Overview

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Build the mathematical toolkit you need for materials science and engineering. This course covers three essential areas, multivariable calculus, infinite series, and complex numbers, and connects each topic directly to physical and engineering applications. You'll start with multiple integration across Cartesian, cylindrical, and spherical coordinate systems. You'll learn how the Jacobian and scale factors enable correct transformations between systems, and you'll apply these tools to calculate solid angles, moments of inertia, and other physical properties for symmetric geometries. Next, you'll study infinite series and convergence criteria using the comparison, ratio, and integral tests. You'll work with Taylor and power series to approximate complex physical laws, including relativistic mechanics models, and perform term-by-term calculus operations to solve advanced problems. Finally, you'll explore complex numbers in both Cartesian and polar form, connect them through Euler's formula, and apply complex impedance methods to solve differential equations for LCR electrical circuits. Who this is for: First- and second-year undergraduates in engineering or science, advanced high school students, and working professionals seeking to strengthen their applied mathematics foundations.

Syllabus

Calculus of Many Variables

In this module, you'll learn how to perform multiple integration, starting with geometric volume calculations for spheres and cones, then expanding to more complex geometries. You'll work across three coordinate systems (Cartesian, cylindrical, and spherical) and discover why the Jacobian and scale factors are essential when transforming between them. By the end of the module, you'll apply these tools to calculate solid angles and moments of inertia, physical quantities that appear throughout materials science and engineering.

Infinite Series

This module focuses on the convergence criteria for infinite series, covering essential tools such as the comparison, ratio, and integral tests. You will delve into Taylor and power series to understand how functions are represented as polynomials within a specific radius of convergence. Next you will apply these techniques to approximate complex physical laws in relativistic mechanics and perform term-by-term calculus to solve advanced engineering problems.

Complex Numbers

Complex numbers are far more than abstract math, they're a powerful tool for solving real engineering problems. This module introduces the definition and basic operations of complex numbers, transitioning from Cartesian form to polar representation. Using Euler’s formula, you will explore the profound relationship between complex exponentials and trigonometric functions. You will apply these mathematical tools to analyse LCR electrical circuits, learning to solve differential equations by reducing them to algebraic forms using impedance and phase analysis. This technique is one you will use repeatedly in electrical engineering and physics.

Applied Math for Materials Science and Engineering (Part 2) Final Exam

This module provides a comprehensive evaluation of the core concepts of Applied Math for Materials Science and Engineering. You will demonstrate your ability to integrate across multiple coordinate systems, test series for convergence, apply Taylor approximations, and use complex number techniques to solve engineering problems. The final exam evaluates your proficiency in constructing mathematical models and analysing materials science and engineering systems through diverse coordinate systems and approximation methods.