Multivariable Calculus Course Outline - University of Sargodha

Course Title: Multivariable Calculus

Course Code: MATH-2214
Course Structure: Lectures: 3 / Labs: 0
Credit Hours: 3
Prerequisites: None

Course Objectives:

The goals are to develop the skills to have ground knowledge of multivariate calculus and
appreciation for their further computer science courses.

Course Syllabus:

Multivariable Functions and Partial Derivatives. Multiple Integrals. Laplace Transforms. Fourier
analysis. Power Series, Taylor Series. Power Series. Functions Given by Power Series. Taylor
and Maclaurin Series. Laurent Series. Residue Integration.

Course Outline:

1.     Multivariable Functions and Partial Derivatives: Functions of Several Variables. Limits
and Continuity. Partial Derivatives. Differentiability, Linearization, and Differentials.
The Chain Rule. Partial Derivatives with Constrained Variables. Directional Derivatives,
Gradient Vectors, and Tangent Planes. Extreme Values and Saddle Points. Lagrange
Multipliers. Taylor's Formula. [TB1: Ch. 11]

2.      Multiple Integrals: Double Integrals. Areas, Moments, and Centers of Mass. Double
Integrals in Polar Form. Triple Integrals in Rectangular Coordinates. Masses and
Moments in Three Dimensions. Triple Integrals in Cylindrical and Spherical Coordinates.
Substitutions in Multiple Integrals. [TB1: Ch. 12]

3.      Laplace Transforms: Laplace Transform. Inverse Transform. Linearity. First Shifting
Theorem (s-Shifting). Transforms of Derivatives and Integrals. ODEs. Unit Step
Function (Heaviside Function). Second Shifting Theorem (t-Shifting). Short Impulses.
Dirac's Delta Function. Partial Fractions. Convolution. Integral Equations. Differentiation
and Integration of Transform. Systems of ODEs. Laplace Transform: General Formulas.
Table of Laplace Transforms. [TB2: Ch. 6]

4.      Fourier Analysis: Fourier Series, Arbitrary Period. Even and Odd Function. Half-Rang
Expansions. Forced Oscillations. Approximation by Trigonometric Polynomials. Sturm-
Liouville Problems. Orthogonal Functions. Orthogonal Series. Generalized Fourier
Series. Fourier Integral. Fourier Cosine and Sine Transforms. Fourier Transform. [TB2:
Ch. 11]

5.      Power Series, Taylor Series: Sequences, Series, Convergence Tests. Power Series.

Functions Given by Power Series. Taylor and Maclaurin Series. [TB2: Ch. 15]

6. Laurent Series. Residue Integration: Laurent Series. Singularities and Zeros. Infinity.
Residue Integration Method. Residue Integration of Real Integrals. [TB2: Ch. 16]

Textbook(s):

•          Calculus & Analytic Geometry by Thomas, Wiley; 10^th Edition (August 16, 2011).
ISBN-10: 0470458364

•          Advanced Engineering Mathematics by Erwin Kreyszig, Wiley; 10^th Edition (August 16,
2011). ISBN-10: 0470458364

Reference Material:

•         Multivariable Calculus by James Stewart, Brooks Cole; 7^th Edition (January 1, 2011).
ISBN-10: 0538497874

•         Multivariable Calculus by James Stewart 6^th Edition, 2007, Cengage Learning
publishers.

•         Calculus and Analytical Geometry by Swokowski, Olinick and Pence, 6^th Edition,
1994, Thomson Learning EMEA, Ltd.

•         Elementary Multivariable Calculus by Bernard Kolman William F. Trench, 1971,
Academic Press.

•         Multivariable Calculus by Howard Anton, Albert Herr 5th Edition, 1995, John Wiley.

Note: This content is obtained from official documents of University of Sargodha and applied on BS Computer Science for Main Campus, Sub Campuses, and Affiliated Colleges.