graph as shown in the figure at the right. In principle a value is given to x, and we then compute the corresponding y to form a pair (x, y). This pair determines a point in the x-y plane of the figure. By assembling infinite number of such pairs, a trace, called the graph of the function, emerges. In a practical sense we, of course, cannot calculate an infinite number of such pairs. Fortunately, we know from high school algebra that the function given as y = 2x + 1 represents a straight line, so we only need to compute two pairs to determine two points that the line will pass through, and then the line can be drawn easily by connecting those two points. In the figure shown, we have used the points (1, 3) and (0, 1) as the two special points.


Example 2

The function y = f(x) = x2 is continuous everywhere. It represents a parabola as shown in the figure at the right. To draw the parabola, we draw some points on the graph by computing their corresponding (x, y) pairs, and then connected those points by straight lines. Of course, the actual parabola represented by the function is a smooth curve, but not very different from the approximation shown in the figure.










Example 3

The function
       
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.


Example 4

The function
       
is not continuous at x = 0 since
       
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 x be a variable and f(x) a function of x. We say that f(x) is continuous in the range a≤x≤b (denoted as [a, b]) if f(x)→f(x0) as xx0 for all x0 in the range [a, b].

When a function is continuous at some point x0, it does not matter whether x approaches x0 from above or below, so the notation + or - in the limiting process can be dropped. This continuity statement can then be rewritten using the obvious notation of limiting process as
        
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

Consider 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

        
Thus
        
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.

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)=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
we have



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 as
          
The 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 Δx0. Let us denote the area that we want to calculate as I. Then using the summation notation Σ, we can write
          
where 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 as
          
When this definition of integral is combined with the concept of differentiation, the integral becomes calculable. Suppose we have
          
According to the definition of differentiation, we have
          
When Δx is very small, it is a good approximation to just write
          
Do the same for f(ax), and we will have
          
Thus
          
                        
Continuing for f(a+2Δx), f(a+3Δx) …, we will have
          
                                           
Since
          
we can write as an approximation
          
This approximation also becomes better and better as Δx→0. Substituting this approximation into Eq. (1), we have
          
                                          
In 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. Suppose
         
then
         

Sometimes the boundaries of integration is not needed, and we can just write
         

It is called the indefinite integral.


5. Partial Differentiation and Multivariable Integration

When 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 Equations

In 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 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 is

         

The general solution is
         

where c is 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

         

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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