APPLIED MATHEMATICS - III
|
| UP Polytechnic/Diploma In Computer Science And Engineering Applied Mathematics - III Syllabus - EdufyHUB 4U |
RATIONALE
Contents of this course provide understanding of some elementary and advanced mathematics algorithms and their applications of solving engineering problems. Content of this course will enable students to use some advanced techniques like Beta-Gamma function, Fourier series, Laplace transform and probability distributions in solving complex engineering problems.
LEARNING OUTCOMES
After undergoing this course, the students will be able to:
- Understand matrix operations and uses of matrix in different problems.
- Apply elementary row and column operations in finding inverse of a matrix.
- Find Eigen values, Eigen vectors of a matrix and their different properties.
- Understand degree/order of differential equations and their solution techniques.
- Use differential equations in engineering problems of different areas.
- Find Fourier series expansion of a function.
- Apply Laplace transform and their applications in solving engineering problems.
- Understand concept of probability distribution and their applications.
DETAILED CONTENTS
1. Matrices
(16 Periods)
- 1 Algebra Of Matrices
- Inverse
- Addition
- Multiplication Of Matrices
- Null Matrix And A Unit Matrix
- Square Matrix
- Symmetric
- Skew Symmetric
- Hermitian
- Skew Hermition
- Orthagonal
- Unitary
- Diagonal And Triangular Matrix
- Determinant Of A Matrix.
- Definition And Computation Of Inverse Of A Matrix.
- 2 Elementary Row/Column Transformation
- Meaning And Use In Computing Inverse And Rank Of A Matrix.
- 3 Linear Dependence
- Rank Of A Matrix
- Linear Dependence/Independence Of Vectors
- Definition And Computation Of Rank Of Matrix.
- Computing Rank Through Determinants
- Elementary Row Transformation And Through The Concept Of A Set Of Independent Vectors
- Consistency Of Equations.
- 4 Eigen Pairs & Cayley-Hamilton Theorem
- Definition And Evaluation Of Eigen Values And Eigen Vectors Of A Matrix Of Order Two And Three
- Cayley-hamilton Theorem (Without Proof) And Its Verification
- Use In Finding Inverse And Powers Of A Matrix.
2. Differential Calculus (15 Periods)
- 1 Function Of Two Variables
- Identification Of Surfaces In Space
- Conicoids
- 2 Partial Differentiation
- Directional Derivative
- Gradient
- Use Of Gradient F
- Partial Derivatives
- Chain Rule
- Higher Order Derivatives
- Euler’s Theorem For Homogeneous Functions
- Jacobians.
- 3 Vector Calculus
- Vector Function
- Introduction To Double And Triple Integral
- Differentiation And Integration Of Vector Functions
- Gradient
- Divergence And Curl
- Differential Derivatives.
3. Differential Equation (15 Periods)
- 1 Formation & Order
- Degree
- Types
- Solution Formation Of Differential Equations Through Physical
- Geometrical
- Mechanical And Electrical Considerations
- Degree Of A Differential Equation
- Linear
- Nonlinear Equation.
- 2 First Order Equations
- Variable Separable
- Equations Reducible To Separable Forms
- Homogeneous Equations
- Equations Reducible To Homogeneous Forms
- Linear And Bernoulli Form Exact Equation And Their Solutions.
- 3 Higher Order Linear Equation
- Property of solution, Linear differential equation with constant coefficients (PI for X=eax, Sinax, Cosax, Xn, eaxV, XV).
- 3 Simple Applications
- LCR Circuit
- Motion Under Gravity
- Newton's Law Of Cooling,
- Radioactive Decay
- Population Growth
- Force Vibration Of A Mass Point Attached To Spring With And Without Damping Effect
- Equivalence Of Electrical And Mechanical System
4. Integral Calculus-II
(12 Periods)
- 1 Beta And Gamma Functions
- Definition Us
- Relation Between The Two
- Their Use In Evaluating Integrals.
- 2 Fourier Series
- Fourier Series Of f(x)
- -n <x <n
- Odd And Even Function, Half Range Series.
- 3 Laplace Transform
- Definition
- Basic Theorem And Properties
- Unit Step And Periodic Functions
- Inverse Laplace Transform
- Solution Of Ordinary Differential Equations
5. Probability and Statistics
(12 Periods)
- 1 Probability
- Introduction
- Addition And Multiplication Theorem And Simple Problem.
- 2 Distribution
- Discrete And Continuous Distribution
- Binomial Distribution
- Poisson Distribution
- Normal Distribution.
INSTRUCTIONAL STRATEGY
The content of this course is to be taught on a conceptual basis with plenty of real-world examples. The basic elements of Laplace transform, Differential equations, and Applications of differential equations can be taught with engineering applications of the relevant branch.
MEANS OF ASSESSMENT
- Assignments and Quiz/Class Tests
- Mid-term and End-term Written Tests
- Model/Prototype Making
RECOMMENDED BOOKS
- Elementary Engineering Mathematics by BS Grewal, Khanna Publishers, New Delhi
- Engineering Mathematics, Vol I & II by SS Sastry, Prentice Hall of India Pvt. Ltd.
- Applied Mathematics-III by Chauhan and Chauhan, Krishna Publications, Meerut
- Applied Mathematics-II by Kailash Sinha and Varun Kumar; Aarti Publication, Meerut
- E-books/e-tools/relevant software to be used as recommended by AICTE/ NITTTR, Chandigarh
SUGGESTED DISTRIBUTION OF MARKS
| Topic No. | Time Allotted (Periods) | Marks Allotted (%) |
|---|---|---|
| 1 | 16 | 24 |
| 2 | 15 | 20 |
| 3 | 15 | 20 |
| 4 | 12 | 18 |
| 5 | 12 | 18 |
| Total | 70 | 100 |