Tathagata

On this page, we solve few problems on the topic of the change of variables in double integrals using Jacobian method. Questions- Answers on Change of Variables $\boxed{\color{blue}\textbf{Q 1}:}$ Evaluate the double integral I = $\displaystyle \int \displaystyle \int_R x dx dy$ where R = $\{(x,y) \in \mathbb{R}^2: 1 \leq x(1-y)\leq 2, 1 \leq xy … Read more

On this page, we solve some integrals using the change of order of integration. Solved Problems $\boxed{\color{blue}\textbf{Q 1}:}$ Evaluate $\displaystyle \int_{x=0}^1 \displaystyle \int_{y=0}^{\sqrt{1-x^2}} \sqrt{1-y^2} dy dx$ Answer: We will use the change of order of integration. Note R:= $\begin{cases} 0 \leq y \leq \sqrt{1-x^2} \\ 0 \leq x \leq 1. \end{cases}$ So 0 ≤ y2 … Read more

On this page, you will find Engineering Mathematics practice problems. Study Materials: nth Derivative Coplanar Vectors Indeterminate Forms and L’Hospital Rule Characteristic Equation of a Matrix Rolle’s Theorem Cayley-Hamilton Theorem Taylor Series Expansion Volume generated by revolving a curve Eulers Theorem for Homogeneous Functions Area generated by revolving a curve Vector Algebra Change of Order … Read more

Coplanar vectors lie on the same plane. In this page , we learn about coplanar vectors, conditions to be coplanar along with some solved and practice problems. Coplanar Vectors Definition: A set of vectors are said to be coplanar if they lie on the same plane. In other words, any linear combination of this set … Read more

Euler’s theorem for homogeneous functions states that for a homogeneous function of degree n, the sum of each variable multiplied by its partial derivative equals n times the function itself. Homogeneous Function Definition: A function f(x,y) is said to be homogeneous of degree n if f(tx,ty) = tn f(x,y) for all real numbers t. Euler’s … Read more

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