# TEW Linear Algebra & Multivariable Calculus (2019)

 Venue: Mepco Schlenk Engineering College,  Sivakasi, Tamil Nadu Date: 13th, May 2019 to 18th, May 2019

 School Convener(s) Name K.N Raghavan R.Ratha Jeyalakshmi Mailing Address Professor,IMSc, C. I. T. CampusChennai 600 113 Head, Department of Mathematics, Mepco  Schlenk Engineering College,Sivakasi, Tamilnadu - 626005

Speakers with their affiliations:

 Name of the speaker Position Affiliation Anirban Mukhopadhyay IMSc, Chennai K. N. Raghavan Professor IMSc, Chennai S. Sundar IMSc, Chennai

Syllabus:

The topics covered:

Differential forms, following chapter 10 of baby Rudin:

• Definition, examples, basic form, standard presentation, addition and multiplication.

• Differentiation of forms, examples, differentiation of product of forms.

• Change of variables.

• Transformation properties.

• Oriented simplices and their boundaries.

• Stokes’ theorem

• Applications: Green’s theorem, Gauss’ theorem; gradient, divergence, and curl; area and volume computation.

K. N. Raghavan

Linear Algebra, following notes (https://www.imsc.res.in/~knr/past/linalg_ed.pdf) copies of which were distributed. Emphasis was laid on problem solving and the utility of raising questions. Much time was devoted to discussion of the exercise sets in the notes.

• Row reduced echelon forms: existence and uniqueness, properties and uses.

• Equality of row and column ranks; various spaces associated with a matrix and relations between them.

• Solving linear equations; geometry of the solution set.

• Orthogonal projections; approximate solution to an over-determined system of linear equations; application to the line of best fit.

• Spectral theorem for real symmetric matrices; positive definite and semi-definite matrices.

• Singular value decomposition.

S. Sundar

Multivariate Calculus

• Basics of metric spaces, norms induced by inner products, Cauchy Schwarz inequality, equivalence of norms on finite dimensional vector spaces.

• Derivative as a linear map, examples from first principles, chain rule, directional derivative and partial derivative, examples.

• Sufficient condition for existence of total derivative, equality of mixed partial derivatives, Taylor’s theorem up to second order and applications to maxima and minima.

• Diagonalising a symmetric operator, min max theorem, application to the proof of Sylvester’s criterion (for positive definiteness).

Time Table

 Time 09:30 to 11:00 11:15 to 12: 45 14:00 to15:30 15:45 to 16:45 Mon 13th KNR TEA SS LUNCH AM TEA Tutorials(KNR & SS) SNACKS Tue 14th AM KNR SS Tutorials(AM & SS) Wed 15th KNR AM SS Tutorials(KNR & SS) Thur 16th KNR SS AM Tutorials(SS & AM) Fri 17th KNR AM AM Tutorials(KNR & AM) Sat 18th KNR AM KNR Tutorials(AM & KNR)

• KNR - K.N.Raghavan