calculus for dummies pdf
Algebra | coordinate geometry | Topology for dummies | Symmetry | Bertrand Russell philosophy | Sir issac newton | Topology for dummies | Albert Einstein
Calculus is the study of how things change. It provides a framework for modeling systems in which there is change,
and a way to deduce the predictions of such models.
I have been around for a while, and know how things change, more or less. What can calculus add to that?
I am sure you know lots about how things change. And you have a qualitative notion of calculus. For example the concept of speed of motion is a notion straight from calculus, though it surely existed long before calculus did and you know lots about it.
So what does calculus add for me?
It provides a way for us to construct relatively simple quantitative models of change, and to deduce their consequences.
To what end?
With this you get the ability to find the effects of changing conditions on the system being investigated. By studying these, you can learn how to control the system to do make it do what you want it to do. Calculus, by giving engineers and you the ability to model and control systems gives them (and potentially you) extraordinary power over the material world.
The development of calculus and its applications to physics and engineering is probably the most significant factor in the development of modern science beyond where it was in the days of Archimedes. And this was responsible for the industrial revolution and everything that has followed from it including almost all the major advances of the last few centuries.
Are you trying to claim that I will know enough about calculus to model systems and deduce enough to control them?
If you had asked me this question in 1990 I would have said no. Now it is within the realm of possibility, for some non-trivial systems, with your use of your laptop or desk computer.
OK, but how does calculus models change? What is calculus like?
The fundamental idea of calculus is to study change by studying "instantaneous " change, by which we mean changes over tiny intervals of time.
And what good is that?
It turns out that such changes tend to be lots simpler than changes over finite intervals of time. This means they are lots easier to model. In fact calculus was invented by Newton, who discovered that acceleration, which means change of speed of objects could be modeled by his relatively simple laws of motion.
This leaves us with the problem of deducing information about the motion of objects from information about their speed or acceleration. And the details of calculus involve the interrelations between the concepts exemplified by speed and acceleration and that represented by position.
So what does one study in learning about calculus?
To begin with you have to have a framework for describing such notions as position speed and acceleration.
Single variable calculus, which is what we begin with, can deal with motion of an object along a fixed path. The more general problem, when motion can take place on a surface, or in space, can be handled by multivariable calculus. We study this latter subject by finding clever tricks for using the one dimensional ideas and methods to handle the more general problems. So single variable calculus is the key to the general problem as well.
When we deal with an object moving along a path, its position varies with time we can describe its position at any time by a single number, which can be the distance in some units from some fixed point on that path, called the origin of our coordinate system. (We add a sign to this distance, which will be negative if the object is behind the origin.)
The motion of the object is then characterized by the set of its numerical positions at relevant points in time.
The set of positions and times that we use to describe motion is what we call a function. And similar functions are used to describe the quantities of interest in all the systems to which calculus is applied.
The course here starts with a review of numbers and functions and their properties. You are undoubtedly familiar with much of this, so we have attempted to add unfamiliar material to keep your attention while looking at it.
I will get bogged down if I read about such stuff. Must I?
I would love to have you look at it, since I wrote it, but if you prefer not to, you could undoubtedly get by skipping it, and referring back to it when or if you need to do so. However you will miss the new information, and doing so could blight you forever. (Though I doubt it.)
And what comes after numbers and functions?
A typical course in calculus covers the following topics:
1. How to find the instantaneous change (called the "derivative") of various functions. (The process of doing so is called "differentiation".)
2. How to use derivatives to solve various kinds of problems.
3. How to go back from the derivative of a function to the function itself. (This process is called "integration".)
4. Study of detailed methods for integrating functions of certain kinds.
5. How to use integration to solve various geometric problems, such as computations of areas and volumes of certain regions.
There are a few other standard topics in such a course. These include description of functions in terms of power series, and the study of when an infinite series "converges " to a number.
functionsFunctions are the relation between two variables : one is dependent variable and other is independent variable.
Limit definitionLimit is the value of a function when its variable approaches another number. It can be defined using inequality . First continuity of function needs to be defined. A function is continuous when the following condition holds:
So for every value of δ there is always an infinitesimal ε such that absolute value |f(x)-L| is always <= ε . This condition must always hold true.
If this is the case then the function f(x) is continuous and differentiable at the point a.
So continuity is the sole condition for the existence of limit. This condition is the same as existence of both side limits :
The left hand limit and right hand limit must exist and equal at the point where the function is continuous . What we have learned is that limit and continuity is the two sides of the same coin. Limit exist when continuity exists and vice versa. The summary of the above discussion can be put like this :
A cartoon can make it much clear:
DifferentiationDifferentiation is defined as the limit of a fraction. It is rate of change of the function with respect the dependent variable. The mathematical expression is:
The geometrical interpretation is :
Product rule of differentiationWhen we differentiate product of two functions we will follow this rule
Some difficult differentiation problems.
PiHow many ways π can be written ? π is a mysterious number. It can be formulated in various ways. The value of π is exactly the ratio between the circumference of a circle to its diameter.
functionalFunctional is function of a function. When we differentiate the functional a different method is applied.
Example can be given :
Leibniz's rule of differentiationLeibniz developed a special rule for differentiation under integration sign as below:
Vectors and geometric calculusThe equation given below is basically Stoke's theorem. It says that integral of some differential form F over the boundary dM of manifold is equal to integral of the exterior derivative over the whole of the manifold M. In language of calculus it is written as :
The integration of differential form is defined in the following way:
Exact definition of differential form is unavailable at the moment. But it relates the multivariable calculus. One varraibe dx is known as 1-form. Two variable expression dx1.dx2 is known as 2-form and so on. The calculus involving the differential forms are known as exterior calculus.
The definition of integral is :
Which is the same thing as this expression below:
This method of finding area can be shown alternatively with two graphs: one for the function f(x) and other for the integral g(x) :
Liebniz integral lawLiebniz defined differentiation of a integral as follows:
The caculation of volume of an object involves this formula.
Biot - Savart LawApplication of calculus is seen almost everywhere in science. One application is apparent in the formulation of Biot-savart law. Biot - Savart law is applicable to current carrying wire. We can find the magnetic field outside such a current carrying wire.
Differentiation vs integralDifferentiation is the inverse process of integration. So if we differentiate the integral we get back the original function.
Finding are between curvesIf two curves intesect more than once then there are several portions of area in common to them. Here is the example :
Euler- Maclaurin formulaAnother formula that is of a great importance is the Euler-Maclaurin formula. It can be put in this form:
Where B(2p) is the Bernaulli numbers. There is an explicit formula which can be used to compute these numbers.
Differential equations solution methodsA simple method for solving second order linear differential equation is given here.
Important calculus formulaSome geometric and algebraic formulas are related to calculus intimately. These can be mentioned.
Some important rules for integration are :
Some more formulas are :
The equations that form parabolic trajectory are :
Finding of area under the curve
Trigonometric formula cheat sheetTrigonometry is the study of triangle and its angles.
Food for thoughtsThere is a paradox named "Zeno's paradox' which goes like this :
What could be the resolution of this paradox?
Fractional order calculusFractional order calculus can be important sometimes. It is the differentiation process where we differentiate with respect to fractional power of dependent variable. As example :
Example can be given
More formula for integrationMethod of substitution is used in all cases..
The function f (z) = 1/z is in fact one for which different answers are obtained when the paths are not homologous(not deformable into one other). We can see why this must be so from what we already know about logarithms. Towards the end of the previous chapter, it was noted that log z is an indefinite integral of 1/z. (In fact, this was only stated for a real variable x, but the same reasoning that obtains the real answer will also obtain the corresponding complex answer. This is a general principle, applying to our other explicit formulae also.) We therefore have
We know that there are different alternative 'answers' to a
complex logarithm. More to the point is that we can get continuously from
one answer to another. To illustrate this, let us keep a fixed and allow b to
vary. In fact, we are going to allow b to circle continuously once around
the origin in a positive (i.e. anticlockwise) sense (see Figure a), restoring it
to its original position. Remember, that the imaginary part of
log b is simply its argument (i.e. the angle that b makes with the positive
real axis, measured in the positive sense; see Figure b). This argument
increases precisely by 2π in the course of this motion, so we find that log b
has increased by 2π. Thus, the value of our integral is
increased by 2π when the path over which the integral is performed winds
once more (in the positive sense) about the origin.
We can rephrase this result in terms of closed contours, the existence of which is a characteristic and powerful feature of complex analysis. Let us consider the difference between the second and the first of our two paths, that is to say, we traverse the second path firrst and then we traverse the Wrst path in the reverse direction (Figure c). We consider this difference in the homologous sense, so we can cancel out portions that 'double back' and straighten out the rest, in a continuous fashion. The result is a closed
path or contour that loops just once about the origin (see Figure d), and it is not concerned with the location of either a or b. This gives an example of a (closed) contour integral, usually written with the symbol ∫ , and we find, in this example
Of course, when using this symbol, we must be careful to make clear which actual contour is being used or, rather, which homology class of contour is being used. If our contour had wound around twice (in the positive sense), then we would get the answer 4π. If it had wound once around the origin in the opposite direction (i.e. clockwise), then the answer would have been -2π.
Applications of differential equationsForced oscillator can be described by a differential equation of order two. In forced oscillator there is a force f(t) which affects the motion of the oscillator. Here is the explicit formula.
Here the external force is f(t) = Fcos(wt).
If we examine the solution for amplitude A(0) we see that the amplitude is maximum for ω = ω(0). This frequency is called the resonance frequency. At this resonance frequency the system oscillates at a maximum amplitude and energy transfer also beomes maximum. The graph of this amplitude can be plotted accordingly.
On the other hand damp oscillation is slighly different phenomena. In damped oscillation the amplitude decays exponentially but stays within an envelop. The equation contains a damping force proportional to the velocity.
Where b is the damping factor which value affects the rate of the damping. The solution of this damped oscillation is
Where ω = ω` when damping b is small. The graphical analysis would come up as follow :
There are a lot of applications of differential equations in electrical circuits too.
Predatory prey modelPredator-prey relations refer to the interactions between two species where one species is the hunted food source for the other. The organism that feeds is called the predator and the organism that is fed upon is the prey. Here is the mathematical representation of predatory prey system :
So any desired solution can be found from the equation given above.
this website is about theoretical physics and mathematics, theory of relativity general, theories of phyiscs