**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 = x _{0}* is the value, to which
the function approaches, as

_{}

This
looks like a very simple concept. Usually it is just the value f(x_{0}).
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 *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

_{}

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= Ax^{n}

_{}

_{}

Here
O(Δ^{2}) 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= e^{t}

_{}

For
small Δ, e^{Δ }= 1 + Δ + O(Δ^{2}),
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 Δ*f*(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