Real Analysis-II & Numerical Analysis UOR B.Sc fourth Sem Math book

Unit

Topics

Lectures

1

Properties of derivable functions, Darboux’s and Rolle’s theorem. Notion of limit, continuity and differentiability for functions of two variables. Directional derivative, total derivative, expression of total derivative in terms of partial derivatives.

vodyt Qyuksa ds xq.k] MkckSZDl vkSj jksys ds çes;A nks pj okys Qyuksa ds fy, lhek] lkarR;rk vkSj vodyuh;rk dh /kkj.kkA fnd~ vodyt] dqy vodyt] vkaf'kd vodyt ds lkis{k esa dqy vodyt dk O;atdA

15

2

Riemann integration – Lower and Upper Riemann integrals, Riemann integrability, Mean value theorems of integral calculus, Fundamental theorem of integral calculus. Functions of bounded variations.

jheku lekdyu & fuEu vkSj mPp jheku lekdyu] jheku lekdyuh;rk] lekdyu dyu ds ek/; eku çes;] lekdyu dyu ds ewyHkwr çes;A ifjc) fopj.k ds QyuA

15

3

Differences. Relation between differences and derivatives. Differences of a polynomial. Newton’s formulae for forward and backward interpolation. Divided differences. Newton’s divided difference, Lagrange’s interpolation formula. Numerical Differentiation. Derivatives from interpolation formulae.

varjA varj vkSj vodytksa ds chp laca/kA cgqin ds varjA vkxzkarj vkSj i'pkarj ds baVjiksys’ku ds fy, U;wVu ds lw=A foHkkftr varjA U;wVu dk foHkkftr varj] ySxzsat dk baVjiksys’ku lw=A la[;kRed vodyuA baVjiksys’ku lw=ksa ls vodytA

15

4

Numerical integration, Derivations of general quadrature formulas, Trapezoidal rule. Simpson’s one-third, Simpson’s three-eighth and Gauss’s quadrature formulae. Numerical solution of Algebraic and Transcendental equations: Bisection method, secant method, Regula-Falsi method, Iteration method, Newton- Raphson Method. Numerical solutions of ordinary differential equations of first order with initial conditions using Euler and modified Euler’s method.

la[;kRed lekdyu] lkekU; prqHkqZt lw=ksa dh O;qRifÙk] leyEckdkj fu;eA flEilu dk ,d&frgkbZ] flEilu dk rhu&vkBok¡ vkSj x‚l dk prqHkqZt lw=A chth; vkSj vchth; lehdj.kksa dk la[;kRed gy& f}Hkktu fof/k] lsdsaV fof/k] jsxqyk&QkYlh fof/k] iqujko`fÙk fof/k] U;wVu&jkQlu fof/kA vkW;yj rFkk la'kksf/kr vkW;yj fof/k dk mi;ksx djds çkjafHkd fLFkfr;ksa ds lkFk izFke dksfV ds lk/kkj.k vody lehdj.kksa ds la[;kRed gyA

15

Total

60