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Master Variational Calculus & Advanced Mathematical Methods

Name: Master Variational Calculus & Advanced Mathematical Methods
Rating: 4.3 (27 reviews)

Variational calculus, Tensors, Fourier Transforms, Residues, Path integrals, Functional calculus, groups, Optimization

Created byEmanuele Pesaresi

Last updated 5/2026

English

What you'll learn

Master the Calculus of Variations: Students will learn to derive and solve Euler-Lagrange equations, optimizing functionals to address real-world problems
Apply Integral Transforms: Students will gain proficiency in using Laplace, Fourier, and other transforms to simplify and solve complex differential equations
Understand and Utilize Tensor Analysis: Students will acquire a deep understanding of tensor theory, transformation rules, operations with tensors, and more.
Apply tensors to real-world scenarios, such as determining geodesics, understanding curvature, and calculating stress and strain in various frames
Implement Complex Analysis and Residue Calculus: Students will learn to evaluate complex integrals using residue theorems and apply these techniques to problems
Learn to effectively use Fourier transforms, Laplace transforms, and residue theory for analyzing signals, solving differential equations, and more
Apply Path Integrals and Functional Derivatives in Physics – Explore path integral formulation and functional derivatives
Analyze Physical Systems with Canonical Transformations and Hamiltonian Mechanics – Understand canonical transformations, Poisson brackets, Hamiltonian dynamics
Analyze the characteristics of the Lorentz group and Lie Algebra
Understand the Lie algebra of the Lorentz group, and how it is strictly related to the operators of momentum, angular momentum
Understand the mathematics and physical importance of such objects as spinors

Course content

13 sections • 133 lectures • 33h 37m total length

Principle of least action18:46
Variation principle derived from Newton's second law4:58
Derive the variational principle from Newton's second law for a single particle, showing the Lagrangian L = T−V yields the stationary action ∫(T−V) dt.
Hamiltonian5:22
problem 1: double pendulum14:59
Derive the Lagrangian for a two-link double pendulum with masses m1, m2 and lengths L1, L2 by combining kinetic energy and gravity-based potential energy expressed in angles phi1 and phi2.
problem 2: simple pendulum9:08
problem 3: simple pendulum attached to circle17:19
problem 4: system of rods10:13

Intro to the section on tensors1:25
Overview on how laws of physics change in different reference frames7:44
Lorentz transformations and the invariant element22:13
Another proof of the invariance of the infinitesimal length element in SR11:57
Lorentz transformations as derived by Einstein28:40
The physical meaning that Einstein attributes to Lorentz transformations7:05
invariant length element (additional insights)2:02
Explore invariant length element ds^2 = c^2 dt^2 − dx^2 and the moving-clock relation d tau = dt sqrt(1 − v^2/c^2). See how it underpins special relativity and general relativity.
Lagrangian in special relativity7:18
Momentum and Hamiltonian of a relativistic particle, E equals mc squared10:15
Equivalence principle: the starting point of General Relativity6:18
Why differential geometry ?3:15
invariant in tensor notation7:35
Transformation Rules for Tensors10:40
Some basic operations with tensors7:22
How Einstein shows that the inverse of the metric tensor is also a tensor3:12
How Einstein shows the invariance of the infinitesimal volume element4:54
Einstein shows the invariance of the infinitesimal volume element by relating the determinant of the metric tensor to its transformed determinant via a Jacobian, yielding sqrt(-g') = J sqrt(-g).
How Einstein derives the equation of a geodetic line10:24
A derivation of the geodetic equation without using the Variational Principle13:41
How Einstein derives the concept of covariant derivative14:45
Rule of differentiation of determinants (important in GR)12:58
How Einstein derives the concept of covariant divergence16:49
Other important tensor expressions derived by Einstein in his paper on GR29:47
How Einstein derives the Riemann tensor in his paper on General Relativity11:09
How Einstein derives the field equations in the absence of matter4:54
How Einstein derives his equations from the action principle,energy conservation26:42
How Einstein derives the field equations in the presence of matter26:17
How Einstein derives his General Relativity theory from Hamilton's principle47:53

Derivation of the strain tensor from the theory of elasticity11:15
Classical Theory of Thin Plates derived from the Action Principle30:39
Derivation of the strain tensor in spherical coordinates28:51
Gradient in spherical coordinates (using tensors and matrices)20:07
Laplacian in Spherical Coordinates (using tensors and matrices)7:11
Derive the Laplacian in spherical coordinates by formulating the gradient with tensors and matrices, then translate to index notation and verify with a Matlab implementation.

Relativistic Hamilton Jacobi equation19:06
Review of Classical Field Theory8:23
Functional derivative and Euler Lagrange equations in Classical Field Theory10:19
Application of the functional derivative3:55
Minimal surfaces in Calculus of Variations12:10
Explains finding minimal surfaces with fixed boundary by optimizing the area functional for z(x,y). Presents two nontrivial solutions: the catenoid and the helicoid.
Exercise on the variation of the determinant of the metric tensor1:06
Trace of the logarithm of a matrix and the determinant5:28
Proof of the Jacobi identity13:26
Neumann series3:21
Physics of thin plates derived from the variational principle41:39
Heuristic derivation of the path integral from the classical theory of fields18:33
From the Hamilton Jacobi equation to the Schrodinger equation9:01
Functional derivatives and generalization of Euler Lagrange equations in QFT19:51

Functions of a complex variable part 18:55
Functions of a complex variable part 220:20
Introducing the derivative in complex calculus19:53
Introducing integrals of complex functions and Cauchy theorem19:01
Extension of Cauchy theorem19:22
Cauchy integral formula part 121:07
Cauchy integral formula part 29:18
Explore Cauchy's integral formula and its generalization, deriving higher order derivatives as contour integrals of f(w)/(w - z)^{n+1} and noting winding number effects.
Laurent series26:03
Explore Laurent series in complex calculus, a generalization of Taylor and Maclaurin series near z0, through contour integrals on gamma curves and Cauchy integral formula.
Laurent Series in compact form8:29
Fourier series derivation from Laurent series14:56
Fourier series generalization to any generic period T5:13
Taylor series derivation from Laurent series6:13
Derive the Taylor series from the Laurent expansion for a function continuous at z0 by using a contour integral around gamma once, showing that the coefficients c_n equal f^(n)(z0)/n!.
Concept of Residue7:03
Residue Theorem10:39
Calculation of residues and coefficients of the Laurent series22:56
Evaluation of a real integral using complex integration25:19
Use contour integration around the real axis with a small indentation at zero to compute the principal value of e^{i ω z}/z, yielding i π and its cos/sin implications.
representation of the Dirac delta32:11
Another integral representation of the Dirac Delta17:27
Prove the Dirac delta integral representation by applying the Fourier transform to 1/(t(t - x)) and evaluating the resulting integral via contour integration, using the signum of k.
Evaluation of complex Gaussian integrals18:01
Solving an Integral from Feynman's Book (more elegantly than ChatGPT)21:56
Derive a simple, intuitive method to evaluate the third Gaussian-type integral from Feynman's path integral appendix, handling a 1/x^2 term and complex numbers.

Fourier Series and Fourier Transform12:05
Notation for the Fourier Transform6:00
Explain notational conventions for the Fourier transform and Fourier series, including space versus time domains and conjugate variables such as k and omega.
The importance of the Dirac Delta in defining the Inverse Fourier Transform16:17
Derivation of the Uncertainty "principle"16:54
More on the Dirac delta and its relation to the Inverse Fourier Transform13:08
Fourier Transform applied to the derivative of a function3:52
Gibbs phenomenon and its relation to the Fourier series30:35
Exercise on multiple convolutions and Fourier Transforms15:56
Energy of signals and Fourier Transforms (signals with limited bandwidth)17:51
Intro to the Laplace Transform from the Fourier Transform8:22
Solution to the diffusion equation using complex calculus (Laplace transform)13:26
2nd order non-homogeneous differential equation solved via Fourier Transform37:41
Calculation of Inverse Laplace Transform with residues15:58
Interesting Fourier series in 2 dimensions13:20
Fourier series expansion for thin plates12:11
Represent q(x,y) and w(x,y) with a double Fourier sine series for a thin plate with zero boundary loads, solving nabla^4 w = q/d and noting high modes decay.
How convolutions in the path integral lead to the Schrodinger equation20:37
From Path Integrals to Perturbation Theory and then to Feynman Diagrams37:03
From the path integral to the effective action, renormalization group, couplings52:41
Asymptotics and divergent series17:07

Requirements

Learners should have a strong understanding of undergraduate-level calculus, including differentiation, integration, differential equations, vector calculus
Familiarity with Linear Algebra: A good grasp of linear algebra concepts such as matrices, vectors, and eigenvalues is recommended.
While prior experience with the specific topics covered is not required, learners should be comfortable with abstract mathematical reasoning and eager to tackle challenging material.
Understanding of Classical Mechanics – Familiarity with Newtonian mechanics will be helpful, as many examples are drawn from physics.
Prior Exposure to Complex Numbers and Basic Complex Calculus – Knowledge of complex numbers and introductory concepts in complex calculus (e.g., Euler's formula, polar form) is recommended

Description

Overview

This course brings together several advanced mathematical tools that often appear separately in physics, engineering, and applied mathematics, but which are deeply connected when studied in the right order.

We will move through the calculus of variations, integral transforms, tensor analysis, complex analysis with residue calculus, the intuition behind path integrals and quantization, and a final part on constrained optimization. Along the way, the course also touches on Hamiltonian mechanics, Poisson brackets, the Schrödinger equation, Feynman diagrams, Lorentz transformations, spinors, and Lie algebras.

The goal is not only to present formulas, but to build mathematical intuition. I try to show why these tools were introduced, how they are used, and how they connect different areas of classical and quantum physics.

This course is intended for students, professionals, researchers, and anyone with a solid mathematical background who wants to strengthen their understanding of advanced mathematical methods and their applications in physics and engineering.

What You Will Learn

Calculus of Variations

We begin with the idea of optimizing functionals, which is one of the key mathematical structures behind classical mechanics, field theory, geometry, and many engineering problems.

You will study the Euler-Lagrange equations, boundary conditions, variational principles, and applications such as geodesics and mechanical systems. The emphasis is on understanding not only how to use the equations, but also where they come from.

Integral Transforms

The course then develops important tools such as Fourier and Laplace transforms. These methods are essential for solving differential equations, studying signals, and transforming difficult problems into more manageable ones.

The aim is to make these techniques feel natural rather than purely formal, with examples showing how they simplify calculations and reveal hidden structure.

Tensor Analysis

A significant part of the course is devoted to tensors and their applications. Tensors are introduced as mathematical objects that naturally appear in geometry, continuum mechanics, relativity, and field theory.

The presentation follows a pedagogical path similar to what one might encounter in a course on General Relativity. We discuss tensor transformations, covariant notation, the metric tensor, and the role of tensors in describing physical laws independently of the chosen coordinates.

Complex Analysis and Residue Calculus

The course also includes complex analysis, with particular attention to residues and contour integration.

Residue calculus is one of the most elegant and useful tools in mathematical physics. We will see how complex functions can be used to evaluate real integrals and solve problems that would otherwise be much harder to approach directly.

Path Integrals and Quantization

Another important part of the course is devoted to the intuition behind path integrals and the quantization of classical theories.

The purpose here is not to replace a full course in quantum field theory, but to help students understand how ideas from classical mechanics can be reformulated in a way that leads naturally toward quantum mechanics and, eventually, field theory.

Mathematical Connections Between Classical and Quantum Physics

The course also builds bridges between classical mechanics and quantum physics through Poisson brackets, Hamiltonian mechanics, the path integral approach, the Schrödinger equation, and the first ideas behind Feynman diagrams.

This part is especially useful for students who already know some classical mechanics and want to understand how the language of modern theoretical physics begins to emerge from it.

Lorentz Algebra, Lie Groups, and Spinors

Two recent sections have been added to the course, focusing on the Lorentz group, its Lie algebra, spinors, Pauli matrices, Dirac matrices, operators, and intrinsic angular momentum.

These topics are closely related to quantum mechanics, relativity, and quantum field theory. They also show beautifully how tensors, transformations, and symmetry ideas fit together.

Constrained Optimization

The final part of the course introduces constrained optimization problems, especially through the method of Lagrange multipliers.

Here, too, the focus is not only on the standard procedure, but also on developing the intuition behind it. The goal is to understand why the method works and how it can be applied in practical mathematical and physical problems.

Additional Comments

This course covers a broad range of topics, but the intention is not to collect unrelated material. The common thread is the use of advanced mathematical methods to understand physical systems.

We start from variational principles and classical mechanics, then move toward tensors, geometry, transforms, complex analysis, and finally some ideas that point toward quantum mechanics and quantum field theory.

Several examples are included throughout the course, such as the double pendulum, geodesics, strain tensors, Fourier transform applications, and other problems from physics and engineering. These examples are meant to make the abstract theory more concrete and to show how the same mathematical ideas appear in different contexts.

Some topics, such as Einstein’s field equations, Poisson brackets, the Maupertuis principle, and residue calculus, are treated in more detail because they are especially important for building a deeper understanding of theoretical physics.

Who Should Enroll

This course is suitable for:

Advanced undergraduate and graduate students in mathematics, physics, and engineering.

Professionals and researchers who want to strengthen their understanding of advanced mathematical methods.

Students interested in the mathematical foundations of classical mechanics, relativity, quantum mechanics, and field theory.

Anyone with a strong background in calculus, linear algebra, and differential equations who wants to study these topics in a connected and structured way.

Course Features

The course includes theoretical explanations, step-by-step derivations, worked examples, and selected applications.

Whenever possible, I try to make the lectures self-contained, so that students can follow the logic of each section without constantly needing to consult external material.

The course is also designed so that students may skip some sections or focus only on the topics that are most relevant to their own studies. For example, someone interested mainly in tensors and relativity may follow those sections directly, while another student may focus more on integral transforms, complex analysis, or path integrals.

Prerequisites

A solid understanding of undergraduate calculus, linear algebra, and differential equations is strongly recommended.

Some familiarity with classical mechanics and basic physics is helpful, especially for the sections involving variational principles, Hamiltonian mechanics, tensors, and quantum physics. However, I try to introduce the main ideas gradually and with enough context to make the material accessible to motivated students.

Note on Course Structure and Overlap

Part of the material in this course also serves as supplementary mathematical background for some of my other physics courses.

Some overlap may therefore exist between this course and those physics courses. However, the purpose here is different. In this course, the mathematical ideas are collected, organized, and developed in a more systematic way (well, this is my desire, although it is not always easy to do...), so that students can study them independently from the specific physical applications.

In other words, this course is meant to help students build a stronger mathematical foundation, while the physics courses use some of these tools in more specific physical contexts.

New Material

New material was added on 30 November 2024.

The new sections include topics such as the Lorentz group, Lie algebras, spinors, Pauli matrices, Dirac matrices, operators, and related concepts. These additions expand the connection between tensors, relativity, and quantum physics.

References

This course reflects my own way of organizing and presenting the material, with the aim of giving students a clear and logical path through several important areas of advanced mathematics and theoretical physics.

Some of the material was inspired by the following sources:

L. Landau and E. Lifshitz, Mechanics, Vol. 1

L. Landau and E. Lifshitz, The Classical Theory of Fields, Vol. 2

A. Einstein, The Foundation of the General Theory of Relativity, 1916

A. Einstein, Hamilton’s Principle and the General Theory of Relativity, 1916

A. Einstein, Cosmological Considerations on the General Theory of Relativity, 1917

B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry — Methods and Applications, Part 1

L. Landau and E. Lifshitz, Theory of Elasticity, Vol. 7

L. S. Schulman, Techniques and Applications of Path Integration

D. Tong, Quantum Field Theory, lecture notes, University of Cambridge

D. Skinner, Quantum Field Theory II, lecture notes, University of Cambridge

Who this course is for:

Advanced Students in Mathematics, Physics, and Engineering: Ideal for upper-level undergraduates, graduate students, and doctoral candidates who seek to deepen their understanding of advanced mathematical methods.
Professionals and Researchers: Engineers, physicists, data scientists, and researchers in academia or industry who need to apply sophisticated mathematical tools to solve complex problems in their fields.
Aspiring Academics and Educators: Those pursuing a career in teaching or academic research who wish to gain a thorough mastery of topics like calculus of variations, tensor analysis, and integral transforms.
Enthusiasts with a Strong Mathematical Background: Individuals passionate about mathematics and theoretical sciences who want to explore higher-level concepts and their practical applications.

Master Variational Calculus & Advanced Mathematical Methods

What you'll learn

Explore related topics

Course content

Lagrangian, Hamiltonian, Equations of motions + Problems7 lectures • 1hr 21min

Problems on Conservation laws and theoretical considerations on Reynolds theorem4 lectures • 41min

Liouville theorem, canonical transformations, Poisson brackets6 lectures • 50min

Maupertuis principle, the Brachistochrone curve, Classical Jacobi equation3 lectures • 59min

Tensors explained by Einstein's papers + exercises27 lectures • 5hr 57min

Geodesics, surfaces, Gauss Theorema Egregium, different types of curvatures4 lectures • 1hr 14min

Application of tensors and action principle to the theory of elasticity5 lectures • 1hr 38min

Relativistic field theory, functional derivatives13 lectures • 2hr 46min

Residues and Basics of Complex Analysis20 lectures • 5hr 34min

Fourier series and Transforms19 lectures • 6hr 1min

Requirements

Description

Who this course is for: