# Third Year B.Sc.

## Paper I : REAL ANALYSIS and MULTIVARIABLE CALCULUS

Unit 1: Riemann Integration, Double and Triple Integrals

(a) Uniform continuity of a real valued function on a subset of R (brief discussion)

(i) Definition.

(ii) a continuous function on a closed and bounded interval is uniformly continous (only statement).

(b) Riemann Integration.

(i) Partition of a closed and bounded interval [a, b], Upper sums and Lower sums of a bounded real valued function on [a, b]. Refinement of a partition,

Definition of Riemann integrability of a function. A necessary and sufficient condition for a bounded function on [a, b] to be Riemann integrable. (Riemann's Criterion)

(ii) A monotone function on [a; b] is Riemann integrable.

(iii) A continuous function on [a; b] is Riemann integrable.

A function with only finitely many discontinuities on [a; b] is Riemann integrable.

Examples of Riemann integrable functions on [a; b] which are discontinuous at all rational numbers in [a; b]

(c) Algebraic and order properties of Riemann integrable functions.

(i) Riemann Integrability of sums, scalar multiples and products of integrable functions. The formulae for integrals of sums and scalar multiples of Riemann integrable functions.

(ii) If $f : [a, b] \Rightarrow \mathbb{R}$ is Riemann integrable and $f(x) \geq 0$ for all $x \in [a b]$, then $\int_{a}^{b} f(x) dx \geq 0$

(iii) If f is Riemann integrable on [a, b], and a < c < b, then f is Riemann integrable on [a, c] and [c, b], and $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$

(d) First and second Fundamental Theorem of Calculus.

(e) Integration by parts and change of variables formula.

(f) Mean Value Theorem for integrals.

(g) The integral as a limit of a sum, examples.

(h) Double and Triple Integrals

(i) The definition of the Double (respectively Triple) integral of a bounded function on a rectangle (respectively box).

(ii) Fubini's theorem over rectangles.

(iii) Properties of Double and Triple Integrals:

(1) Integrability of sums, scalar multiples, products of integrable functions, and formulae for integrals of sums and scalar multiples of integrable functions.

(2) Domain additivity of the integrals.

(3) Integrability of continuous functions and functions having only finitely (countably) many discontinuities.

(4) Double and triple integrals over bounded domains.

(5) Change of variables formula for double and triple integrals (statement only).

Reference for Unit 1:

1. Real Analysis Bartle and Sherbet.

2. Calculus, Vol. 2: T. Apostol, John Wiley.

Unit 2. Sequences and series of functions

(a) Pointwise and uniform convergence of sequences and series of real-valued functions. Weierstrass M-test. Examples.

(b) Continuity of the uniform limit (resp: uniform sum) of a sequence (resp: series) of real-valued functions. The integral and the derivative of the uniform limit (resp: uniform sum) of a sequence (resp: series) of real-valued functions on a closed and bounded interval. Examples.

(c) Power series in $\mathbb{R}$. Radius of convergence. Region of convergence. Uniform convergence.

Term-by-term differentiation and integration of power series. Examples.

(d) Taylor and Maclaurin series. Classical functions defined by power series: exponential, trigonometric, logarithmic and hyperbolic functions, and the basic properties of these functions.

Reference for Unit 2:

Methods of Real Analysis, R.R. Goldberg. Oxford and International Book House (IBH) Publishers, New Delhi.

Unit 3. Differential Calculus

(a) Limits and continuity of functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ (Vector fields)

Basic results on limits and continuity of sum, difference, scalar multiples of vector fields. Continuity and components of vector fields.

(b) Differentiability of scalar functions:

(i) Derivative of a scalar field with respect to a non-zero vector.

(ii) Direction derivatives and partial derivatives of scalar fields.

(iii) Mean value theorem for derivatives of scalar fields.

(iv) Differentiability of a scalar field at a point (in terms of linear transformation). Total derivative. Uniqueness of total derivative of a differentiable function at a point. (Simple examples of finding total derivative of functions such as $f(x, y) = x^2 + y^2, f(x, y, z) = x + y + z,$ may be taken). Differentiability at a point implies continuity, and existence of direction derivative at the point. The existence of continous partial derivatives in a neighbourhood of a point implies differentiability at the point.

(v) Gradient of a scalar field. Geometric properties of gradient, level sets and tangent planes.

(vi) Chain rule for scalar fields.

(vii) Higher order partial derivatives, mixed partial derivatives. Sufficient condition for equality of mixed partial derivative. Second order Taylor formula for scalar fields.

(c) Differentiability of vector fields.

(i) Definition of differentiability of a vector field at a point. Differentiability of a vector field at a point implies continuity.

(ii) The chain rule for derivative of vector fields.

Reference for Unit 3:

(1) Calculus, Vol. 2, T. Apostol, John Wiley.

(2) Calculus. J. Stewart. Brooke/Cole Publishing Co.

Unit 4. Surface integrals

(a) (i) Parametric representation of a surface.

(ii) The fundamental vector product, definition and it being normal to the surface.

(iii) Area of a parametrized surface.

(b) (i) Surface integrals of scalar and vector fields (definition).

(ii) Independence of value of surface integral under change of parametric representation of the surface.

(iii) Stokes' theorem, (assuming general form of Green's theorem) Divergence theorem for a solid in 3-space bounded by an orientable closed surface for continuously differentiable vector fields.

Reference for Unit 4:

(1) Calculus. Vol. 2, T. Apostol, John Wiley.

(2) Calculus. J. Stewart. Brooke/Cole Publishing Co.

## Paper II: ALGEBRA

Unit 1: Linear Algebra

Review of vector spaces over R:

(a) Quotient spaces:

(i) For a real vector space V and a subspace W, the cosets v + W and the quotient space V=W. First Isomorphism theorem of real vector spaces (Fundamental theorem of homomorphism of vector spaces.)

(ii) Dimension and basis of the quotient space V=W, when V is finite dimensional.

(b) (i) Orthogonal transformations and isometries of a real finite dimensional inner product space. Translations and reflections with respect to a hyperplane. Orthogonal matrices over R.

(ii) Equivalence of orthogonal transformations and isometries fixing origin on a finite dimensional inner product space. Characterization of isometries as composites of orthogonal transformations and isometries.

(iii) Orthogonal transformation of $\mathbb{R}^2$. Any orthogonal transformation in $\mathbb{R}^2$ is a reflection or a rotation.

(c) Characteristic polynomial of an $n \times n$ real matrix and a linear transformation of a finite dimensional real vector space to itself. Cayley Hamilton Theorem (Proof assuming the result $A adj(A) = I_n$ for an $n \times n$ matrix over the polynomial ring R[t].)

(d) Diagonalizability.

(i) Diagonalizability of an $n \times n$ real matrix and a linear transformation of a finite dimensional real vector space to itself. Definition: Geometric multiplicity and Algebraic multiplicity of eigenvalues of an $n \times n$ real matrix and of a linear transformation.

(ii) An $n \times n$ matrix A is diagonalisable if and only if $\mathbb{R}^n$ has a basis of eigenvectors of A if and only if the sum of dimension of eigenspaces of A is n if and only if the algebraic and geometric multiplicities of eigenvalues of A coincide.

(e) Triangularization - Triangularization of an $n \times n$ real matrix having n real characteristic roots.

(f) Orthogonal diagonalization

(i) Orthogonal diagonalization of $n \times n$ real symmetric matrices.

(ii) Application to real quadratic forms. Positive definite, semidefinite matrices. Classification in terms of principal minors. Classification of conics in $\mathbb{R}^2$ and quadric surfaces in $\mathbb{R}^3$.

Reference for Unit 1:

(1) S. Kumaresan, Linear Algebra: A Geometric Approach.

(2) M. Artin. Algebra. Prentice Hall.

(3) T. Banchoff and J. Wermer, Linear Algebra through Geometry, Springer.

(4) L. Smith, Linear Algebra, Springer.