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Calculus of Variations 2

4.9 Filled star Filled star Filled star Filled star Half star
Based on 17 ratings
Ross McGowan

This is the second course in a series of courses on The Calculus of Variations which covers the second variation and some generalised functionals.

I would recommend having completed the first course or alternatively have a good grounding on the basics of Calculus of Variations.

We will cover the fundamentals of the second variation which will consist of a thorough grounding in conjugate points and envelopes. Then we will move onto looking at some more generalised functionals.

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This is the second course in a series of courses on The Calculus of Variations which covers the second variation and some generalised functionals.

I would recommend having completed the first course or alternatively have a good grounding on the basics of Calculus of Variations.

We will cover the fundamentals of the second variation which will consist of a thorough grounding in conjugate points and envelopes. Then we will move onto looking at some more generalised functionals.

This is a difficult course but you will learn a lot of mathematics at a graduate level that you would only get taught at a University. I have worked through all of the derivations myself and have rewritten then in a manner in which I think will add value to your understanding and will also give you really useful methods and way to understand advanced Calculus.

Don't worry of you don't get it at first. I often find when trying to understand an advanced topic it is best to leave it sometimes for a few days and then go back to it afresh. Also remember to get in contact with me if you run into any problems or there is something you just don't get.

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What's inside

Learning objectives

  • Second variation
  • Legendre test
  • Jacobi accessory equation
  • Conjugate points
  • Generalised functionals
  • Green's theorem

Syllabus

Introduction
This is a revision video from the first course covering the first variation
Derivation of Euler Lagrange Equation (The First Variation)
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Career center

Learners who complete Calculus of Variations 2 will develop knowledge and skills that may be useful to these careers:
Theoretical Physicist
A theoretical physicist develops mathematical models and theories to explain physical phenomena, often at the most fundamental level. "Calculus of Variations 2" provides foundational, graduate-level knowledge in the second variation, Jacobi accessory equation, and generalized functionals, which are essential for understanding the stability of physical systems and deriving fundamental equations from action principles. This course is directly relevant to the core methodologies of theoretical physics, crucial for disciplines like quantum field theory or general relativity. This role typically requires an advanced degree. It helps build a foundation for contributing to the most fundamental understanding of the universe.
See salaries and explore the career path for Theoretical Physicist
Mathematical Modeler
A mathematical modeler constructs conceptual and computational models to represent complex systems across diverse scientific, engineering, and economic domains. "Calculus of Variations 2" is profoundly relevant for this role, as it provides an advanced, graduate-level understanding of how to formulate and solve problems by optimizing functionals. The course's detailed treatment of the second variation, conjugate points, and generalized functionals equips modelers with sophisticated tools for deriving governing equations and analyzing the stability and behavior of their models. This course helps build a foundation for creating accurate and insightful representations of real-world phenomena.
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Applied Mathematician
An applied mathematician uses advanced mathematical principles to address complex real-world challenges in science, engineering, and industry. "Calculus of Variations 2" is exceptionally relevant, offering a deep, graduate-level understanding of deriving and analyzing optimal solutions to problems described by functionals. The course's focus on the second variation, Jacobi accessory equation, and generalized functionals provides sophisticated tools for modeling physical systems and optimizing processes. For a prospective applied mathematician, this course helps build a foundation for translating intricate problems into solvable mathematical frameworks, providing a distinctive advantage in developing innovative solutions.
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Numerical Analyst
A numerical analyst develops and evaluates computational methods for solving complex mathematical problems, particularly those involving differential equations and optimization. "Calculus of Variations 2" offers a deep theoretical grounding in the properties of functionals, their extremals, and stability analysis, including the second variation and conjugate points. This understanding is critical for designing and analyzing numerical methods used in solving partial differential equations derived from variational principles or optimizing complex systems. For an aspiring numerical analyst, this course helps build a foundation for understanding the behavior and accuracy of computational solutions, essential for robust algorithm development.
See salaries and explore the career path for Numerical Analyst
Quantitative Analyst
A quantitative analyst develops sophisticated mathematical models for financial markets, pricing complex derivatives, and managing risk. The "Calculus of Variations 2" course provides a graduate-level understanding of advanced mathematical principles, particularly the optimization of functionals. This includes detailed study of the second variation, Legendre test, and generalized functionals, which are crucial for analyzing stability and optimality in dynamic financial systems. This rigorous training in advanced calculus for those aiming to excel as a quantitative analyst helps build a foundation for developing cutting-edge quantitative strategies and solving complex financial problems with profound mathematical insight.
See salaries and explore the career path for Quantitative Analyst
Computational Physicist
A computational physicist uses advanced mathematical and numerical methods to model and simulate complex physical phenomena. "Calculus of Variations 2" offers critical insights into the underlying principles of many physics models, particularly those derived from variational principles such as Lagrangian mechanics. Understanding the second variation, conjugate points, and generalized functionals is essential for developing stable and accurate numerical schemes for complex physical systems. For an aspiring computational physicist, this graduate-level mathematical rigor helps build a foundation for formulating and solving challenging problems in physics, from quantum mechanics to astrophysics.
See salaries and explore the career path for Computational Physicist
Research Scientist
A research scientist explores new frontiers, developing theories and models across diverse scientific and engineering domains. "Calculus of Variations 2" offers critical, graduate-level expertise in advanced mathematical modeling, focusing on the second variation, conjugate points, and generalized functionals. This knowledge is essential for deriving fundamental equations governing physical phenomena, optimizing complex designs, or developing novel algorithms in various research fields. For those aspiring to be a research scientist, this course provides the mathematical rigor needed to conceive and validate innovative research, enhancing their ability to tackle some of the most challenging unsolved problems.
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Financial Engineer
A financial engineer applies sophisticated quantitative methods and engineering tools to design and analyze financial products and strategies. "Calculus of Variations 2" offers a graduate-level understanding of advanced optimization techniques for functionals, including the second variation and generalized functionals. This rigorous mathematical foundation is highly relevant for deriving and analyzing optimal strategies in areas like portfolio optimization, option pricing, and dynamic asset allocation, which are central to financial engineering. This course helps build a foundation for developing and implementing cutting-edge financial models and tools to solve complex market challenges.
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Research Engineer
A research engineer conducts research and development in engineering fields, applying scientific principles and advanced mathematics to solve complex technical problems. "Calculus of Variations 2" offers a graduate-level foundation in optimizing functionals and analyzing their stability, including the second variation and generalized functionals. This knowledge is highly relevant for deriving governing equations, optimizing designs, or developing theoretical models in fields such as materials science, structural mechanics, or fluid dynamics, where variational methods are often employed. This course helps build a strong analytical foundation for innovative problem-solving in engineering research.
See salaries and explore the career path for Research Engineer
Control Systems Engineer
A control systems engineer designs and implements systems that regulate dynamic processes to achieve optimal performance and stability. "Calculus of Variations 2" is highly relevant, providing a graduate-level understanding of optimizing functionals, including the analysis of second variations and conjugate points. These concepts are foundational for developing advanced optimal control strategies and ensuring system robustness across various industrial and scientific applications. For those aiming to become a control systems engineer, this course may be useful, helping build a strong analytical foundation for designing sophisticated and efficient control solutions for complex systems.
See salaries and explore the career path for Control Systems Engineer
Operations Research Analyst
An operations research analyst applies mathematical modeling, optimization, and statistical analysis to improve decision-making and efficiency within organizations. "Calculus of Variations 2," with its advanced treatment of optimizing functionals and analyzing their stability through the second variation and conjugate points, provides a sophisticated mathematical toolkit. This course may be useful for understanding and developing complex optimization models, particularly those involving continuous processes or dynamic systems, thus enhancing an operations research analyst's problem-solving capabilities in logistics, resource allocation, and strategic planning.
See salaries and explore the career path for Operations Research Analyst
Aerospace Engineer
An aerospace engineer designs and develops aircraft and spacecraft, often tackling complex optimization challenges in trajectory planning and structural design. "Calculus of Variations 2" offers a rigorous mathematical framework essential for analyzing and solving these problems, covering the second variation, conjugate points, and generalized functionals. This graduate-level understanding of extremals and their stability is crucial for designing efficient and robust aerospace systems. This course may be useful for those pursuing a career as an aerospace engineer, providing deep analytical skills to optimize performance and ensure the integrity of flight systems.
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Algorithm Developer
An algorithm developer designs, implements, and optimizes computational procedures for diverse applications, from scientific computing to artificial intelligence. "Calculus of Variations 2," with its focus on advanced mathematical optimization, including the second variation and generalized functionals, provides a deep theoretical grounding. Understanding how to derive and analyze extremals for functionals may be useful for designing algorithms that minimize errors, optimize performance, or solve complex inverse problems in fields like computer vision or scientific computing. This course helps build a foundation for creating robust and efficient algorithms for challenging computational tasks.
See salaries and explore the career path for Algorithm Developer
Data Scientist
A data scientist analyzes vast datasets, builds predictive models, and develops algorithms to extract insights and solve business problems. While core data science often focuses on statistics and machine learning, "Calculus of Variations 2" offers a deep understanding of advanced optimization principles, including the analysis of functionals and their stability. This foundational knowledge, especially in generalized functionals and their derivatives, may be useful for those developing novel optimization algorithms, understanding the behavior of complex models, or tackling advanced problems in areas like image processing within data science. It helps build a strong mathematical toolkit for complex analytical challenges.
See salaries and explore the career path for Data Scientist
Actuary
An actuary assesses and manages financial risks, primarily within the insurance and pension industries, utilizing statistical and mathematical modeling. While heavily reliant on probability and statistics, a deep understanding of advanced optimization principles, as explored in "Calculus of Variations 2," may be useful for intricate modeling problems. These include premium optimization or designing complex benefit structures where functionals could play a role in advanced risk analysis. This course's rigorous mathematical training helps build strong analytical and problem-solving skills, broadening an actuary's capacity to tackle highly complex and novel financial risk challenges.
See salaries and explore the career path for Actuary

Reading list

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Advanced Calculus
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Rating
4.9 Filled star Filled star Filled star Filled star Half star
Effort
6.5 total hours
Via
Udemy
Instructor
Ross McGowan
Language
English
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