Math by Tathagata
This is the page for Vector Algebra for Engineering Mathematics. Modulus The modulus/magnitude or the length of a vector $\vec{A}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ is defined by the following quantity: $|\vec{A}|=\sqrt{a_1^2+a_2^2+a_3^2}$. Unit Vector A vector is called a unit vector if it has modulus 1. Note, given a vector $\vec{A}$, the vector $\dfrac{\vec{A}}{|\vec{A}|}$ is always a unit vector. It … Read more
An orthogonal polynomial sequence is a family of sequences in which any two distinct polynomials are orthogonal with respect to an inner product. In this page, we will study about orthogonal polynomial sequence with its definition and examples. Orthogonal Polynomial Sequence Definition 1: A sequence of polynomials {Pn(x)} is called orthogonal over the interval (a,b) … Read more
Here we study Hermite polynomials along with its generating function and recurrence relations. Note that eax = $\displaystyle \sum_{n=0}^\infty \dfrac{(ax)^n}{n!}$ …(I) Generating Function of Hermite Polynomials Theorem: The generating function for the Hermite Polynomials Hn(x) is given by $G(x, t) = e^{2tx-t^2}$. That is, $e^{2tx-t^2} = \displaystyle\sum_{n=0}^{\infty} H_n(x) \dfrac{t^n}{n!}$ Proof: One need to prove that … Read more
Here we study Charlier polynomials along with its generating function and orthogonality relation. Before doing this, let us recall the series notation of exponential and binomial functions. eax = $\displaystyle \sum_{n=0}^\infty \dfrac{(ax)^n}{n!}$ …(∗) Also, recall how the Cauchy product of two infinite series. $\displaystyle \sum_{n=0}^\infty a_n \times \displaystyle \sum_{n=0}^\infty b_n$ $=\displaystyle \sum_{n=0}^\infty c_n$ where cn … Read more
In this page, we will learn Taylor series expansion formula along with its remainder form and solved examples. Taylor Series Formula The Taylor series expansion formula of f(x) about x=a is given by f(x) = f(a) + (x-a)f'(a) + $\dfrac{(x-a)^2}{2!}f”(a)$ + $\dfrac{(x-a)^3}{3!}f”'(a) + \cdots$)f'(a) + $\dfrac{(x-a)^2}{2!}f”(a)$ + $\dfrac{(x-a)^3}{3!}f”'(a) + \cdots$ Maclaurin Series Formula Taylor series … Read more
Indeterminate Form and L’Hospital Rule: L’Hôpital’s rule in Calculus is a powerful tool for solving indeterminate limits of the forms like 0/0 or ∞/∞ by taking the derivatives of the top and bottom simultaneously and then re-evaluating the limit of the resulting quotient. This rule is repeatedly applied till the limit exists, and this value … Read more
Answer: The derivative of f(x) defined by f(x) = x2sin(1/x) if x≠0 and 0 if x=0 is equal to 0. That if, if f(x) = $\begin{cases} x^2 \sin \left(\dfrac{1}{x} \right) & \text{ if } x \neq 0 \\ 0 & \text{ if } x=0 \end{cases}$, then $f'(0)=0$. Differentiate x^2sin(1/x) Question: Find $f'(0)$ where f(x) is … Read more
Engineering Mathematics Assignments. This page contains a list of assignments for Engineering Mathematics. Unit I and IV: Differential Calculus & Vector Algebra Q1: Show that the following limits do not exist. Q2: If $y=(ax+b)^m$, then find $y_n$. For this, you study the article nth Derivative. Q3: If $y=e^{m \sin^{-1}x}$, then show that $(1-x^2)y_{n+2} -(2n+1)xy_{n+1}$ $- … Read more
Special functions practice problems. A list of practice problems on Special Functions are given here. Problems on Beta Gamma Functions Q1: For what values of m and n, the integral $\displaystyle \int_0^1 x^{m+1}(1-x)^{n-3}\,dx \quad$ converges. Q2: Find the Value of $B\left(\dfrac{1}{2}, \dfrac{1}{2} \right)\quad$ [Solution] Q3: What are the values of $\Gamma(1)$ and $\Gamma(5)$? Q4: Find … Read more
Engineering Mathematics Syllabus: In this page, the syllabus of Engineering Mathematics I is provided. Study Material Books: Engineering Mathematics I: Best Books, Practice Problems Engineering Mathematics I Syllabus Unit I: Differential Calculus Introduction to limit, continuity, derivative for function of one variable; Successivedifferentiation, Leibnitz’s theorem; Rolle’s theorem, Lagrange’s mean value theorem, Taylor’and Maclaurin’s theorems with … Read more