## calculus for dummies review

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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.
And so?
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.

## functions

Functions are the relation between two variables : one is dependent variable and other is independent variable.

Parameterized surface is represented by the mapping ψ : R^2 -> R^3 as follows ψ(x, y) = (f(x,y), g(y,x), h(x,y)).
An example of a parameterized surface is the representation of the graph f(x,y) in the three dimension having height z=f(x,y) at each point. ψ(x,y) = (x, y, f(x,y)) . This way a surface is always paremeterized by two parameters.

## Limit definition

Limit 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. Similiarly limit of multivariable function can be defined.

The summary of the above discussion can be put like this :

A cartoon can make it much clear:

## Limit theorems

There are many theorems of limit. Limit obeys these theorems. These are :

When we can not determine limit of a function in usual ways some special techniques are applied. One of these is the L'Hospital rule .

For example the following limit can not be evaluated . So L'hostpital rule needs to be applied:

## Differentiation

Differentiation 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 :

The derivative of sinx can be calculated using the basic definition of calculus as

## Product rule of differentiation

When we differentiate product of two functions we will follow this rule

Some difficult differentiation problems.

## Pi

How 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.

Differential calculus has a lot of applications in both mathematics and calculus. Here is the use of finding the slope or tangent to curves.

this is how logarithm is defined

We can find the differentiation of logarithm using fundamental theorem of differentiation.

Why differentiation of a constant is zero?

Sometimes center of gravity of an object is calculated using caculus and integration.

Here are some more important theorems of calculus.

Leibniz developed a formula to differentiate under integral sign.

First order partial differential equations have some common characteristics

A second order non-homogeneous differential equation can be solved in a special way :

In case of finding particular solution of the above equation W no logner depend on t but the coefficients C1, C2 depend on t.

## Green's function

Green's function is often encountered in quantum field theory equations. A green's function G(x, s ) is a solution of a linear differential operator L = L(x) acting on distributions over subset R^n is the solution of the equation

Where Δ is the dirac delta function. The distritions are linear functional that map a set of test function into a set of real numbers. This property of the green function can be used to solve differential equation. L(x)U = f(x). Sophus Lie stated the importance of differential calculus in a qoute.

What is pi ?

## functional

Functional is function of a function. When we differentiate the functional a different method is applied.

Example can be given :

## Leibniz's rule of differentiation

Leibniz developed a special rule for differentiation under integration sign as below:

## Vectors and geometric calculus

The 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.

## Integral 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) :

There is a special rule of integration of rational function

We can use integral calulus to find the surface area of a solid figure as

## Liebniz integral law

Liebniz defined differentiation of a integral as follows:

The caculation of volume of an object involves this formula.

## Biot - Savart Law

Application 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 integral

Differentiation is the inverse process of integration. So if we differentiate the integral we get back the original function.

## Finding are between curves

If two curves intesect more than once then there are several portions of area in common to them. Here is the example :

## Euler- Maclaurin formula

Another 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 methods

A simple method for solving second order linear differential equation is given here.

## Important calculus formula

Some geometric and algebraic formulas are related to calculus intimately. These can be mentioned.

## Directional derivative

The concept of directional derivative is oftentimes important in physics, especially in theoretical physics. The directional derivation is similar to line integral in some respect. Here we are concerned with a vector in whose direction we compute the derivative. The partial derivative of a real valued function f on R(n) measures the rate of change along the coordinate axes. It does not measure the rate in other direction like y = x. These are measured by directional derivative. Let V = (v1, v2, v3,.. vn) is a vector. Then the directional derivative along V at a point X is

An example is given

## More formulas

Some important rules for integration are :

Some more formulas are :

The equations that form parabolic trajectory are :

## Trigonometric formula cheat sheet

Trigonometry is the study of triangle and its angles.

## Food for thoughts

There is a paradox named "Zeno's paradox' which goes like this :

What could be the resolution of this paradox?

## Fractional order calculus

Fractional order calculus can be important sometimes. It is the differentiation process where we differentiate with respect to fractional power of dependent variable. As example :

## Implicit Differentiation

Example can be given

## More formula for integration

Method of substitution is used in all cases..

## Contour integral

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.

## Numerical calculus

Numerical integration is the method of computing integral using discrete process or by summing finite number of terms instead of infinite number of terms. Both differentiation and integration can be calculated using numerical method.
First method to be discussed is the Newton's devided difference formula.

Now the first divided difference is

The second difference is

and so on..

## Applications of differential equations

Forced 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.
The integral of motion can be found by coupled first order equation

## Predatory prey model

Predator-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.

## Euler Identity

Euler Identity is perhaps the best equation in mathematics. It includes four mysterious numbers in one equation. It can be proved easily using power series expansion of cosine and sine.

## Line integral

In calculus of integration line integral is a special kind of integral evaluated along the path or curve. In physics work W is calculated using line integral. This is because an object is usually taken along some specific route from one point on the curve to another. The mathematical expression turns out to be :

It also happens that path or line integral is sometimes path independent. If the vector field F is the gradient of a scalar field φ that is &del; φ = F then it can be proved that

That is it only depends on the end points of the path. So the path integral is actually path independent.

## Complex calculus

Calculus on complex plane is slightly different than the usual calculus on real numbers. There is a theorem relating to the complex integration. Thia is called the fundamental theorem of calculus of complex variable. The path independence is the essense of this theorem.

## Total derivative

Sometimes the concept of total derivative becomes very important. It is the best linear approximation of a function near the point its derivative is evaluated. Unlike partial derivative , the total derivative approximate the function with respect to all of its arguments , not just a single varaible. Usually aother variable other than the usual variables with with the function varies. The mathematical expression for the total derivative is then

The application of total derivative can be found in field equations derivation.

## Arch length formula

An arch is a curve to be precise. It can be any cord of a circle , it can be any part of parabol or hyperbola. The length of arch can be determined by elementary calculus . It is very simple : take the derivative of the position vector of points on the arch and take the modulus of this vector. We then integrate it to find the distance or length of it ;

## Feynman integral

Feynman was a clever person, at least, when it comes to integration. Here is an example

## Proving wave equation

First , wave is a function of space and time. Wave carries energy as disturbances in the medium in which it propagates. Wave equation is a second order partial differential equation. In 1974 De Alembert first discovered wave equation in one-dimension. Later Euler discovered three-dimensional wave equation.
Wave equation can be derived from Hook's law :
Let us imagine an array of little weights of mass m interconnected with massless spring of length h. The spring has a spring constant k.

Here the dependent variable u(x) measures the distance from the equilibrium of the mass situated at x, so that u(x) essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The forces exerted on the mass m at the location x + h are:

The equation of the motion of the weight at the location x+h is given by equation these two forces.

Where time dependence of u(x) on time t has been made explicit. If the arrangement of weights consists of N weights spaced evenly over the length L = Nh of total mass M = Nm, and the total spring constant of the array K = k/N we can write the above equation as:

Which is a second order differential equation.
The constant KL^2/M is the speed of the wave. In case of electromagnetism it is the speed of light. Wave equation has a lot of importance in physics and astronomy.

## Calculus in determining the population growth

Calculus can be used to describes the dynamics of population in a country. Holden and Right first invented mathematical theory of population and reproduction process. Let us explain this :
If the quantity of a species is X and population is N then this condition holds X = ∫Ndt when N = dx/dt and N is considered a function of time. Here small x is the unit of population. If the ratio of male and female is constant then αN is the number of male and βN is the number of female and α + β = 1 . The number of sexual intercourse between male and female in unit time is then
αβN.N = αβN^2 and further, if n sexual intercoures produces m new species of the same kind then total number of new birth is
Kαβ(m/n)N^2 = λN^2 where K and N are some proportionality constants.
If mortality coefficient is η then in time dt we have dN = λN^2 dt - ηNdt; that is
dN/dt = (λN - η)N which is a differential equation of first order. Now get by simplification process
[1/η][{λ/(λN - η)} - dN/N ] = dt . Integrating both side we get
N/[λN - η] = e^ [-η(t+c)] where c is an integrating constant.
At t = 0 , if population No then we get ,
e^[cη] = No(λNo - e)^(-1) so N = ηNo[(η-Noλ)e^ηt + Noλ]^(-1)
If we let [Noλ/η] h we get further N = No[(1-h)e^(ηt) + h]^(-1) . If h = 1 or η - λNo = 0 , then population N is constant and No = η/N .
But if h < 1 then η - λNo > 0 then as t tends to infinity(∞) , N tends towards zero(0). Again if h > 1 then η - λNo < 0 then population N tends to infinity as t tends to [1/η]log[(Noη)/(Noη-t) although this event is unreal.
As a result, for population No = η/N is a critical value above which population honestly increases and below which population decreases to zero(0).

## Helmholtz equation

Helmholtz equation is a kind of wave equation. It is second order partial differential equation. It has the following form

## Heaviside equation

Heaviside developed many equations related to electrical and electronics engineering. He also transformed Maxwell's equation into modern form. Here is an exampl of his contribution

Where R and C are the resistances and G is the membrane leakage conductance.

## Lebesque integral

Lebesque integral is a variant of Riemann integration. It is used to find the area of a region under a curve same as the Riemann integral. But the method is slightly different. Here we need some concepts of measure theory
First any function can be approximated by simple functions. Simple functions are real-valued function over a subset of the real line, similar to a step function. A basic example of simple function is a floor function over a half-interval [0, 9) , whose values are {1, 2, 3, 4, 5, 6, 7, 8} . The more precise and formal statement is that it is linear combination of indicator functions of measureable sets ( sets which are measureable) as

An indicator function 1 is a function defined on the set X , that indicates the membership of an element in subset A of X , having 1 for all the elements of A and 0 for all the elements of X not in A. The proper formal definition is

The Lebesque integral of a function f(x) with respect to a measure μ is defined as

The method is the partition the are under the curve into a large number horizontal strips. In case of Riemann integral the whole of the area is divided into vertical strips of rectangles. The formal definition can be given explicitly.

A measure μ assigns to each set A a measure μ(A) representing the size of A. The elementary area of the thin rectangle contained in y and y - dt is just

The Lebesque integral is then defined as

So instead of defining the integral as the integral of the function f*(t) we used a measure μ . The outcome is the same which can be obtained using Riemann integral.
On the other hand the integral of the indicator function is defined as

Where S is a measurable set. When coefficients a(k) are non-negative

## Arch length formula

Arc laength formula is used to measure distance or length of an arc

## Heron's formula

It is a formula of the area of a triangle expressed in terms of perimeter of it.

## Finite element method

Finite element method is used to solve a complex problem by breaking it down into smaller and simple forms which can be called finite elements. It is widely used to solve engineering and mathematical problems. The finite elements are created by discretizing process. The solution region of the equation to be solved in divided into large finite parts. These finite parts when summed give the approximate solution of the original equation. It is a method used to solve differential equation. The formulation is

In two dimensional case the field variable φ(x, y) used in a trianglular element has three degrees of freedom. Three nodal values can be used to approximate any value of the variable inside the triangle.

Where N1 , N2 and N3 are the shape functions or the bending functions. This is a special case of the approximation of the function u(x) in terms of other functions

The more the number of shape functions is, the more precise the final solution would be.

this website is about theoretical physics and mathematics, theory of relativity general, theories of phyiscs

## Basic Trigonometry

Calculus sometimes involves manipulations of trigonometric functions and identities. Tigonometry basics start with the coordinate axes divided into four quadrants. Different quadrants have different properties associated with the trigonometric functions.

For exampl in first quadrant all the trigonometric functions ( cosines, sine, tan) are positive. In the second quadrant sine is positive. In the third quadrant tangent (tan) is positive and in the fourth quadrant cosine is positive. Then we have the three trigonometric identities are defined as follows:

The basic trigonometric formulas are

The most common values of trigonometric functions

Law of cosines and sines are sometimes very useful.

Where s is the perimeter of the triangle in question.
S = (a + b + c )/2.

## Finding area of a cirle using calculus

We can find the area of a cirle easily using calculus. The first rule is to estimate the ifinitesimal differential area dA using rules of algebra :

After trigonometric substitution we can then evaluate the integral.

## Perseval's identity

Perseval's odentity involves the integral of a function to its fourier expansion coefficients. It formally states.

## Diffusion equation

Differential equations are quite ubiquitous in mathematics and physics. One of the examples is the diffusion equation. It describes the smoke flow

Maxwell's equations in differential and exterior form can be written as

# Computer science and Engineering

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### Reference materials:

Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf
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