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Mathematics-II [2nd Semester, First Year] Course Description Offered by Department Credits Status Code Mathematics 4-0-0, (4) EPR MA101002MA [Pre-requisites: Intermediate Mathematics and Mathematics I] Course Objectives To expose student to understand the basic importance of matrices, Ordinary Differential Equations, Multiple Integrals and vector calculus in science and engineering. Course Content Unit-1 Matrices Real vector space, Subspace, Linear span, Linear dependence and linear independence of vectors, Basis, Dimension, Linear transformation, Matrix associated with a linear transformation, Rank and inverse by elementary transformation (Gauss Jordan method), System of linear equations, Eigenvalues and eigenvectors, Cayley-Hamilton theorem, Diagonalization of matrices. Unit-2 Ordinary Differential Equations Exact differential equations; reducible to exact form; first order differential equations (linear and non -linear); Existence and Uniqueness of solutions. Picard's theorem (Statement only), Linear differential equations of higher order with constant coefficients, Method of variation of parameters, Cauchy's and Euler's equations Unit-3 Multiple Integrals a Convergence of improper integrals, tests of convergence, Beta and Gamma functions - elementary properties, Double and triple integrals, change of order of integration, Application to area and volume. Scalar and vector fields, Vector operator, Directional derivative, Gradient, Divergence and curl, Line, Surface anc Volume integrals, Green’s, Gauss’s & Stoke’s theorem (without proof) and applications. Course Materials Required Text: Text books 1. Higher Engineering Mathematics by B. S. Grewal, Khanna Publishers. 2. Advanced Engineering Mathematics by Erwin Kreyszig, John Wiley & Sons. 3. Higher Engineering Mathematics by B. V. Ramana, TMH. Optional Materials: Reference Books 1. Advanced Engg. Mathematics by R. K. Jain and S. R. K. lyengar, Narosa Publishing House. 2. Linear Algebra by M. Thamban Nair and Arindama Singh, Springer. 3. Differential Equations with Applications and Historical Notes by George Simmons, TMH. 4. Higher Engineering Mathematics by Ravish R Singh and Mukul Bhatt, TMH. Unit3.... 1) beta and gama function. 2) double integration in cartesian form 3) double integration in polar t form - 4) change the order of integration 5) evaluation of integral-by changing to polar co-ordinate form 6) evaluation of triple integral 7) finding area as a double integral 8) finding volume as a double integral 9) volume as a triple integral 4 *., 0x 6)! “Lind fhe value of | % Sl Tye Thi hn , Using Phat forint : Fis Fei. tye bp 514 = 47 = 4 Put is | In Ig 7 7 $0, pera fits pa o fk © © ol ‘ : = 4] ar f eee Pde 0 o bag a) / ® Li wo op fol that. fe (2) \"e" he) | ° Je C ° — - 1.7 !7 6 ath q 1" xh Pl | ; a MoT Me = ip Ant LL Sf ££ aaa a uy a, a, 2 ae a en a | 2a | ~ (P -an J | x6)! pre shat I, Ke Fis NM = (a? ie » Qolhs- fee sinbu die = J: 1 E™ lney pari] 4 dt) 1 , ; = fives pare Jy am ol fy ’ = Imag pas] uf i: toi y AK a a bmog. post) f (Pa U)X ra dip. yf oR hl ia le io ares | | Kr} = beg P er - = Im pase 3 Gea icaias =e pout en abe) 4b = 4a - (2a4 (a-¥) qa (at) aw. LybbouvdvdvUUEVTUUVUUUYUUY we . ii, . Duphicatten 1 formula | Proof A ian (P14 NITRR) _ {tn lw) mle me ws se Hiyou - WJ amt and MS fon): of ings) 4B = Hing woe = pe Sa a) > “W)2 = Bln) = af) Kano. ws Me A Tn Tn — a. (te = Ta geet [sia —— © = yup 20=T > do< Jat > Shen ag go tt ad 6= Noi $=. =e firm _ ‘ 7 — i Tm Im aaa (int fe pat a Tam an io) }o. i: = = gb 2)" int ea at = a, 8. ic rif hess Sars me ge 1,8 BS .. T ghy OT Sea jay Re ot [am ae Tate ron Se ; =f K | OKT) Drove Hab 4 i TR Pa | Tht HA Tbh h oe i Th ne [ni puslicetioh “Tan * Tan Lace dogmas a a a fl ag Jv. Th I 1 rl aA €x (5); Evaluate fax. ( “Pe Yi- x4 Lfineé Kolhe— dt eS vay, niet | ¥= ie wait X=0,[=0 | oe ow at dee | 4, | y2 1 dy | + a4 tte Kail df. fron ise ~ SF Ine “if eete pene a +)"4 7 “afb ote - 1 p(3,t\ pitt - ob Val eT = al im M4. dr Ic as tT 416. Lug : C7] 77 w w +77 ~ uw 7 ww 7 w w 7] wy uw we u wv LF] wy Dw Ue ou u v v vv Ivy Vb 1 v W | to (" [-tar'o)"! —a-tano.seeD dB » Jo (tatopth y I = 0 fe ebanaytm ‘ cero dp © (Geb) aman “t! } ; 7 . : sf [> (tino Iceco)'™! sot a) : = (geroprrmisn 2 (gan? (aap AB - B (inn) = LH iVbaove fat ("t= — (Ty (4-W») rw Tan a No 1 -lonf’=T = 4 = fant 2) dO= 4 dt ee (+> nia 2 | and ind = atk | then charge the wade of integrable wo || attent phe edit, We oll four Jimids ve enpliert then deabte ad prdut ye vingle a e en¢jat and id WY) 16 integral com be waitfeh tnt caval 4. Se &@g Evaluate Uf no ery a he Ssh oT Cleauly,, hae iG fn) Wad cfm 0 to Fee cl (a enn vom yo J he | ! d li. Very t) rovemejad r= fee L= a! yas I, ine crak dM ps [nae (07 — tax") Ay ~ tl hs + i" fs 1 [eg dere} | 1-3 ee —$—$_—__— Ses = » Frm ily | way 't = 4 teal | (W | ta Jay = joa (4+ fi [ney ON ex Cyaluste [tov “p day whue Lie the -(riangle woth yuulteed (0,9) (19 iI), (1!) SolM§— Qhven veut ey ey {ioe (00) (orl) (a1) —Linding ogh 4 OA OE 6/4), AC )1) Sate ee ie 18. -0 ait=0 t= (1, a ade jp I vee oo fa ey |") —findin ht 9B 4— 8 jo) LIoy plac), Bly!) 5 “ya = 8 = |45u aad 4-0 \-D — V,tey v= ie y= fl a oe oe 8 3) ERRCERRAKKMKK KAKA wy wee & ee © ee ee Se eX ~Y He He He He eX Oe we we we we ewe ewe we ey we we we we wwewewwy LkO); (- val tude | ePIdudy vet the Snougle pounded by fhe Lined J p, Yeo ond yrye| Lolhg- Hare, phe sey 4 oA BO. ‘suppote that ref M 18 R then Ri sexe], veyet nr int ogee | iy) 18 trae ‘ (0,1) mf 
