Math by Tathagata
Tathagata The Pigeonhole Principle states that if n containers are occupied by more than n items, then at least one container must contain more than one item. In this page we will study Pigeonhole Principle with its statement and examples. Also, learn about the generalised version of pigeonhole principle. Pigeonhole Principle Statement If n containers are … Read more
Answer: The area of the surface generated by revolving the parabola y2=2ax about x-axis bounded by x=a is equal to 2πa2(3√3-1)/3 unit. In this page, we study the surface area generated by revolving a curve (parabola) about x-axis with formula. The formulas for the volume are given below: Formula • The formula for the area A … Read more
Answer: The volume generated by revolving the parabola y2=2ax about y-axis is equal to (2√2πa3)/5 unit. In this page, we study the volume generated by revolving a curve (parabola) about y-axis with formula. The formulas for the volume are given below: Formula • The formula for the volume V generated by revolving a curve y=f(x) from … Read more
In Complex Analysis, zeroes are points where the function vanishes while singularities are points where the function loses its analytic property (differentiability). Here we study zeros and singularities along with their types, examples, residues and related theorems. Zero of a Function Definition: Let f(z): D → ℂ be a function. A point z=a is called … Read more
In graph theory, the degree of a vertex is the number of edges connected to it. In this article, we will study the degree of a vertex in a graph with its definition, examples, and related theorems, such as Handshaking lemma. Degree of a Vertex Definition: In an undirected graph, the number of edges connected … Read more
In this page, we will show that a simple graph with n vertices and k components have at most (n-k)(n-k+1)/2 edges. Question: Prove that a simple graph of $n$ vertices and $k$ components can have at most $\dfrac{(n-k)(n-k+1)}{2}$ number of edges. Answer: Let the $i$-th component of the graph G is denoted by Gi. Suppose … Read more
The sequences (-1)n and (-1)n/n are special type of sequences. In this page, we will study the convergence of the sequences (-1)^n and (-1)^n/n. Convergence of (-1)n Question: Discuss the convergence of the sequence {(-1)n}. Answer: The sequence {(-1)n} can be written as follows: {(-1)n} = {-1, 1, -1, 1, -1, 1, …}. Therefore, the … Read more
To find the residue at pole it is very important to know the order of pole. In this page, we will learn when a singular point can be a pole and how to find the order of that pole. Definition of a Pole A point z=a is called a pole of a complex function f(z) … Read more
In this page, we list Real Analysis problems that can be treated as a sample question for the upcoming final exams. Real Analysis Practice Problems Notation: Q1: Define a rational number. Show $\sqrt{3}$ is an irrational number. Q2: Prove that $\mathbb{N}$ has no limit point. Q3: Define an enumerable set. Discuss the enumerability of the … Read more
An orthogonal trajectory is a curve that intersects another family of curves at right angles. In this post, we study orthogonal trajectory along with a few questions and answers. Definition of Orthogonal Trajectory An orthogonal trajectory of a family of curves is a curve that intersects every member of that family at a right angle. … Read more