BS-136A Calculus and Ordinary Differential Equations L T P Credit Major Minor Total Time Test Test 3 1-4 75 5 1 3 h Purpose To familiarize the prospective engineers with techniques inmultivariate integration, ordinary and partial differential equations and comple variables. Course Outcomes CO1 To introduce effective mathematical tools for the solutions of differential equations that model physical processes. CO To acquaint the student with mathematical tools needed in evaluating multiple integrals and their usage. CO 3 To introduce the tools of differentiation of functions of comple variable that are used in various techniques dealing engineering problems. CO 4 To study various fundamental concepts of Vector Calculus. CO 5 CO 6 To introduce the tools of integration of functions of comple variable that are used in various techniques dealing engineering problems. To provide the conceptual knowledge of Engineering mathematics. UNIT-I (1 hrs) First order ordinary differential equations: Eact, linear and Bernoulli s equations, Euler s equations, Equations not of first degree: equations solvable for p, equations solvable for y, equations solvable for and Clairaut s type. Ordinary differential equations of higher orders: Second order linear differential equations with constant coefficients, method of variation of parameters, Cauchy and Legendre s linear differential equations. UNIT-II (1 hrs) Multivariable Calculus (Integration): Multiple Integration: Double integrals (Cartesian), change of order of integration in uble integrals, Change of variables (Cartesian to polar) Applications: areas and volumes; Triple integrals (Cartesian), orthogonal curvilinear coordinates, Simple applications involving cubes, sphere and rectangular parallelepipeds. UNIT-III (1hrs) Vector Calculus: Introduction, Scalar and Vector point functions, Gradient, divergence & Curl and their properties, Directional derivative. Line integrals, surface integrals, volume integrals, Theorems of Green, Gauss and Stokes (without proof). UNIT-IV (1 hrs) Comple Variable Differentiation: Differentiation, Cauchy-Riemann equations, analytic functions, harmonic functions, findingharmonic conjugate; elementary analytic functions (eponential, trigonometric, logarithm) andtheir properties; Comple Variable Integration:Contour integrals, Cauchy-Goursat theorem (without proof), Cauchy Integral formula (withoutproof), Taylor s series,zeros of analytic functions, singularities, Laurent s series; Residues, Cauchy Residue theorem (without proof). Suggested Books: 1. G.B. Thomas and R.L. Finney, Calculus and Analytic geometry, 9th Edition, Pearson,Reprint,.. Erwin kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley & Sons, 6.
3. Erwin kreyszig and SanjeevAhuja, Applied Mathematics- II, Wiley India Publication, 15. 4. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary ValueProblems, 9th Edn., Wiley India, 9. 5. S. L. Ross, Differential Equations, 3rd Ed., Wiley India, 1984. 6. E. A. Coddington, An Introduction to Ordinary Differential Equations, Prentice HallIndia, 1995. 7. E. L. Ince, Ordinary Differential Equations, Dover Publications, 1958. 8. J. W. Brown and R. V. Churchill, Comple Variables and Applications, 7th Ed., Mc-Graw Hill,4. 9. N.P. Bali and Manish Goyal, A tet book of Engineering Mathematics, Lami Publications, Reprint, 8. 1. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 36th Edition, 1. Note: The paper setter will set the paper as per the question paper templates provided. LECTURE PLAN Subject Name Subject Code Course:- Semester:- Calculus and Ordinary Differential Equations BS-136A B.Tech. (ME) nd Semester Lecture L1 L L3 L4 L5 L6 L7 L8 L9 L1 L11 L1 L13 L14 L15 L16 L17 L18 L19 L L1 L L3 L4 Topic Eact Differential Equation linear and Bernoulli s equations Euler s equations Equations solvable for p,equations solvable for y, equations solvable for Clairaut s type Second order linear differential equations with constant coefficients Method of variation of parameters Cauchy and Legendre s linear differential equations Double integrals (Cartesian) Change of order of integration in uble integrals Change of variables (Cartesian to polar) Areas and Volumes Triple integrals (Cartesian) Orthogonal curvilinear coordinates Simple applications involving cubes Sphere and rectangular parallelepipeds Scalar and Vector point functions, Gradient and its properties, Directional derivative
L5 L6 L7 L8 L9 L3 L31 L3 L33 L34 L35 L36 L37 L38 L39 L4 L41 L4 L43 L44 L45 L46 L47 L48 divergence & Curl and their properties Line integrals, surface integrals, volume integrals Green Theorem Gauss Theorem Stokes Theorem Differentiation, Cauchy-Riemann equations Analytic functions, harmonic functions, finding harmonic conjugate Elementary analytic functions (eponential, trigonometric, logarithm) and their properties Contour integrals, Cauchy-Goursat theorem (without proof) Cauchy Integral formula (withoutproof) Taylor s series Zeros of analytic functions Singularities Laurent s series; Residues, Cauchy Residue theorem (without proof) Revision Tutorial Sheet-1 Q.1 Solve y y y d y y dy 4 Q. Solve D D 1y cos Q.3 Solve d y dy y e sin d d Q.4 Solve d y dy 4 y e d d d y dy 3 3 1y 6 d d Q.5 Solve Tutorial Sheet- Q.1 Calculate r 3 drd over the area include between the circles r = sin ө & r =4sin ө.
Q. Evaluate the following integrals. i 1 1 dyd 1 y ii yddy over the positive quadrant of the Circle y a Q.3 Calculate the area included between the curve r = a (sec ө + cos ө) and its asymptote. 1 1 y Q.4 Evaluate the following Integrals. i 1 yzddydz ii a y e yz dzdyd Q.5 Find the volume bounded by the cylinder +y =4 and the planes y+z=4 and z = Q.6 Evaluated e y ddy by changing to polar coordinates. Tutorial Sheet-3 Q.1 In what direction from (3,1,-) is the directional derivative of ө = y z 4 is maimum? Find also the magnitude of this maimum. Q. Find div F and curl F where F = grad ( 3 +y 3 +z 3-3yz). Q.3 Find the value of a if the vector (a y + yz)i + (y z )J + (yz- y )K has zero divergence. Find the curl of the above vector which has zero divergence. Q.4 Show that. Q.5 Verify Green s theorem for y y C d dy where C is bounded by y = and y =. Q.6 Verify Stoke s theorem for the vector field F = (-y)i yz J-y zk over the upper half surface of +y +z = 1 bounded by its projection on the y- plane. Q.7 Verify divergence theorem for F = ( -yz)i+(y -z)j+(z -y)k takes over the rectangular parallelopiped a, y b, z c. Tutorial Sheet-4 Q.1 Show that the function f z y is not analytic at the origin even though C-R equations are satisfied at that point. Q. If f(z) is a regular function of z. Prove that y f ' z 4 f z
Q.3 Find the regular function whose imaginary part is e sin y ycos y Q.4 Show that the polar form of Cauchy-Riemann equations u 1 v v 1 u, r r r r u r 1 u r r 1 u r. Deduce that Sample Paper BT-II Calculus and Ordinary Differential Equations Paper: BS-136A Time: Three Hours Maimum marks: 75 Note: The Eaminers will set eight questions, taking two from each unit. The students are required to attempt five questions in all selecting at least one from each unit. All questions will carry equal marks. Q.1 (a) Solve yd dy 3 y e d (b) Solve p 3 p( y y d y d 3 UNIT-I ) y( y) dy d Q. (a) Solve 3 3 1y 6 (b) Solve by the Method of Variation of parameters: d y dy y e log d d UNIT-II Q.3(a) Calculate r 3 drd over the area include between the circles r = sin ө & r =4sin ө. a a y log a (b) Evaluate y ddy y Q.4 (a) Find the volume bounded by the cylinder +y =4 and the planes y+z=4 and z =.
(b) Evaluate 1 1 1 y 1 ddydz y z by changing to spherical coordinates. UNIT-III '' ' Q.5 (a) Prove that f r f r f r (b) Find Interpretation of divergence. Q.6(a) Verify Green s theorem for y y C r d dy where C is bounded by y = and y =. (b) Verify Stoke s theorem for the vector field F = (-y)i yz J-y zk over the upper half surface of +y +z = 1 bounded by its projection on the y- plane. UNIT-IV Q.7(a) Show that the function f z y is not analytic at the origin even though C-R equations are satisfied at that point. (b) If f(z) is a regular function of z. Prove that y f ' z 4 f z Q.8(a) Find the regular function whose imaginary part is e sin y ycos y (b) Show that the polar form of Cauchy-Riemann equations u 1 v v 1 u, r r r r u r 1 u r r 1 u r. Deduce that