Mathematical Preliminaries

 

Since the physics courses for engineers at the university level are calculus based you need to be familiar with its basic concepts such as limit, derivative and integral. I will give short explanations for them. These will not satisfy a mathematician but hopefully will be useful in this course.  

 

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Limit

 

Limit of a function f(x) at x = x0 is the value, to which the function approaches, as x approaches x0. We show it as

 

 

This looks like a very simple concept. Usually it is just the value f(x0). However often it saves us from many situations like  or . This is what makes it useful.

 

Lets look at the simple function,

 

f (x)= 1

 

This function has value 1 everywhere and at x=0. As x approaches 0, f(x) stays at 1. So we say that the limit of this function at x=0 is 1. The value of the function f(0) is also 1.

 

Now lets look at,

,

 

Here, if could cancel the x's above and below the line we would get f(x)=1 again, however at x=0 we can not divide by x. The formula gives 0/0. Now the limit to the rescue. We take a series of x's each smaller than the rest but none of them really 0. For each x in the series, f(x) gives 1. In this case

 

 

even though the function f is undefined at x=0.

 

Similarly,

 

is undefined (0/0) at x=1. For we can cancel (x-1) from top and bottom and get

 

 for .

 

with x very close to but not really equal to 1 we see that f(x) approaches 2 as x approaches 1.

we say that

 

.

 

Sometimes even the limit does not exist. For example  approaches -1 and +1 as we approach x=0 from left and right. Similarly  does not have a limit value as x approaches 0. Fortunately we will not meet these bad functions in Physics I or II.

Derivative

 

Derivative of a function, f, shows the slope of its curve. It gives how much it changes when its variable is changed by a small amount. If the variable is x it is shown as,

 

or sometimes as

      

If the variable is t then it is shown as,

or sometimes as

                 

 

For example a road going up the hill climbs y=1m for every 20m of horizontal distance, x, traveled. Then the derivative is 1/20. Hence it is the slope of the road. The derivative always gives the slope of a curve.

 

 

 

 

 

In the curve below the slope is not constant. The lines ad, ac and ab all have different slopes. These are the average slopes of the curve in these intervals. When we talk about the slope of the curve at point, a, we mean the ratio of the height difference to horizontal distance for a very very small region around a. Since we know limits we can do this. It is the limit of the slope of the triangle as its base goes to 0.

 

 

 

 

In physics we meet derivatives often. Velocity, v, is the derivative of displacement,  and acceleration, a, is the derivative of velocity, .

 

Some examples:

 

1)        

           

           

           

           

 

Note that letting D go to 0 is postponed until cancelling the D factors from the numerator (top) and denumerator (bottom). After the cancellation there is no 0/0 type indeterminacy and doubt about the result. Patience pays off.

 

2)         y= Axn

                  

 

Here O(Δ2) means terms as small as or smaller than the square of the small quantity Δ. Neglecting these small terms,

 

 

 

 

3)         y= sin(ωt)

 

 

Again neglecting the small terms so that
sin(ωΔ)=ωΔ and cos(ωΔ)=1 we get

 

 

4)         y= et

 

           

 

For small Δ, eΔ = 1 + Δ + O(Δ2), so that the derivative is, 

 

               


Integral

 

Integration is the operation of finding the area under a curve. If the function is just a constant  then the area under this line between and is . If the function is not a constant, but changes from  to within the interval we do not know which value (, or something in between) to use. 


Again we use the limit concept. We divide the area into thin rectangular slices. Each slide has width Δ, and its height is the value of the function at the left end of the slice. The errors in this process decrease as we take greater number of thinner slices. In the limit   the errors disappear and the sum of the areas of the slices approaches the area under the curve.

 

When Δ is the whole of (a-b) only the dark gray area is included. When it is halved the light gray area comes in too. When it is halved again the hatched area is also included. So in the limit    the sum of the areas of the slices approaches the total area under the curve.

 

The integral is denoted by the “ ” sign.      denotes the integral of the function f from t=a to t=b.

If the curve lies above the x axis, it makes positive addition to the integral. If it lies below the curve then it adds negative values. For example for and for . Thus its integral is positive in the first interval and negative in the second. In the interval  the negative and positive parts cancel out and the integral is 0.

 

 

The integral is the inverse of the derivative: We can see that the integration is just the opposite of derivation. If we extend the integral to b+Δ The new area added is just Δf(b).

 

 

The integrals at the right hand side are the areas between (a and b+Δ) and (a and b). Obviously the difference is the area from (b to b+Δ). But the area of this thin slice is just f(b)Δ and the derivative of the integral is again the function f(t) evaluated at b.

This means that the integration formulas are exactly the opposites of the derivation formulas. It also means that velocity is the integral of acceleration and displacement is the integral of the velocity.

 

Examples:

 

 

 

  

 

 

 

 

 

 

 

In all these examples C is an arbitrary constant that comes in because the limits of the integration are left undetermined.

 

When we do not have to integrate: As we saw above when the function is a constant the integral is just the product of the constant value with the size of the base. This simple case is met very often in physics. (But it is not the only case.)

Examples:

Constant speed:                                    changing speed:

Constant force:                       changing force:        

 

In Physics you will see many integrals. In many cases these integrals can be replaced by a simple product. 

 

Integrals and derivatives of vector functions: A vector function, such as velocity , can also be differentiated with respect to time. In this operation each component is treated separately and differentiated as a scalar function. These form the components of the final derivative vector.

For example if

 

Then the derivative of the x, y and z components of the velocity are,

 

                           

                           

                           

 

So that the acceleration vector is,

 

                                  

 

Similarly a vector can be integrated over a scalar variable component by component. Using the same example the displacement is found as,

 

                         

                          

                         

                         

                              

 

Thus integrating and differentiating vectors over scalar variables like time are done the same way we do similar operations for scalars. Integrating a vector along a vector path is a new thing you will learn later this year.

 

Ahmet Giz