 SCI 211 - Mathematical Methods

Content
SCI 211 - Mathematical Methods is one of the two 200-level Mathematics courses offered at University College Roosevelt; SCI 211 is offered bi-annually and alternates with the course SCI 212 - Theory of Statistics and Data Analysis.

SCI 111 - Mathematical Ideas & Methods in Context introduces basic mathematical methods and techniques, such as limits, differentiation, integration, Taylor series, complex numbers, difference and differential equations and matrix algebra. SCI 211 expands on these topics and discusses a number of new techniques. Many of these methods and techniques are strongly related to physics and engineering.

The course is divided into three parts. The first part of the course deals with vector differential and vector integral calculus. Vector algebra is introduced; including the scalar product, the cross product and the scalar triple product. The Nabla operator formalizes the concept of vector differentiation. The Nabla operator applied to a scalar function amounts to the Gradient, the Nabla operator in combination with scalar and cross product of a vector field results in the Divergence and Curl. This completes the discussion on vector differential calculus. Vector integral calculus combines the techniques for integration from SCI 111 and the knowledge on vectors gained so far. The discussion is limited to path integrals, including path independence, and surface integrals by means of Green's Theorem. Gauss' and Stokes' Theorem are discussed in SCI 221 - Electromagnetism.

The second part of the course addresses Fourier series, integrals and transforms. Periodic functions can be represented by an infinite trigonometric series. Each term of this series is the sum of a sine and cosine with a multiple of a fundamental frequency as argument. This allows for representing periodic functions in terms of the coeffcients of such a trigonometric series, or in short a Fourier Series. Stretching the period to in nity amounts to the Fourier integral. The complex form of the Fourier integrals yields the Fourier Transform. Fourier series, Fourier Transforms, but also the Laplace and the Z-Transform will be discussed in much greater detail in the course SCI 311 - Signals & Systems. SCI 211 applies these techniques to partial differential equations, to be specific, the one-dimensional wave equation and the one-dimensional heat equation.

The third and final part of SCI 211 deals with complex analysis. Complex  numbers, introduced in SCI 111, are reviewed and this knowledge is extended with the concepts of analytic and harmonic functions and conformal mapping. The importance of the exponential function in complex analysis will become clear. These techniques are used to study complex integration; line integrals and Cauchy's integral formulas will be discussed. Finally, complex analysis is applied to potential theory.

The following course is required in order to take this course:

• SCI 111 Mathematical Ideas and Methods in Contexts

This course is required in order to take the following course:

• SCI 311 Signals and Systems
• SCI 312 Advanced Mathematics