Course Description This course is designed for first year of Four years B.Sc. program. The main aim of this course is to provide knowledge of Calculus.
Course Objectives: The objective of this course is to acquaint students with the concepts of Calculus and differential equations and their applications. It aims at enabling students to build knowledgebase in Calculus.
Unit 0. Review of Elementary Calculus:
Functions, Graphs, Evaluations of limits, Continuity, Discontinuity, Test of continuity and properties of continuous functions. [10 Lectures]
Unit 1. Tangents and Normals:
Tangents and Normals, Subtangents, Subnormal and their lengths, Derivatives of arc length, Polar equations of subtangents and subnormal, Angle between radius vector and tangent, Length of perpendicular from pole on tangent, Pedal equations and Angle between two curves. [10 Lectures]
Unit 2. Higher Order Derivatives and Mean Value Theorems:
Higher order derivatives, Following theorems (without proofs): Rolle's theorem, Langrange's theorem Cauchy Mean Value theorem, Maclaurin's theorem and Taylor's theorem and their applications in solving problems. [15 Lectures]
Unit 3. Application of Derivatives:
Indeterminate forms, L'Hospital's rule (without proof), Asymptotes, Types of asymptotes, Asymptotes of algebraic curves, Curve tracing techniques, Standard curves and their tracing, Curvature, Chord of curvature, Curvature at origin, Center and circle of curvature. [15 Lectures]
Unit 4. Partial Differentiations and Maxima and Minima of Functions of 2 and 3 Variable:
Basic ideas of limits and continuity of functions of 2 and 3 variables, Partial derivatives and their geometrical interpretation, Higher order partial derivatives, Homogeneous functions, Euler's theorem (proof for 2 variables only), Total differentials, Extreme values, Stationary points, Criteria for maxima and minima, Subsidiary conditions, Lagrange's method of undetermined multipliers. [15 Lectures]
Unit 5. Integration and Definite Integrals:
Integration concepts, Integration techniques and standard formulae, Integration of rational functions and hyperbolic functions, Integration as the limit of a sum, Definite integral and fundamental theorem of integral calculus (without proof), properties of definite integral. [10 Lectures]
Unit 6. Beta and Gamma Functions and Reduction formulae:
Improper integrals, Beta and Gamma functions and their properties, Reduction formulae. [8 Lectures]
Unit 7. Rectification and Quadrature, Volume and Surface Area of Solid of Revolution:
Rectification notion, Length formulae, Idea of quadrature and area formula, Volume and surface area of solid of revolution. [10 Lectures]
Unit 8. Double Integrals:
Double and iterated integrals in rectangular coordinates, Changes of variables in double integrals( to polar coordinates and curvilinear coordinates), Computing area and volume using double integrals, Application of double integrals in mechanics: mass and static moments of a lamina, centre of gravity, moments of inertia of a lamina. [10 Lectures]
Unit 9. Vector Calculus:
Vector Fields, Gradient Fields, Line Integrals, Line Integrals in Space, Line Integrals of Vector Fields, The Fundamental Theorem for Line Integrals, Independence of Paths, Conservation of Energy, Green‟s Theorem, Extended Version of Green‟s Theorem, Curl and Divergence, Vector forms of Green‟s Theorem. [10 Lectures]
Unit 10. Vector Calculus (Contd.):
Parametric Surfaces and their Areas, Surface Integrals, Surface Integrals of Vector Fields, Stoke‟s Theorem, The Divergence Theorem. [10 Lectures]
Unit 11. Differential Equations of the First Order and the First Degree:
Introduction, standard form, Variables-separable equations, Homogeneous equations, Equations reducible to homogeneous equations, Non(homogeneous equation of the first order, Exact differential equation, Condition for exactness, Integrating factors first order, Exact differential equation, Condition for exactness, Integrating factors and techniques, Linear differential equations and equations reducible to linear forms. [15 Lectures]
Unit 12. Linear Differential Equations with Constant Coefficient:
Linear equations with constant coefficients, Linear equations solvable using symbolic operators, Symbolic operation techniques, Particular integrals and complementary function, Homogeneous linear equations, Equations reducible to homogeneous form. [12 Lectures]
Text books
1. M.B. Singh and B.C. Bajracharya; Differential Calculus, Sukunda Pustak Bhandar, Kathmandu, 1995.
2. G.D. Pant and G.S. Shrestha; Integral Calculus and Differential Equations, Sunita Prakashan, Kathmandu 1994.
3. James Stewart, Calculus Early Transcendentals, Cengage Learning, 7th Edition Metric Version, 2015.
Reference books
4. Anton, Bivens and Davis, Calculus, Wiley, 7th Edition, 2012.
5. D.A. Murray; Introductory Course in differential Equations, Oriental Longman.
6. T.M. Apostal; Calculus Vol I & II, Wiley Eastern Ltd, New Delhi, 1986.
7. S.M. Maskey; Calculus, Ratna Pustak Bhandar, Kathmandu, 2008
