( Mathematics) Integral Calculus समाकलन गणित ( Paper -I )

Integral Calculus

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Units

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Topics

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I

Definite integrals as limit of the sum, Riemann integral, Integrability of continuous and monotonic functions, Fundamental theorem of integral calculus, Mean value theorems of integral calculus, Differentiation under the sign of Integration. ¼;ksx dh lhek ds :i esa fuf'pr lekdyu] jheku lekdyu] 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)

II

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)

III

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)

IV

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)