This course offers a brief introduction to the multivariate calculus required to build many common machine learning techniques. It deals with calculus and functions of several variables. Students should know the basic concepts and technique of multivariable calculus and vector calculus. Some knowledge on linear algebra, such as matrix notations and calculations, is preferred. we learn how to calculate vectors that point up hill on multidimensional surfaces and even put this into action using an interactive game. We will learn how it applied in linear regression models etc. It would not be unusual for a machine learning method to require the analysis of a function with thousands of inputs.

Intended Learning Outcomes: Upon successful completion of the course, a student will be able to:

1. Sketch three-dimensional surfaces using cross-sections and contours; apply the chain rule to partial derivatives.
2. Solve simple Partial Differential Equations by using a change of variables.
3. State the definitions, results and formulae presented in lectures; apply and adapt these to solve suitable problems.
4. Take a function with multiple inputs and determine the influence of each of them separately.

Contents

1. Multivariable Functions and Partial Derivatives: Functions of Several Variables.
2. Limits and Continuity. Partial Derivatives. Differentiability, Linearization, and Differentials.
3. The Chain Rule. Partial Derivatives with Constrained Variables. Directional Derivatives,
4. Gradient Vectors, and Tangent Planes. Extreme Values and Saddle Points. Lagrange Multipliers. Taylor's Formula. [TB1: Ch. 11]
5. Multiple Integrals: Double Integrals. Areas, Moments, and Centers of Mass.
6. Double Integrals in Polar Form. Triple Integrals in Rectangular Coordinates. Masses and Moments in Three Dimensions.
7. Triple Integrals in Cylindrical and Spherical Coordinates. Substitutions in Multiple Integrals. [TB1: Ch. 12]
8. Laplace Transforms: Laplace Transform. Inverse Transform. Linearity. First Shifting Theorem (s-Shifting). Transforms of Derivatives and Integrals. ODEs. Unit Step Function (Heaviside Function).
9. Second Shifting Theorem (t-Shifting). Short Impulses. Dirac's Delta Function. Partial Fractions. Convolution. Integral Equations. Differentiation and Integration of Transform. Systems of ODEs.
10. Laplace Transform: General Formulas. Table of Laplace Transforms. [TB2: Ch. 6]
11. Fourier Analysis: Fourier Series, Arbitrary Period. Even and Odd Function. Half-Rang Expansions. Forced Oscillations. Approximation by Trigonometric Polynomials.
12. Sturm-Liouville Problems. Orthogonal Functions. Orthogonal Series. Generalized Fourier Series.
13. Fourier Integral. Fourier Cosine and Sine Transforms. Fourier Transform. [TB2:Ch. 11]
14. Power Series, Taylor Series: Sequences, Series, Convergence Tests. Power Series.
15. Functions Given by Power Series. Taylor and Maclaurin Series. [TB2: Ch. 15]
16. Laurent Series. Residue Integration: Laurent Series. Singularities and Zeros. Infinity.
17. Residue Integration Method. Residue Integration of Real Integrals. [TB2: Ch. 16]

Recommended Books

1. Calculus & Analytic Geometry by Thomas, Wiley; 10 th Edition (August 16, 2011).
2.  Advanced Engineering Mathematics by Erwin Kreyszig, Wiley; 10 th Edition

Suggested Books

1. Calculus: One and several variables by Salas, Hille and Etgen, Wiley; 10 th Edition
2. Multivariable Calculus by James Stewart 6 th Edition, 2007, Cengage Learning publishers.
3. Calculus and Analytical Geometry by Swokowski, Olinick and Pence, 6 th Edition, 1994, Thomson Learning EMEA, Ltd.
4. Elementary Multivariable Calculus by Bernard Kolman William F. Trench, 1971, Academic Press.
5. Multivariable Calculus by Howard Anton, Albert Herr 5th Edition, 1995, John Wiley.

RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS

Research project/assignment based on multivariable calculus often involve features which require their solutions can be awarded to students during their course work period. Examples of features are

1. Accelerating and motivating interest to develop algorithms based on calculus.
2. Finding applications and implementing by softwares like Maple
3. Improving existing techniques

Assessment Criteria

• Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)
• Mid-Term Exam: 30 (or Term Paper)
• Final-Term Exam: 50 (or Viva + online)

Key Dates and Time of Class Meeting

Monday/Tuesday                                                           15:30 -17:00 and 17:00 -18:30

Commencement of Classes (Summer Semester)                   October 05, 2020 (Monday)

Mid Term Examination                                                            November 02 to 06, 2020, (Monday to Friday)

Final Term Examination                                                          December 04, 2020 (Friday)

Declaration of Result                                                              December  2020