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Mathematical and Computational Methods
James Freericks
James Freericks
Physicists use math all of the time in nearly everything that they work on. Hence, it is critical that you become efficient in being able to use more advanced math to enable you to work on more advanced physics courses. Most of you are currently what I would call a technician at math. You are able to perform the required manipulations to find and simplify answers. But, in many cases, you are missing the deeper understanding to help you interconnect different math topics and for you to be able to apply the relevant math for whatever problem you will face, based on your knowledge of how the different math topics inter-relate. On the other hand, a practitioner is someone who not only is technically adept at performing math calculations, but also have the insight and deeper understanding to know how to recognize what math applies to what problem. They understand how math is interconnected and recognize that math involves a handful of simple ideas that repeat. They are able to re-derive important formulas from basic principles or know precisely where to look them up and use them.
Read more
Physicists use math all of the time in nearly everything that they work on. Hence, it is critical that you become efficient in being able to use more advanced math to enable you to work on more advanced physics courses. Most of you are currently what I would call a technician at math. You are able to perform the required manipulations to find and simplify answers. But, in many cases, you are missing the deeper understanding to help you interconnect different math topics and for you to be able to apply the relevant math for whatever problem you will face, based on your knowledge of how the different math topics inter-relate. On the other hand, a practitioner is someone who not only is technically adept at performing math calculations, but also have the insight and deeper understanding to know how to recognize what math applies to what problem. They understand how math is interconnected and recognize that math involves a handful of simple ideas that repeat. They are able to re-derive important formulas from basic principles or know precisely where to look them up and use them.
Read more
Physicists use math all of the time in nearly everything that they work on. Hence, it is critical that you become efficient in being able to use more advanced math to enable you to work on more advanced physics courses. Most of you are currently what I would call a technician at math. You are able to perform the required manipulations to find and simplify answers. But, in many cases, you are missing the deeper understanding to help you interconnect different math topics and for you to be able to apply the relevant math for whatever problem you will face, based on your knowledge of how the different math topics inter-relate. On the other hand, a practitioner is someone who not only is technically adept at performing math calculations, but also have the insight and deeper understanding to know how to recognize what math applies to what problem. They understand how math is interconnected and recognize that math involves a handful of simple ideas that repeat. They are able to re-derive important formulas from basic principles or know precisely where to look them up and use them.
The goal of this class is to transform you from a math technician to a math practitioner. Mathematicians take this one step further and actually create new math. We will not focus on how to do that at all in this class.
Register for this course and see more details by visiting:
OpenCourser.com/course/o27efq/mathematical
What's inside
Learning objectives
- Be able to apply techniques of calculus (learned in the first three semesters of a calculus sequence) to solve problems that arise in physics.
- Derive and use the geometric series in calculations.
- Manipulate power series expressions and employ them in physics contexts.
- Calculate taylor polynomials/series of common functions and use them in approximating functions
- Follow the development for how one integrates polynomials, rational functions of polynomials, square roots of quadratics, rational functions with square roots of quadratics, and why the procedure cannot solve integrals with square roots of quartics.
- Solve integrals via parametric methods (differentiating under the integral sign) including techniques for introducing the parameter into the integrand
- Set up and integrate multidimensional integrals with variable mass density and for moments of inertia.
- Solve problems in multivariable integrals via the different integral theorems
- Solve laplace's equation for simple geometries
- Compute the curl or divergence of a vector field and the gradient of a scalar function
- Solve complex numbers arithmetic problems
- Use calculus of residues to solve integrals
- Use row reduction to solve simultaneous linear equations
- Calculate the determinant, inverse, and eigenvalues of matrices
- Construct abstract vector spaces using the definitions of a vector space and generalized inner products
- Solve any first-order linear differential equation
- Solve the six classes of nonlinear first-order differential equations
- Solve higher-order differential equations with constant coefficients (homogeneous and inhomogeneous)
- Construct representations of curves in three dimensions using the frenet-serret apparatus
- Construct fourier series of periodic functions
- Be able to apply techniques of calculus (learned in the first three semesters of a calculus sequence) to solve problems that arise in physics.
- Derive and use the geometric series in calculations.
- Manipulate power series expressions and employ them in physics contexts.
- Calculate taylor polynomials/series of common functions and use them in approximating functions
- Follow the development for how one integrates polynomials, rational functions of polynomials, square roots of quadratics, rational functions with square roots of quadratics, and why the procedure cannot solve integrals with square roots of quartics.
- Solve integrals via parametric methods (differentiating under the integral sign) including techniques for introducing the parameter into the integrand
- Set up and integrate multidimensional integrals with variable mass density and for moments of inertia.
- Solve problems in multivariable integrals via the different integral theorems
- Solve laplace's equation for simple geometries
- Compute the curl or divergence of a vector field and the gradient of a scalar function
- Solve complex numbers arithmetic problems
- Use calculus of residues to solve integrals
- Use row reduction to solve simultaneous linear equations
- Calculate the determinant, inverse, and eigenvalues of matrices
- Construct abstract vector spaces using the definitions of a vector space and generalized inner products
- Solve any first-order linear differential equation
- Solve the six classes of nonlinear first-order differential equations
- Solve higher-order differential equations with constant coefficients (homogeneous and inhomogeneous)
- Construct representations of curves in three dimensions using the frenet-serret apparatus
- Construct fourier series of periodic functions
Good to know
Good to know
Know what's good ,
what to watch for , and
possible dealbreakers
Taught by James Freericks, a lecturer in the physics of materials at the University of California, Irvine
Explores a deeper understanding of math, which is highly relevant to physics
Teaches the practical use of calculus, which is standard in the field of physics
Examines various mathematical techniques, including complex numbers, differential equations, and Fourier series, which are highly relevant to different areas of physics
Develops proficiency in applying mathematical techniques to solve problems in physics, which is a core skill for physicists
Save this course
Save Mathematical and Computational Methods to your list so you can find it easily later:
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Reviews summary
Mathematical methods
Learners say this highly recommended course in mathematical and computational methods is well-received and challenging. Engaging assignments, an experienced instructor, and excellent content are all cited as strengths of this course.
Learners say that the course has both engaging and fun labs, which provide opportunities for hands-on practice.
"Many unexpected but interesting topics included."
"I enjoyed the videos with the experiments/demonstrations."
"The labs were really interesting and fun and probably my favorite part of the course."
Students say this course is challenging but rewarding and that regular practice helps improve comprehension.
"It's a long time since I made a living as engineer so I decided to take this course to catch up with my college calculus knowledge in order to enable me to follow other courses and refresh my understanding of physics."
"Challenging but very rewarding course."
" WARNING: Do not attempt this course without the required mathematics background."
"The big number of exercises really helped practice my skills and improve comprehension."
The instructor is described as knowledgeable, clear, helpful, and available to answer questions.
"This is my second course with him."
"I can't wait to have the third one!"
"Professor Jim Freericks was very helpful and available during the whole class, and he is an excellent teacher, and I enjoyed every week."
Learners say this is a well-received course that teaches a strong mathematical foundation for physics and engineering.
"This is a foundation course for his Quantum Mechanics course which is also just fantastic."
"Highly recommended."
Learners say that the course covers a wide range of topics and material needed for upper-level physics and engineering.
"An excellent course by a great teacher."
"A bold and ambitious course."
"This course is suitable both for those who want to gain a birdview of certain math concepts and for those who want to review the concepts."
"Many unexpected but interesting topics included."
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 and Computational Methods with these
activities:
Attend 'Advanced Calculus for Physics Applications' Workshop
Show steps
Participate in a workshop focused on advanced calculus techniques utilized in physics, providing opportunities for hands-on learning and expert guidance.
Browse courses on
Calculus
Show steps
-
Register for the 'Advanced Calculus for Physics Applications' workshop.
-
Attend the workshop sessions and engage in discussions.
Review 'Mathematical Methods for Physics and Engineers'
Show steps
Reinforce understanding of mathematical concepts essential for physics by reviewing a foundational textbook, 'Mathematical Methods for Physics and Engineers'.
View
Mathematical Methods for Physics and...
on Amazon
Show steps
-
Read selected chapters relevant to the course.
-
Review solved examples and practice exercises.
Solve calculus problems
Show steps
Solving calculus problems will help you master the techniques and concepts covered in the course.
Browse courses on
Calculus
Show steps
-
Identify the type of problem you are solving.
-
Apply the appropriate calculus techniques.
-
Check your answer.
Ten other activities
Expand to see all activities and additional details
Show all 13 activities
Watch video tutorials on math topics
Show steps
Watching video tutorials on math topics can help you reinforce your understanding of the concepts covered in the course.
Browse courses on
Calculus
Show steps
-
Find video tutorials on the math topic you are studying.
-
Watch the video tutorials and take notes.
-
Try to solve practice problems related to the topic.
Join a study group
Show steps
Joining a study group can help you learn from your peers and improve your understanding of the course material.
Browse courses on
Calculus
Show steps
-
Find a study group that meets your needs.
-
Attend study group meetings regularly.
-
Participate in discussions and ask questions.
Apply Calculus to Physics Problems
Show steps
Collaborate with peers to apply calculus concepts to real-world physics problems, fostering teamwork and enhancing problem-solving abilities.
Browse courses on
Calculus
Show steps
-
Identify a physics problem that requires calculus.
-
Form a study group with 2-3 classmates.
-
Work together to apply calculus techniques to solve the problem.
Integrate Polynomials and Rational Functions
Show steps
Practice integrating polynomials and rational functions of polynomials to improve fluency and recall of integration techniques.
Show steps
-
Review integration rules for polynomials and rational functions.
-
Complete 20 practice problems integrating polynomials and rational functions.
Visualize Vector Fields
Show steps
Deepen understanding of vector fields by creating visual representations using software or online tools, reinforcing concepts of divergence and curl.
Browse courses on
Vector Calculus
Show steps
-
Explore vector field visualization tools and select one to use.
-
Obtain or create a dataset representing a vector field.
-
Visualize the vector field using the chosen tool, experimenting with different parameters.
Write a math paper
Show steps
Writing a math paper will help you develop your mathematical writing skills and deepen your understanding of the course material.
Browse courses on
Calculus
Show steps
-
Choose a math topic to write about.
-
Research the topic and gather information.
-
Write an outline for your paper.
-
Write the first draft of your paper.
-
Revise and edit your paper.
Create a math portfolio
Show steps
Creating a math portfolio can help you showcase your skills and knowledge to potential employers or graduate schools.
Browse courses on
Calculus
Show steps
-
Gather your math work, including assignments, projects, and papers.
-
Select the best pieces to include in your portfolio.
-
Organize your portfolio in a logical way.
Solve Laplace's Equation for Simple Geometries
Show steps
Enhance understanding of Laplace's Equation and its applications by following guided tutorials and solving problems in simple geometries.
Show steps
-
Watch video tutorials on solving Laplace's Equation.
-
Attempt step-by-step walkthroughs of solving Laplace's Equation for rectangular and circular geometries.
Build a Fourier Series Representation
Show steps
Enhance comprehension of Fourier series by constructing a representation of a given periodic function, reinforcing analytical and programming skills.
Browse courses on
Fourier Series
Show steps
-
Choose a periodic function to represent.
-
Derive the Fourier series coefficients for the chosen function.
-
Implement a program to calculate and plot the Fourier series representation.
Contribute to Sympy's Calculus Module
Show steps
Deepen understanding of calculus and programming by contributing to the open-source Sympy library's calculus module, enhancing problem-solving and coding skills.
Browse courses on
Calculus
Show steps
-
Familiarize with the Sympy library and its calculus module.
-
Identify an area in the calculus module that needs improvement or extension.
-
Implement a code solution and submit a pull request to the Sympy repository.
Attend 'Advanced Calculus for Physics Applications' Workshop
Show steps
Participate in a workshop focused on advanced calculus techniques utilized in physics, providing opportunities for hands-on learning and expert guidance.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Register for the 'Advanced Calculus for Physics Applications' workshop.
- Attend the workshop sessions and engage in discussions.
Review 'Mathematical Methods for Physics and Engineers'
Show steps
Reinforce understanding of mathematical concepts essential for physics by reviewing a foundational textbook, 'Mathematical Methods for Physics and Engineers'.
View
Mathematical Methods for Physics and...
on Amazon
Show steps
Show steps
Show steps
- Read selected chapters relevant to the course.
- Review solved examples and practice exercises.
Solve calculus problems
Show steps
Solving calculus problems will help you master the techniques and concepts covered in the course.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Identify the type of problem you are solving.
- Apply the appropriate calculus techniques.
- Check your answer.
Ten other activities
Expand to see all activities and additional details
Show all 13 activities
Watch video tutorials on math topics
Show steps
Watching video tutorials on math topics can help you reinforce your understanding of the concepts covered in the course.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Find video tutorials on the math topic you are studying.
- Watch the video tutorials and take notes.
- Try to solve practice problems related to the topic.
Join a study group
Show steps
Joining a study group can help you learn from your peers and improve your understanding of the course material.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Find a study group that meets your needs.
- Attend study group meetings regularly.
- Participate in discussions and ask questions.
Apply Calculus to Physics Problems
Show steps
Collaborate with peers to apply calculus concepts to real-world physics problems, fostering teamwork and enhancing problem-solving abilities.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Identify a physics problem that requires calculus.
- Form a study group with 2-3 classmates.
- Work together to apply calculus techniques to solve the problem.
Integrate Polynomials and Rational Functions
Show steps
Practice integrating polynomials and rational functions of polynomials to improve fluency and recall of integration techniques.
Show steps
Show steps
Show steps
- Review integration rules for polynomials and rational functions.
- Complete 20 practice problems integrating polynomials and rational functions.
Visualize Vector Fields
Show steps
Deepen understanding of vector fields by creating visual representations using software or online tools, reinforcing concepts of divergence and curl.
Browse courses on
Vector Calculus
Show steps
Show steps
Show steps
- Explore vector field visualization tools and select one to use.
- Obtain or create a dataset representing a vector field.
- Visualize the vector field using the chosen tool, experimenting with different parameters.
Write a math paper
Show steps
Writing a math paper will help you develop your mathematical writing skills and deepen your understanding of the course material.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Choose a math topic to write about.
- Research the topic and gather information.
- Write an outline for your paper.
- Write the first draft of your paper.
- Revise and edit your paper.
Create a math portfolio
Show steps
Creating a math portfolio can help you showcase your skills and knowledge to potential employers or graduate schools.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Gather your math work, including assignments, projects, and papers.
- Select the best pieces to include in your portfolio.
- Organize your portfolio in a logical way.
Solve Laplace's Equation for Simple Geometries
Show steps
Enhance understanding of Laplace's Equation and its applications by following guided tutorials and solving problems in simple geometries.
Show steps
Show steps
Show steps
- Watch video tutorials on solving Laplace's Equation.
- Attempt step-by-step walkthroughs of solving Laplace's Equation for rectangular and circular geometries.
Build a Fourier Series Representation
Show steps
Enhance comprehension of Fourier series by constructing a representation of a given periodic function, reinforcing analytical and programming skills.
Browse courses on
Fourier Series
Show steps
Show steps
Show steps
- Choose a periodic function to represent.
- Derive the Fourier series coefficients for the chosen function.
- Implement a program to calculate and plot the Fourier series representation.
Contribute to Sympy's Calculus Module
Show steps
Deepen understanding of calculus and programming by contributing to the open-source Sympy library's calculus module, enhancing problem-solving and coding skills.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Familiarize with the Sympy library and its calculus module.
- Identify an area in the calculus module that needs improvement or extension.
- Implement a code solution and submit a pull request to the Sympy repository.
Career center
Learners who complete Mathematical and Computational Methods will develop knowledge and skills
that may be useful to these careers:
Astronomer
Astronomer
An Astronomer may use advanced mathematics and computational methods to analyze data and make predictions about the universe. For example, this course could help an Astronomer who wants to develop new models of the evolution of galaxies or design new instruments to study the cosmos. This course would be especially useful for an Astronomer interested in computational cosmology, which uses mathematical and computational techniques to model the large-scale structure and evolution of the universe.
Mechanical Engineer
Mechanical Engineer
A Mechanical Engineer may use advanced mathematics and computational methods to design and test machines, engines, and other mechanical systems. This course would be especially useful for a Mechanical Engineer who wants to develop new methods for simulating the behavior of a machine or engine or for designing new materials for use in mechanical applications.
Chemical Engineer
Chemical Engineer
A Chemical Engineer may use advanced mathematics and computational methods to design and test chemical plants and processes. This course would be especially useful for a Chemical Engineer who wants to develop new methods for simulating the behavior of a chemical plant or process or for designing new materials for use in chemical engineering applications.
Civil Engineer
Civil Engineer
A Civil Engineer may use advanced mathematics and computational methods to design and test bridges, buildings, and other civil structures. This course would be especially useful for a Civil Engineer who wants to develop new methods for simulating the behavior of a bridge or building or for designing new materials for use in civil engineering applications.
Aerospace Engineer
Aerospace Engineer
An Aerospace Engineer may use advanced mathematics and computational methods to design and test aircraft, spacecraft, and other vehicles. This course would be especially useful for an Aerospace Engineer who wants to develop new methods for simulating the flight of an aircraft or spacecraft or for designing new materials for use in aerospace applications.
Electrical Engineer
Electrical Engineer
An Electrical Engineer may use advanced mathematics and computational methods to design and test electrical circuits and systems. This course would be especially useful for an Electrical Engineer who wants to develop new methods for simulating the behavior of an electrical circuit or system or for designing new materials for use in electrical applications.
Physicist
Physicist
A Physicist may use advanced mathematics and computational methods in nearly all aspects of their work. Most especially, this course may be useful for a Physicist who wants to write or analyze advanced simulations in their field of study. There are a number of methods taught in this course that can help build a foundation in computational physics.
Operations Research Analyst
Operations Research Analyst
An Operations Research Analyst may use advanced mathematics and computational methods to solve problems in logistics, supply chain management, and other business operations. This course would be especially useful for an Operations Research Analyst who wants to develop new methods for solving business problems or for designing new algorithms for use in operations research applications.
Software Engineer
Software Engineer
A Software Engineer may use advanced mathematics and computational methods to design and test software systems. This course would be especially useful for a Software Engineer who wants to develop new methods for simulating the behavior of a software system or for designing new algorithms for use in software engineering applications.
Financial Analyst
Financial Analyst
A Financial Analyst may use advanced mathematics and computational methods to analyze financial data and make investment recommendations. This course would be especially useful for a Financial Analyst who wants to develop new methods for analyzing financial data or for designing new models for use in financial analysis applications.
Computer Scientist
Computer Scientist
A Computer Scientist may use advanced mathematics and computational methods to design and test computer systems and software. This course would be especially useful for a Computer Scientist who wants to develop new methods for simulating the behavior of a computer system or software or for designing new algorithms for use in computer science applications.
Systems Engineer
Systems Engineer
A Systems Engineer may use advanced mathematics and computational methods to design and test complex systems, such as computer systems, communication systems, and transportation systems. This course would be especially useful for a Systems Engineer who wants to develop new methods for simulating the behavior of a complex system or for designing new algorithms for use in systems engineering applications.
Data Scientist
Data Scientist
A Data Scientist may use advanced mathematics and computational methods to analyze data and make predictions. This course would be especially useful for a Data Scientist who wants to develop new methods for analyzing data or for designing new algorithms for use in data science applications.
Mathematician
Mathematician
A Mathematician may use advanced mathematics and computational methods to solve problems in a wide variety of fields, such as physics, chemistry, biology, and engineering. This course would be especially useful for a Mathematician who wants to develop new methods for solving problems in a particular field or for designing new algorithms for use in mathematics applications.
Computational Scientist
Computational Scientist
A Computational Scientist may use advanced mathematics and computational methods to solve problems in a wide variety of fields, such as physics, chemistry, biology, and engineering. This course would be especially useful for a Computational Scientist who wants to develop new methods for solving problems in a particular field or for designing new algorithms for use in computational science applications.
For more career information including salaries, visit:
OpenCourser.com/course/o27efq/mathematical
Reading list
We've selected 15 books
that we think will supplement your
learning. Use these to
develop background knowledge, enrich your coursework, and gain a
deeper understanding of the topics covered in
Mathematical and Computational Methods.
Classic physics textbook that provides a comprehensive overview of nuclear and particle physics. It good choice for students who need to learn the basics of nuclear and particle physics or who want to learn more about the subject.
Classic physics textbook that provides a comprehensive overview of quantum mechanics. It good choice for students who need to learn the basics of quantum mechanics or who want to learn more about the subject.
Classic physics textbook that provides a comprehensive overview of statistical physics. It good choice for students who need to learn the basics of statistical physics or who want to learn more about the subject.
Save
Classic physics textbook that provides a comprehensive overview of electromagnetism. It good choice for students who need to learn the basics of electromagnetism or who want to learn more about the subject.
Classic physics textbook that provides a comprehensive overview of optics. It good choice for students who need to learn the basics of optics or who want to learn more about the subject.
Classic physics textbook that provides a comprehensive overview of solid state physics. It good choice for students who need to learn the basics of solid state physics or who want to learn more about the subject.
Provides a comprehensive overview of the mathematical methods used in physics and engineering, including calculus, linear algebra, differential equations, and complex analysis. It valuable reference for students and professionals in these fields.
Classic physics textbook that provides a comprehensive overview of the subject. It good choice for students who need to learn the basics of physics or who want to learn more about the subject.
Save
Classic physics textbook that provides a comprehensive overview of classical mechanics. It good choice for students who need to learn the basics of classical mechanics or who want to learn more about the subject.
Classic calculus textbook that provides a thorough grounding in the subject. It good choice for students who need to review the basics of calculus or who want to learn more about the subject.
Provides a clear and concise introduction to linear algebra. It good choice for students who need to learn the basics of linear algebra or who want to learn more about the subject.
Provides a comprehensive overview of differential equations. It good choice for students who need to learn the basics of differential equations or who want to learn more about the subject.
Provides a comprehensive overview of complex variables. It good choice for students who need to learn the basics of complex variables or who want to learn more about the subject.
Provides a comprehensive overview of the mathematical methods used in the physical sciences. It good choice for students who need to learn the basics of the mathematical methods used in the physical sciences or who want to learn more about the subject.
Provides a comprehensive overview of the mathematical methods used in physics. It good choice for students who need to learn the basics of the mathematical methods used in physics or who want to learn more about the subject.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/o27efq/mathematical
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Effort
7 to 10 hours per week for 18 weeks
Level
Advanced
Via
edX
Institution
Georgetown University
Instructor
James Freericks
Session
January 14, 2024
- May 26, 2024
Language
English
Transcript
English
This course belongs to a
collection
called
Foundations of Quantum Sensing. It's comprised of one courses:
- Mathematical and Computational Methods (viewing)
Good to know
Taught by James Freericks, a lecturer in the physics of materials at the University of California, Irvine
Explores a deeper understanding of math, which is highly relevant to physics
Teaches the practical use of calculus, which is standard in the field of physics
Examines various mathematical techniques, including complex numbers, differential equations, and Fourier series, which are highly relevant to different areas of physics
Develops proficiency in applying mathematical techniques to solve problems in physics, which is a core skill for physicists
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