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LESSON PLAN: BS-M101(Mathematics-1A for CSE & IT)
Module-I
CALCULUS (INTEGRATION) (8 lectures)
CONTENTS
Calculus (Integration):
Evolutes and involutes; Evaluation of definite and improper integrals; Beta and Gamma
functions and their properties; Applications of definite integrals to evaluate surface areas and
volumes of revolutions.
Module Objectives:
Broad Objectives of this module is to
i) learn evaluation techniques and use of integrals.
Lecture Serial Topics of Discussion
Lecture-1. Evolutes and Involutes – Formula for radius of curvature in Cartesian equation
(Explicit Function: y=f(x) or x=f(y)) and Equation of circle of curvature with co-
ordinate of centre of curvature (Cartesian coordinates only). Discussion with
related problems.
Lecture-2. Evolutes and Involutes – Concept of Evolute and Involute and their
determination. Related problems (Cartesian Coordinates only)
Lecture-3. Evaluation of Definite Integral and Improper Integrals – Review of basic
properties of definite integral. Introduction to Improper Integral. Types of
Improper Integral. Necessary and sufficient condition for convergence of
Improper integral (Statement only). Related problems.
Lecture-4. Beta and Gamma Functions- Definition of Gamma Function. Proof of basic
properties of Gamma function : Γ(1) = 1, Γ(x+1) = x Γ(x), Γ(n+1) = n! and other
properties( proof not required). Problems on gamma function.
Lecture-5. Beta and Gamma Functions- Definition of Beta Function. Derivation of various
∞ tx−1
forms of Beta function. [B(x,y) = B(y,x), B(x,y)= ∫ x+y dt , B(x,y)= 2
0 (1+t)
π
2
∫sin2x−1θcos2y−1θdθ and other properties( proof not required). Relation
0
between Beta and Gamma function (Statement only). Problems on Beta and
Gamma functions.
Lecture-6. Reduction Formulae for both indefinite and definite integrals of types
π
n 2
∫sin xdx , ∫sinnxdx .
0
π
n 2
∫cos xdx , ∫cosnxdx .
0
π
m 2
∫sin xcosnxdx & ∫sinmxcosnxdx
0
π
m 2
∫cos xsinnxdx & ∫cosmxsinnxdx
0
and related problems.
Lecture-7. Surface areas - Quadrature of Plane area: Cartesian coordinates. Calculation of
area of some standard curves in Cartesian coordinates. (e.g. Circle, Parabola,
Ellipse, Hyperbola, Catenary, Folium of Descartes, Astroid, Cycloid).
Lecture-8. Volume of revolution: Volumes of solids of revolution: Rotation of a curve
around x-axis/ y-axis. Problems on Volume of sphere, ellipsoid, paraboloid,
catenary (Cartesian forms only).
Tutorial Assignment—1
Module-II
CALCULUS (Differentiation) (6 lectures)
CONTENTS
Calculus (Differentiation):
Rolle’s Theorem, Mean value theorems, Taylor’s and Maclaurin theorems with remainders;
indeterminate forms and L'Hospital's rule; Maxima and minima.
Broad Objectives of this module is:
i) Solve and model many core engineering problems with applications of one variable
differential calculus.
Lecture Serial Topics of Discussion
Lecture-1. Leibnitz’s Theorem: Successive differentiation, Leibnitz theorem and related
problems.
Lecture-2. Laws of Mean- Rolle’s Theorem, Lagrange’s and Cauchy’s Mean Value
theorems (statement only) and geometrical interpretations.
Lecture-3. Laws of Mean(contd.)- Discussion of problems and applications.
Lecture-4. Taylor’s Theorem- Taylor’s theorem with Lagrange’s and Cauchy’s form of
remainders and its applications. Maclaurin’s Theorem with problems.
Lecture-5. Indeterminate form- L’Hospital’s Rule. Different indeterminate forms e.g.
0 , ∞ , ∞ , 0×∞ , ∞−∞ , 0 , ∞ . Related problems.
0 ∞ 1 0 ∞
Lecture-6. Maxima and Minima- Concept of local and global Maxima and Minima.
Necessary and sufficient conditions for the existence of extreme value at a
particular point. Applications.
Tutorial Assignment—2
Module-III
Matrices [ 7 Lectures]
CONTENTS
Matrices:
Matrices, vectors: addition and scalar multiplication, matrix multiplication; Linear systems of
equations, linear Independence, rank of a matrix, determinants, Cramer’s Rule,
inverse of a matrix, Gauss elimination and Gauss-Jordan elimination.
Broad Objectives of this module is:
i) Acquire knowledge of matrices and determinants and its evaluation
ii) to learn and apply techniques of matrices to find solution of system of equations.
Lecture Topics of Discussion
Serial
Lecture-9. Determinant of a square matrix-Minors and Cofactors, Laplace’s method of
expansion of determinant- elementary properties of determinant and their
applications towards evaluation of determinants-solution to related problems.
Product of two determinants. Cramer’s Rule.
Lecture-10. Inverse of a non-singular Matrix- Properties of invertible matrices- Adjoint of a
determinant. Singular and Non-Singular Matrix, Adjoint of a matrix, –.
Determination of inverse of a non-singular matrix by finding Adjoint.
Lecture-11. Introduction to special Matrices-
Symmetric and skew symmetric matrices.
Orthogonal matrices.
Idempotent matrices.
Unitary matrices
Hermitian& skew Hermitian matrices
Lecture-12. Rank of a matrix- Elementary row and Column operation of matrices.
Determination of rank by reducing it to triangular matrix –different approaches for
introduction of the notion of rank. Rank-nullity theorem..
Lecture-13. System of simultaneous linear equations: Consistency and inconsistency-
Solution of system of linear equations by matrix inversion method.
Lecture-14. Matrix inversion: Gauss elimination method and Gauss Jordan elimination
method. Solving problems using these two processes.
Lecture-15. Matrix Algebra – Introduction to Matrix Algebra-Related Problems.
Identification of matrix as vectors with respect to addition and scalar
multiplication.
Tutorial Assignment—3
Module-IV
Vector Spaces: (9 lectures)
CONTENTS
Vector Spaces:
Vector Space, linear dependence of vectors, basis, dimension; Linear transformations (maps), range
and kernel of a linear map, rank and nullity, Inverse of a linear transformation, rank-nullity theorem,
composition of linear maps, Matrix associated with a linear map.
Broad Objectives of this module is to be
1. familiar with the linear spaces, its basis and dimension
2. to learn and apply the technique of linear transformation and its associated matrix
form for solving system of linear equations.
Lecture Serial Topics of Discussion
Lecture-16. Vector spaces: Concept of internal and external law of compositions. Definition
n
of vector spaces over a real field. Examples of vector spaces ( R , C, P , R etc.)
n mxn
Elementary properties of vector spaces.
Lecture-17. Subspace:Subspaces. Criterion for a vector space to be a subspace (statement
only). Examples. Notion of some important subspace of a vector space.
Lecture-18. Linear dependence of vectors: Linear combination of vectors and linear span.
Linearly dependent and independent set of vectors. Elementary properties and
related problems.
Lecture-19. Basis and dimension: Definition of basis and dimension. Replacement theorem.
Related problems. Dimension of finite and infinite vector spaces. Related
problems.
Lecture-20. Basis and dimension (Contd.): Any two bases of a finite dimensional vector
space have same number of vectors. Extension theorem (statement only). Related
problems.
Lecture-21. Linear Transformation:Definition of linear transformation. Examples. Kernel
and Image of a linear map. Dimension of Ker T and Image T. Nullity and Rank of
linear map. Statement of nullity of T + Rank of T = dim V. Related problems.
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