Mathematical Techniques for Problem Solving in Engineering and Science from edX

Mathematics is the most essential tool in any STEM professional’s toolbox. In this course, we will provide you with an introduction to linear algebra, multivariable calculus, and differential equations, through exploring the main definitions, theorems and practical examples required.

Can we use linear algebra to do data compression? What’s the meaning of an eigenvalue and an eigenvector in a mechanical system? How do vector fields help to describe wind flow? How can you make optimal parameter choices in industrial processes?

We aim to answer all these questions and more, so that you can use these mathematical techniques when tackling problems in your own field of study.

We will use examples, graphic representations, applets, and exercises to exemplify the various theorems and definitions.

You will acquire the skills to cope with matrix-formulated problems typically arising from applications in science and technology. Not only will you be able to use practical algorithms, solve systems of equations and differential equations, compute the singular value and eigenvalue decomposition, and solve optimisation problems, you will also acquire a set of properties that will assist in simplifying and understanding mathematical problems.

The course will give you the tools to transform optimisation problems and differential equations into matrix language. Most importantly, you will learn that matrix computations are ubiquitous in science and engineering.

What you'll learn

What vector spaces are and how their elements can be represented by coordinate vectors with respect to a basis
Linear transformations between vector spaces and how to represent them in matrix notation
To compute inner products, norms, and orthogonal projections
To define and calculate eigenvalues and eigenvectors and their algebraic and geometric multiplicities
To calculate the singular value decomposition
To understand the concepts of a real function of multiple variables, partial and directional derivatives and the multivariate chain rule
To determine critical points and identify extrema of multivariate functions
To understand the concepts of (conservative) vector fields and be able to calculate and simplify their line integrals
To understand what gradient, divergence, and curl operators are and how to calculate them
To classify and solve (systems of) first-order differential equations
To understand and apply linear algebra techniques to solve linear systems of differential equations with constant coefficients and analyse their stability

What's inside

Learning objectives

What vector spaces are and how their elements can be represented by coordinate vectors with respect to a basis
Linear transformations between vector spaces and how to represent them in matrix notation
To compute inner products, norms, and orthogonal projections
To define and calculate eigenvalues and eigenvectors and their algebraic and geometric multiplicities
To calculate the singular value decomposition
To understand the concepts of a real function of multiple variables, partial and directional derivatives and the multivariate chain rule

To determine critical points and identify extrema of multivariate functions
To understand the concepts of (conservative) vector fields and be able to calculate and simplify their line integrals
To understand what gradient, divergence, and curl operators are and how to calculate them
To classify and solve (systems of) first-order differential equations
To understand and apply linear algebra techniques to solve linear systems of differential equations with constant coefficients and analyse their stability

What vector spaces are and how their elements can be represented by coordinate vectors with respect to a basis
Linear transformations between vector spaces and how to represent them in matrix notation
To compute inner products, norms, and orthogonal projections
To define and calculate eigenvalues and eigenvectors and their algebraic and geometric multiplicities
To calculate the singular value decomposition
To understand the concepts of a real function of multiple variables, partial and directional derivatives and the multivariate chain rule
To determine critical points and identify extrema of multivariate functions
To understand the concepts of (conservative) vector fields and be able to calculate and simplify their line integrals
To understand what gradient, divergence, and curl operators are and how to calculate them
To classify and solve (systems of) first-order differential equations
To understand and apply linear algebra techniques to solve linear systems of differential equations with constant coefficients and analyse their stability

Syllabus

Vector Spaces

Basis and Coordinates

Fundamental Spaces

Linear Transformations

Inner Product Spaces

Inner Product and Norm

Projection and Orthogonal Bases

Least Squares

The Eigenvalue Decomposition

Eigenvalues and Eigenvectors

Theorem and Properties

The Eigenvalue Decomposition

Properties of Symmetric Matrices

The Singular Value Decomposition

Optimisation

Real Functions of n Real Variables

Curves in Rn

Partial Derivatives and Gradient

Extrema

Integral Theorems

Vector and Scalar Fields

Conservative Vector Fields

Line Integrals of Vector Fields

Double Integrals

Gradient, Divergences, Curl Operators

Theorem of Green

Differential Equations

First Order Differential Equations

Linear Systems of First Order Differential Equations

Non-Linear Autonomous Systems

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Taught by experts in their field, this course offers credibility and depth

Develops core STEM skills necessary for success in various industries and fields

Builds a strong foundation in essential mathematical concepts and techniques

Provides real-world examples and applications, making the learning process more engaging and relevant

Covers a wide range of topics, giving learners a comprehensive overview of the subject matter

May require a strong mathematical background and comfort level with abstract concepts

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Mathematical Techniques for Problem Solving in Engineering and Science with these activities:

Organize and Review Course Content

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Establish a solid foundation by organizing and reviewing course materials in advance.

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Gather all course materials, such as syllabus, lecture notes, and assignments.
Create a system for organizing and storing the materials.
Review the materials regularly to reinforce your understanding.

Review Matrix Operations

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Strengthen your foundation in matrix algebra before starting the course.

Browse courses on Linear Transformations

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Review your notes or textbooks on matrix operations.
Practice solving matrix problems involving addition, subtraction, multiplication, and inversion.
Check your answers using an online matrix calculator or software.

Read 'Linear Algebra Done Right'

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Supplement your course materials with a comprehensive textbook on linear algebra.

View Linear Algebra Done Right (Undergraduate Texts... on Amazon

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Purchase or borrow the book.
Read the book thoroughly, taking notes and working through the examples.
Attend office hours or online forums to discuss the book's content.

Four other activities

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Practice Linear Algebra Problems

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Solve practice problems to reinforce your understanding of linear algebra concepts.

Browse courses on Vector Spaces

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Find practice problems online or in textbooks.
Solve the problems step-by-step, showing your work.
Check your answers against the provided solutions.

Attend a Mathematics Conference

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Network with professionals in the field and learn about the latest advancements in mathematics.

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Find a mathematics conference that aligns with your interests.
Register for the conference and attend the sessions.
Network with other attendees and exchange ideas.

Create a Vector Space Visualization

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Deepen your understanding of vector spaces by creating a visual representation.

Browse courses on Vector Spaces

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Choose a vector space to visualize.
Select a software or tool for creating the visualization.
Create the visualization, ensuring that it accurately reflects the vector space's properties.

Develop a Differential Equation Model

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Apply your understanding of differential equations by creating a model for a real-world problem.

Browse courses on Differential Equations

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Identify a problem that can be modeled using differential equations.
Develop the mathematical model using differential equations.
Solve the differential equations to obtain the solution.
Validate the model by comparing it to real-world data.

Career center

Learners who complete Mathematical Techniques for Problem Solving in Engineering and Science will develop knowledge and skills that may be useful to these careers:

Data Scientist

Data Scientists use mathematical techniques to solve complex problems and make informed decisions. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for data analysis and modeling. You'll learn how to manipulate and visualize data, perform statistical analysis, and build predictive models. This course will give you a solid foundation for a successful career as a Data Scientist.

See salaries and explore the career path for Data Scientist

Financial Analyst

Financial Analysts use mathematical techniques to evaluate investments and make financial decisions. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding financial markets and developing financial models. You'll learn how to analyze financial data, forecast trends, and make recommendations for investment decisions. This course will give you a competitive edge in the financial industry.

See salaries and explore the career path for Financial Analyst

Quantitative Analyst

Quantitative Analysts use mathematical techniques to develop and implement financial models. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding financial markets and developing quantitative models. You'll learn how to analyze financial data, forecast trends, and make recommendations for investment decisions. This course will give you a competitive advantage in the financial industry.

See salaries and explore the career path for Quantitative Analyst

Software Engineer

Software Engineers use mathematical techniques to design and develop software systems. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding software architecture and developing efficient algorithms. You'll learn how to analyze software requirements, design software systems, and implement software solutions. This course will help you build a strong foundation for a successful career as a Software Engineer.

See salaries and explore the career path for Software Engineer

Operations Research Analyst

Operations Research Analysts use mathematical techniques to improve the efficiency and effectiveness of operations. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding optimization models and developing operational strategies. You'll learn how to analyze data, develop models, and implement solutions to improve operational performance. This course will give you a competitive edge in the field of Operations Research.

See salaries and explore the career path for Operations Research Analyst

Actuary

Actuaries use mathematical techniques to assess and manage risk. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding risk models and developing insurance products. You'll learn how to analyze data, calculate premiums, and develop risk management strategies. This course will give you a strong foundation for a successful career as an Actuary.

See salaries and explore the career path for Actuary

Physics

Physicists use mathematical techniques to understand the physical world. The course will provide you with a strong foundation in linear algebra, multivariable calculus, and differential equations, which are essential for understanding physical laws and developing mathematical models. You'll learn how to solve problems in mechanics, electromagnetism, and quantum mechanics. This course will prepare you for a successful career as a Physicist.

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Statistician

Statisticians use mathematical techniques to collect, analyze, and interpret data. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding statistical models and developing data analysis methods. You'll learn how to design experiments, collect data, perform statistical analysis, and make inferences. This course will give you a strong foundation for a successful career as a Statistician.

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Mathematician

Mathematicians use mathematical techniques to solve complex problems in various fields. The course will provide you with a strong foundation in linear algebra, multivariable calculus, and differential equations, which are essential for understanding mathematical concepts and developing mathematical models. You'll learn how to prove theorems, solve problems, and develop new mathematical theories. This course will prepare you for a successful career as a Mathematician.

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Economist

Economists use mathematical techniques to analyze economic data and develop economic models. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding economic theories and developing economic models. You'll learn how to analyze economic data, forecast trends, and make policy recommendations. This course will give you a competitive edge in the field of Economics.

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Computer Scientist

Computer Scientists use mathematical techniques to design and develop computer systems. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding computer science principles and developing computer science solutions. You'll learn how to analyze problems, design algorithms, and implement software. This course will give you a strong foundation for a successful career as a Computer Scientist.

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Engineer

Engineers use mathematical techniques to design and develop products and systems. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding engineering principles and developing engineering solutions. You'll learn how to analyze problems, design systems, and implement solutions. This course will give you a strong foundation for a successful career as an Engineer.

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Business Analyst

Business Analysts use mathematical techniques to analyze business data and make business decisions. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding business principles and developing business models. You'll learn how to analyze data, forecast trends, and make recommendations. This course will give you a competitive edge in the field of Business Analysis.

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Consultant

Consultants use mathematical techniques to solve problems and improve performance. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding business principles and developing business solutions. You'll learn how to analyze problems, develop solutions, and implement strategies. This course will give you a competitive edge in the field of Consulting.

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Data Analyst

Data Analysts use mathematical techniques to analyze data and extract insights. The course will provide you with the necessary skills in linear algebra, multivariable calculus, and differential equations, which are essential for understanding data analysis techniques and developing data analysis models. You'll learn how to collect data, clean data, analyze data, and visualize data. This course will give you a competitive edge in the field of Data Analytics.

See salaries and explore the career path for Data Analyst