For Second Year Physics and Polymer Chemistry Students.

### MTS3C03- MATHEMATICS-3

##### COURSE OUTCOME

• Address the practical problem of evaluation of double and triple integral using Fubini’s theorem and change of variable formula.
•  Realise the advantage of choosing other coordinate systems such as polar, spherical, cylindrical etc. in the evaluation of triple integrals
• Understand the idea of divergence of a vector field,line integral and surface integral and their evaluation and interpretation.
•  Learn  Green’s theorem and Gauss’s theorem  of multivariable calculus and their use in several areas and directions.
• To understand the difference between differentiability and analyticity of a complex function.
• To know a few fundamental results on contour integration theory such as Cauchy’s theorem, Cauchy-Goursat theorem and their applications.
• To understand and apply Cauchy’s integral formula and a few consequences of it such as Liouville’s  theorem and Fundamental Therem of Algebra and so forth in various situations.

##### COURSE CONTENT

MODULE 3

Triple Integral- definition, Evaluation by Iterated Integrals, Applications, Cylindrical Coordinates, Conversion of Cylindrical Coordinates to Rectangular Coordinates, Conversion of Rectangular Coordinates to Cylindrical Coordinates, Triple Integrals in Cylindrical Coordinates, Spherical Coordinates, Conversion of Spherical Coordinates to Rectangular and Cylindrical Coordinates, Conversion of Rectangular Coordinates to Spherical Coordinates, Triple Integrals in Spherical Coordinates .

Divergence Theorem- Another Vector Form of Green’s Theorem , divergence or Gauss’ theorem, ( proof omitted ), Physical Interpretation of Divergence .

Change of Variable in Multiple Integral- Double Integrals, Triple Integrals .

Complex Numbers- definition, arithmetic operations, conjugate, Geometric Interpretation .

Powers and roots-Polar Form, Multiplication and Division, Integer Powers of 𝑧 , DeMoivre’s Formula, Roots .

Sets in the Complex Plane- neighbourhood, open sets, domain, region etc.

Functions of a Complex Variable- complex functions, Complex Functions as Flows, Limits and Continuity, Derivative, Analytic Functions - entire functions .

Cauchy Riemann Equation- A Necessary Condition for Analyticity, Criteria for analyticity, Harmonic Functions, Harmonic Conjugate Functions.

Exponential and Logarithmic function- (Complex)Exponential Function, Properties, Periodicity, (‘Circuits’ omitted), Complex Logarithm-principal value, properties, Analyticity .

Trigonometric and Hyperbolic functions- Trigonometric Functions, Hyperbolic Functions, Properties -Analyticity, periodicity, zeros etc.

MODULE 4:

Contour integral- definition, Method of Evaluation, Properties, MLinequality. Circulation and Net .

Cauchy-Goursat Theorem- Simply and Multiply Connected Domains, Cauchy’s Theorem, Cauchy–Goursat theorem, Cauchy–Goursat Theorem for Multiply Connected Domains.

Independence of Path- Analyticity and path independence, fundamental theorem for contour integral, Existence of Antiderivative .

Cauchy’s Integral Formula-  First Formula, Second Formula-C.I.F. for derivatives. Liouville’s Theorem, Fundamental Theorem of Algebra