Differential Calculus & Integral Calculus (अवकलन एवं समIकलन गणित)

Part-A: Differential Calculus  भाग.।रू  अवकलन गणित)

Units

bdkbZ

Topics

¼fo"k;½

I

Introduction to Indian Ancient Mathematics and Mathematicians should be included under Continuous Internal Evaluation (CIE). (भारतीय प्राचीन गणित एवं गणितज्ञों का परिचय सतत आंतरिक मूल्यांकन (ब्प्म्) के अन्तर्गत शामिल किया जाना चाहिए)

Neighborhood of a point, bounded above sets, bounded below sets, Bounded Sets, Unbounded sets, open sets/intervals, closed sets/intervals, Limit points of a set, Isolated points.

¼,d fcanq dk {ks=] mifjc) leqPp;] fuEu ifjc) leqPp;] lhfer leqPp;] vlhfer leqPp;] foo`r leqPp;@varjky] lao`r leqPp;@varjky] ,d leqPp; ds lhek fcanq] foeqDr fcanqA½

Limit, continuity and differentiability of function of single variable, Cauchy’s definition, Heine’s definition, Equivalence of definition of Cauchy and Heine, Uniform continuity, Borel’s theorem, Boundedness theorem, Bolzano’s theorem, Intermediate value theorem, Extreme value theorem, Darboux’s intermediate value theorem for derivatives, Chain rule, Indeterminate forms.  ¼,dy pj ds Qyu dh lhek] fujarjrk ,oa fHkUurk] dkS’kh dh ifjHkk"kk] gsu dh ifjHkk"kk] dkS’kh rFkk gsu dh ifjHkk"kk dh lekurk] leku fujarjrk] cksjsy dk çes;] ifjc)rk çes;] cksytkuks dk çes;] ek/;eku çes;] loksZPp eku çes;] Mkjc‚Dl ds ek/;eku çes; ds fy, vodyt] J`a[kyk fu;e] vfuf'pr :i½       (07)

II

Rolle’s theorem, Lagrange and Cauchy Mean value theorems, Mean value theorems of higher order, Taylor’s theorem with various forms of remainders, Successive differentiation, Leibnitz theorem, Maclaurin’s and Taylor’s series, Partial differentiation, Euler’s theorem on homogeneous function. ¼jksys dh çes;] ySxzsat ,oa d‚ph ekè; eku çes;] mPp Øe ds ekè; eku çes;] fofHkUu çdkj ds vo'ks"kksa ds lkFk VkW;yj dk çes;] mÙkjksÙkj vodyu] fyCuht çes;] eSdy‚fju vkSj Vsyj dh J`a[kyk] vkaf'kd vodyu] ltkrh; Qyu ij vkW;yj dk çes;½               (07)

III

Tangent and normals, Asymptotes, Curvature, Envelops and Evolutes, Tests for Concavity and Convexity, Points of inflexion, Multiple points, Parametric representation of curves and tracing of parametric curves, Tracing of curves in Cartesian and Polar forms. ¼Li'khZ; ,oa lkekU;] vuarLi'kÊ] oØrk] vkoj.k rFkk fodklt] voryrk ,oa mÙkyrk ds fy, ijh{k.k] foHkfDr Çcnq] cgq Çcnq] oØksa dk iSjkehfVªd izn’kZu vkSj iSjkehfVªd oØksa dk vuqjs[k.k] dkrhZ; rFkk èkzqoh; :iksa esa oØksa dk vuqjs[k.k½                                                                                                                                (07)

IV

Definition of a sequence, Theorems on limits of sequences, Bounded and monotonic sequences, Cauchy’s convergence criterion, Cauchy sequence, Limit superior and Limit inferior of a sequence, subsequence, Series of non-negative terms, Convergence and Divergence, Comparison tests, Cauchy’s integral test, Ratio tests, Root test, Raabe’s logarithmic test, de Morgan and Bertrand’s tests, alternating series, Leibnitz’s theorem, absolute and conditional convergence. ¼vuqØe dh ifjHkk"kk] vuqØeksa dh lhek ij çes;] ifjc) ,oa ,dfn"V vuqØe] dkS’kh ds vfHklj.k ekunaM] dkS’kh vuqØe] vuqØe dh lhek mPp rFkk fuEu lhek] vuqØe] _.ksŸkj inksa dh J`a[kyk] vfHklj.k ,oa fopyu] rqyuk ijh{k.k] dkS’kh dk lEiw.kZ ijh{k.k] vuqikr ijh{k.k] ewy ijh{k.k] jkcs dk y?kqx.kd ijh{k.k] Mh e‚xZu rFkk cVªsUM ds ijh{k.k] çR;korÊ J`a[kyk] fyCuht dh çes;] fujis{k ,oa lçfrcaèkh vfHklj.k½                                                                                                                                         (09)

Part-B: Integral Calculus ¼Hkkx-B: lekdyu xf.kr½

Units

bdkbZ

Topics

¼fo"k;½

V

Concept of partition of interval, Properties of Partitions,  Riemann integral, Criterion of Riemann Integrability of a function, Integrability of continuous and monotonic functions, Fundamental theorem of integral calculus, Mean value theorems of integral calculus, Differentiation under the sign of Integration. ¼varjky ds foHkktu dh vo/kkj.kk] jheSu lekdy] ,d Qyu ds jheSu lekdyuh;rk ds ekunaM] lrr ,oa ,dfn"V Qyu dh lekdyuh;rk] lekdyu xf.kr dk ewy çes;] lekdyu xf.kr dk ekè;eku çes;] lekdyu ds fpUg ds vUrxZr voydu½   (09)                                    

VI

Improper integrals, their classification and convergence, Comparison test, m-test, Abel’s test, Dirichlet’s test, Quotient test, Beta and Gamma functions. ¼vuqfpr lekdy] mudk oxÊdj.k rFkk vfHklj.k] lekurk ijh{k.k] m-ijh{k.k] ,csy dk ijh{k.k] fMfjpysV dk ijh{k.k] HkkxQy ijh{k.k] chVk ,oa xkek Qyu½                                        (07)

VII

Rectification, Volumes and Surfaces of Solid of Revolution, Pappus theorem, Multiple integrals, change of order of double integration, Dirichlet’s theorem, Liouville’s theorem for multiple integrals.  ¼Bksl ds ifjØe.k dk fn"Vdj.k] vk;ru ,oa lrgsa] iIil çes;] ,dkf/kd lekdyu] nksgjs lekdyu ds Øe esa ifjorZu] fMfjpysV dk çes;] cgq lekdyu ds fy, fymfoy dk çes;½   (07)

VIII

Vector Differentiation, Gradient, Divergence and Curl, Normal on a surface, Directional Derivative, Vector Integration, Theorems of Gauss, Green, Stokes and related problems.  ¼lfn’k voydu] ço.krk] vilj.k ,oa dyZ] ,d lrg ij lkekU;] fnd~ vodyt] lfn’k lekdyu] xkWl ds çes;] xzhu] LVksd rFkk lEcafèkr leL;k,a½                                                                (07)