Study Notes

Study Notes on Math, Physics and Chemistry, and other related sciences for students in colleges and those seeking admission in graduate courses.

“What are Significant Figures?”, “What are they used for?”, “What is their significance?” Students often ask these questions because why would you want to identify significant numbers, if you can do mathematical calculations without them, right? Where the idea of using significant figures might come off as unnecessary, it is a crucial concept that you need to learn if you want to improve the accuracy and

A transistor is a semiconductor gadget used for amplifying or switching electronic signals and electrical power. It is built with a single piece of semiconductor material and consists of three terminals connected to the external circuit. If you provide any voltage through one end of a transistor, it changes the current flow through the rest of the terminals. Transistors are an essential part of modern electronics.

I don’t feel that I should explain the importance of kinematics in Physics. Kinematic equations form the very foundation of any question you intend to solve in physics. Be it uniform rectilinear motion or rotational motion, these equations will always help you to find the correct answer. Whether you are preparing for NEET or JEE you must have a strong command in kinematics if you want

Everything is the result of chemical reactions whether you cry or burn wood or dissolve sugar. No process is possible without any chemical reactions occurring in them. Chemical reactions are the basis of each and every activity going on around us. Although there are innumerable activities that involve a chemical reaction, there are certain universal reactions that occur in them. Want to know what types of

Formulas are the most important part of mathematics and as we all know one is the backbone of the latter. Considering there are thousands of mathematical formulas to help people develop analytical approach and solve problems easily — there are some that go beyond. Some formulas aren’t just timesaving but those also do wonders. In this article I have collected some of the finest time-saving formulas

“Blood Facts” – seriously? Like all humans who have red colored blood, there are few organisms which have varying blood colors. Spiders and octopus have blue color blood, cockroaches have white/colorless blood, grasshopper, leeches, and some varieties of earthworm have green color blood and so on. Apart from these, there are more facts about the blood. In this article, few interesting facts about Blood and Blood

Introduction In earlier parts we discussed the basics of integral equations and how they can be derived from ordinary differential equations. In second part, we also solved a linear integral equation using trial method. Now we are in a situation from where main job of solving Integral Equations can be started. But before we go ahead to that mission, it will be better to learn how can integral equations

Linear Space or Vector Space over a field K Definition: The linear space over a field K is a non-empty set along with a function $+ : X \times X \to X$ called linear/vector addition  (or just, ‘addition‘) and another function $ \cdot : K \times X \to X$ called scalar multiplication, such that for all elements $x, y, z, \ldots$ in $X$ and $1, k,

‘Symmetry’ has a special meaning in physics. A picture is said to be symmetrical if one side is somehow the same as the other side. Precisely, a thing is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation. For example, if we look at a base that is left and right symmetrical, then turn it 180°

In this article we will define what are Macrostates and Microstates in Statistical Physics with examples and illustrations. Consider some (4, say) distinguishable particles. If we wish to distribute them into two exactly similar compartments in an open box, then the priori probability for a particle of going into any one of the compartments will be exactly 1/2 as both compartments are identical. If the four

Square Integrable function or quadratically integrable function $\mathfrak{L}_2$ function A function $y(x)$ is said to be square integrable or $\mathfrak{L}_2$ function on the interval $(a,b)$ if $$\displaystyle {\int_a^b} {|y(x)|}^2 dx <\infty$$ or $$\displaystyle {\int_a^b} y(x) \bar{y}(x) dx <\infty$$. For further reading, I suggest this Wikipedia page. $y(x)$ is then also called ‘regular function’. The kernel $K(x,t)$ , a function of two variables is an $\mathfrak{L_2}$ – function if atleast one

In this article I will explain what are Integral Equations, how those are structured and what are certain types of Integral Equations. What is an Integral Equation? An integral equation is an equation in which an unknown function appears under one or more integration signs. Any integral calculus statement like — $ y= \int_a^b \phi(x) dx$ can be considered as an integral equation. If you noticed

Every student or graduate knows how hard the first experience of passing exams is. Preliminary preparation starves the nervous system and the physical condition of the human body, however, the exam itself is always a stressful situation, which requires a candidate a great manifestation of mental and physical abilities. Therefore, just the knowledge of a subject is not enough for the exam. The examinee needs to

This is a continuation of the series of summer projects sponsored by department of science and technology, government of India. In this project work, I have worked to collect and expand what Ramanujan did with Nested Radicals and summarized all important facts into the one article. In the article, there are formulas, formulas and only formulas — I think this is exactly what Ramanujan is known

Sequence of real numbers A sequence of real numbers (or a real sequence) is defined as a function $ f: \mathbb{N} \to \mathbb{R}$ , where $ \mathbb{N}$ is the set of natural numbers and $ \mathbb{R}$ is the set of real numbers. Thus, $ f(n)=r_n, \ n \in \mathbb{N}, \ r_n \in \mathbb{R}$ is a function which produces a sequence of real numbers $ r_n$ .

These study notes on Set Theory, Functions and Real Numbers were written by Gaurav Tiwari when he was studying as a Math undergraduate in 2012-2013. The language is sought to be simple and easy to understand. Further reading material is also provided with this article. If you have any questions, feel free to send a message here. Sets A set is a well defined collection of

The greatest number theorist in mathematical universe, Leonhard Euler had discovered some formulas and relations in number theory, which were based on practices and were correct to limited extent but still stun the mathematicians. The prime generating equation by Euler is a very specific binomial equation on prime numbers and yields more primes than any other relations out there in number theory. Euler told that the