**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
- 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

- Teacher: Benitta Susan Aniyan