### Calculus and Analytical Geometry

Course No.: MTH-104

Full Marks: 90+10

Pass Marks: 36+4

__Nature of course__: Theory

__Course Synopsis__: Preliminaries revision of differentiation and integration; Techniques of integration infinite series; Vectors and analytical geometry in space (differential geometry). Vector valued functions. Multivariable functions and partial derivatives. Multiple integrals and integration in vector fields. Partial derivatives; Equations of First Partial Derivatives.

__Goals__: This course aims at providing students with some advanced topics in undergraduate calculus and fundamental concepts of partial differentiation and P.D.E of second order. It is assured that a student who has done Certificate Level papers in mathematics will be able to study this course.

Download PDFSyllabus Of "Calculus and Analytical Geometry"

**Course Contents:**

**Unit 1.**

__Topics in Differential Calculus and Integral Calculus__[8 Hrs.]

1.1 Functions and Graphs

1.2 Extreme values of functions; graphing of derivatives

1.3 Mean value integers

1.4 Definite integers, Properties and application, Mean value

theory for definite integers

1.5 Fundamental theory of Integral Calculus and application, Improper

integrals

**Unit 2.**

__Infinite Series__[5 Hrs.]

2.1 Infinite sequence and sequence of convergence and divergence

2.2 Integral test, comparison test, ratio and root test

2.3 Absolute and conditional convergence

1.4 Power series, Taylor and Maclaurin series, convergence of Taylor series

**Unit 3.**

__Conic Section__[3 Hrs.]

3.1 Classifying conic sections by eccentricity

3.2 Plane curves, parametric and polar equations, integration in polar coordinates

**Unit 4.**

__Vectors and Vectors Valued Functions__[6 Hrs.]

4.1 Vectors in the space

4.2 Lines and planes in space

4.3 Cylinders and Quadric surfaces

4.4 Cylindrical and Spherical Coordinates

4.5 Vector valued functions and space curves

4.6 Unit tangent vector, curvature and torsion and TNB system

**Unit 5.**

__Multiple Integrals__[5 Hrs.]

5.1 Double integrals in rectangular polar coordinates

5.2 Finding areas, moments and centre of mass

5.3 Triple integrals in rectangular coordinates and application

5.4 Substitutes in multiple integrals

**Unit 6.**

__Multivariate Calculus__[9 Hrs.]

6.1 Functions, limits and continuity of two or more variables

6.2 Partial derivatives

6.3 Differentiability, Differentials, Total Differential Coefficients

6.4 Directional derivatives and gradient vectors

6.5 Extreme values

6.6 Lagrange Multiplies

**Unit 7.**

__Partial Differential Equations__[9 Hrs.]

7.1 Review of Ordinary Differential Equations

7.2 Analysis of P.D.E of 1st and 2nd order

7.3 Linear equations of the 1st order and the general solutions

7.4 P.D.E of 2nd order, its derivation and basic concepts

7.5 Solution of general P.D.E with constant coefficients, complimentary solution and integral solution

7.6 Wave equations and heat equations and their solutions (Chapter II, Section 11.1, 11.2, 11.4, 11.5). Erwin and Kreyszig. 8th edition, John-Wiley Publications.

**Text Books**

Thomas and Fenns: Calculus and Analytical Geometry, 9th Edition, 2004. (Thomas, Jr. G. B., and Finney, Ross L. Publisher: Pearson Education Pvt. Ltd.

Kreyszig, Erwin, Advanced Engineering Mathematics, John- Wiley & Sons (1991). 5th Edition.

**References**

E.W. Swokowski, Calculus with Analytical Geometry, Second Alter Edition.

Sneddan Ian- Elements of Partial Differential Equations.

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