Differential Calculus & Integral Calculus (Mathematics) Book B.Sc 1st Sem U.P. State Universities

Part-A

Differential Calculus

Units

Topics

No. of Lectures

I

Introduction to Indian Ancient Mathematics and Mathematicians should be included under Continuous Internal Evaluation (CIE).

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.

(09)

II

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.

(07)

III

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.

(07)

IV

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.

(07)

Part-B

Integral Calculus

Units

Topics

No. of Lectures

V

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.

(09)

VI

Improper integrals, their classification and convergence, Comparison test, m-test, Abel’s test, Dirichlet’s test, Quotient test, Beta and Gamma functions.

(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.

(07)

VIII

Vector Differentiation, Gradient, Divergence and Curl, Normal on a surface, Directional Derivative, Vector Integration, Theorems of Gauss, Green, Stokes and related problems.

(07)