Fundamentals of calculus chapter 4: More on differential equations

Fundamentals of calculus chapter 4: More on differential equations

Differential equations do not have immediate means to solve them. The reason is, because differential equations are defined from differentiation and so is its integral, it’s easier to find the differential equation rather than it’s solution. If our motive is purely to find the solution of differential equations, then our best approach will be to try and find how the expressions “behave” while trying to differentiate them. Then we may back trace this sequence of steps to find the original expression.

An optimal algorithm for the shortest path in the graph

An optimal algorithm to find the shortest path in the graph

Introduction:

Many real-life solutions rely on shortest path algorithms, especially games rely on them! If a graphic game character traces your moves along a complicated path and finds you, then it is obvious that the game implemented a graph algorithm to find the shortest path to approach you. Is it really that easy to find an optimal path to reach a point in a graph? No not exactly, as the upper bound of the Dijkstra’s algorithm is $O(|V|^2)$. If we assume your game map to be a big grid, then the number of cells in the grid is equal to the number $|V|$. But this is still the worst case, and for most times we don’t need to consider the worst case. Let me describe the Dijkstra’s algorithm. 

Fundamentals of calculus Chapter 3: differential equations of functions involving one independent variable.

Fundamentals of calculus

Chapter 3: differential equations of functions involving one independent variable.

“Everything big is built out of smaller units; and if the smaller units fail to cooperate the bigger ones ultimately fall”

“A good learner never accepts the educative information he comes across unless he knows for sure, that there is no practical and logical way to contradict it”

Calculus as a whole evolved to support something new, a differential equation. One marvellous fact about differential equations are they don’t have any constants to describe the graphical structure they represent. In the previous article ‘chapter 2’, we saw how differentiation eliminates a constant. This fact can be taken advantage of to frame differential equations that elaborate the actual rate of change of different quantities. For example if, $z = ax^2 + bx +c$ then $\frac{dz}{dx} = 2ax + b$ and $\frac{d^2z}{dx^2} = 2a$ and $\frac{d^3z}{dx^3} = 0$.

Fundamentals of calculus Chapter 2: Integration

Fundamentals of calculus

Chapter 2: Integration

Integration is the converse of differentiation. To understand this better, it must be known that during differentiation an important information is lost. Let $\frac{dz(x)}{dx} = z’(x)$. Then, we cannot get back $z(x)$ from $z’(x)$ because, while taking the difference $z(x+\Delta x) – z(x)$ only the information regarding the difference between a point in Z axis and a neighbourhood of that point is retained. But the actual orientation is lost. Throughout this article, unless stated $\Delta x$ is limited to zero. To see how this could cause the loss of a piece of information, let me elaborate this with a simple algebraic example. Consider $a-b = c$ to be true. Then we may write $a – (a-c) = c$, here $b=a-c$.

Fundamentals of calculus Chapter 1: Differentiation.

Fundamentals of calculus
Chapter 1: Differentiation.

Can we really call calculus an art? Moreover, can we call mathematics an art? Most mathematicians will agree to the answer: yes. We express ourselves, our creativity through mathematics. It’s not always about calculation. Read my article ‘why do we need mathematics?’ to better understand what I believe mathematics is really. I will not agree that the explanation I had given there was enough. But anyway, I believe mathematics is an art because it is not based on mere calculations but based on creating new rules, using them to express what is around us in a more precise and defined manner. Most of the time, mathematicians would define these rules not for the sake of answering questions that follow up with our reality. But they just do it, in hope that their discovery might come in handy to some scientist in the near future. Some don’t even care about its necessity and for the sake of not expressing themselves to be someone ignorant of the reality most people scramble through physics equations or some other kind of scientific material to find the right place for their equations or mathematical rules to fit in and prove to be beneficial.

Calculus Vs algebra:-

-By K Sreram

This article was first written on 2013

Introduction:-

This article was written to show the difference between arithmetic algebra and calculus. Differentiation and integration is far different from arithmetic algebra (though arithmetic algebra is used to represent the process of differentiation, differentiated equations, integrated equations or expressions) because, arithmetic algebra does not support the results provided by differentiation or integration process. Arithmetic algebra shows that the result provided by differentiation or integration process is approximated and the error will never be zero; but calculus says exact values.

I have proved all this in this article. This article also shows us what actually calculus is and how to apply them practically. Differential and integral calculus is a modification done to the mathematics at the time when arithmetic (and some geometric) algebra was prominent. It defines a wrong result (according to the ordinary algebra) to be correct (with proper proofs).

How to make a good chess move

How to calculate chess positions?

“The grandmaster's secret” is what every average chess player will want to know. But unfortunately there is no "secret" for evaluating chess positions with high accuracy. We just need to change the view by which we look at the board while evaluating. But that's also not going to be easy. Only extensive practical experience on applying the methods described in this article will improve the player's playing strength. This article is not written by an experienced rated chess player of any sort, the author just play chess and analyses chess to some degree. And the author does not claim that the methods presented in this article are standardized theoretical methods.