Multivariable Calculus and Classical Physics problems from Udemy

In the first part of this course Multivariable Calculus is explained by focusing on understanding the key concepts rather than learning the formulas and/or exercises by rote. The process of reasoning by using mathematics is the primary objective of the course, and not simply being able to do computations. Besides, interesting proofs will be given, such as the Gauss and Stokes theorems proofs.

The prior knowledge requirement is Single variable Calculus (even without a great mastery of it).

I will list some of the most important concepts that we will see here in the following.

partial differentiation. The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. For example, in vector calculus (which we will see), the "del" operator is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. Differential equations containing partial derivatives are called partial differential equations or PDEs. These equations are generally more difficult to solve than ordinary differential equations, which contain derivatives with respect to only one variable (PDEs are not discussed in this course).
Multiple integration. The multiple integral extends the concept of the integral to functions of any number of variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space.
The surface integral and the line integral are used to integrate over curved manifolds such as surfaces and curves. We will see these concepts.

I am available for questions, which I could answer by (possibly) uploading new content to the course, namely videos containing the solution.

The second part of this course is about solving advanced mechanics problems; since multivariable calculus is a staple of this second part, I decided to combine the part on physics problems and the one on multivariable calculus into a single course, where you can therefore find lots of material. This set of problems is taken from the first volume of the course of theoretical physics by Landau and Lifshitz. I have selected some problems from this book and provided a thorough step-by-step solution in the course; the solutions to these problems are also given in the book but they are usually quite terse, namely not many details are provided. Therefore, what we will do in the course is to first construct the necessary theory to deal with the problems, and then we will solve the problems. Some theory is also discussed while solving the problems themselves. Every single formula in this course is motivated/derived.

We will start from the action principle, whose main constituent is the Lagrangian, which is fundamental to dealing with advanced problems in all branches of physics, even if we restrict ourselves to mechanics in this case. We will solve several problems related to how to construct a Lagrangian of a (possibly complex) system, and we will also derive the Hamiltonian from the Lagrangian, which represents the energy of a system, and do some problems on that.

We will also study the kinematics of rigid bodies, and derive formulae for the velocities of points which belong to the bodies, as well as formulae for accelerations. Accelerations are important not just for kinematics, but also for the dynamics of rigid bodies.

As regards the motion of rigid bodies, we will discuss the kinetic energy, which is necessary to obtain the Lagrangian, and solve several problems in three dimensions related to how to find the kinetic energy of a body in motion.

The expression of the kinetic energy is dependent on the angular velocity (which is a concept that we will derive in kinematics), and also depends on the inertia matrix (or inertia tensor), which we will also derive. The formulae will be therefore written in a very general form, and this is useful when tackling difficult problems, since knowing a general method will provide the means to solve them.

The inertia tensor will appear in the expression for the kinetic energy, and it will also appear in dynamics, in the formula for moments; we will see why it appears, and use the theory to solve problems.

We will also discuss non-inertial frames, and find the deflection of a freely falling body from the vertical caused by the Earth's rotation (which makes the Earth a non-inertial frame).

What's inside

Learning objectives

How to intuitively understand multivariable calculus by using the mathematics
Divergence theorem (how to prove it)
Stokes theorem (how to prove it)
Multiple integrals
Line integrals
Surface integrals
Jacobian
How to tackle graduate-level problems in classical mechanics
The kinematics of rigid bodies
How to deal with non-inertial frames of reference

How to calculate the angular velocity of a rigid body
How to calculate the frequency of small oscillations
How to analyze the dynamics of rigid bodies
How to calculate the inertia matrix and moments of inertia
How to construct a lagrangian in classical mechanics
The importance of the lagrange formalism
How to derive the hamiltonian (energy) of a system
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How to intuitively understand multivariable calculus by using the mathematics
Divergence theorem (how to prove it)
Stokes theorem (how to prove it)
Multiple integrals
Line integrals
Surface integrals
Jacobian
How to tackle graduate-level problems in classical mechanics
The kinematics of rigid bodies
How to deal with non-inertial frames of reference
How to calculate the angular velocity of a rigid body
How to calculate the frequency of small oscillations
How to analyze the dynamics of rigid bodies
How to calculate the inertia matrix and moments of inertia
How to construct a lagrangian in classical mechanics
The importance of the lagrange formalism
How to derive the hamiltonian (energy) of a system
Show more
Show less

Syllabus

Introduction

Concept of functions of more than one variable

Understand the concepts of: Directional derivatives, differential of a function, gradient

Career center

Learners who complete Multivariable Calculus and Classical Physics problems will develop knowledge and skills that may be useful to these careers:

Physicist

A physicist investigates the fundamental laws of nature, from subatomic particles to the cosmos, often developing new theories and conducting experiments. This intellectually rigorous career path demands a profound mastery of advanced mathematics and classical mechanics. The Multivariable Calculus and Classical Physics problems course is, by its very nature, an ideal foundation for a physicist. It not only explains multivariable calculus by focusing on intuitive understanding and reasoning with mathematics, but also delves into graduate-level problems in classical mechanics using the Landau and Lifshitz approach. Concepts such as the action principle, Lagrangian, Hamiltonian, non-inertial frames of reference, and proofs of fundamental theorems like Gauss and Stokes are core to the discipline. The course's emphasis on thorough, step-by-step solutions and the derivation of every formula cultivates the precise analytical and problem-solving skills paramount for a career in physics. A PhD is almost always required for research-focused roles in this field.

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Mechanical Engineer

Mechanical engineers design, analyze, manufacture, and maintain mechanical systems, applying principles of motion, energy, and force to a wide range of products from engines to robotics. The Multivariable Calculus and Classical Physics problems course is highly relevant for aspiring mechanical engineers, providing a robust understanding of foundational physics and advanced mathematical tools crucial for analyzing complex mechanical systems. Learners will gain proficiency in rigid body kinematics and dynamics, kinetic energy calculations, and the use of the inertia matrix. The course's exploration of Lagrangian and Hamiltonian formalism, along with practical problems on systems like the 4-bar linkage, directly prepares one for designing and optimizing machinery and dynamic systems. This detailed approach to problem-solving, coupled with an intuitive grasp of multivariable calculus, builds a strong analytical toolkit for addressing real-world engineering challenges. An advanced degree is often pursued for specialized design or research positions in this field.

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Aerospace Engineer

Aerospace engineers are at the forefront of designing, developing, and testing aircraft, spacecraft, missiles, and other flight vehicles. This demanding field requires a deep understanding of aerodynamics, propulsion, and, critically, the complex dynamics of rigid bodies in various environments. The Multivariable Calculus and Classical Physics problems course offers an exceptional preparation for an aerospace engineer by rigorously covering essential concepts such as the kinematics and dynamics of rigid bodies, angular velocity, and the inertia matrix—all fundamental to understanding flight mechanics and spacecraft attitude control. The course’s treatment of non-inertial frames of reference, including detailed problems on phenomena like the Coriolis force, is particularly relevant for analyzing motion within Earth’s rotating frame or during orbital maneuvers. Furthermore, the action principle and Lagrangian formalism are invaluable for modeling complex aerospace systems, providing a powerful framework for advanced analysis. An advanced degree is very common and often required for research and development roles.

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Astrophysicist

Astrophysicists investigate the physical properties and phenomena of celestial bodies and the universe as a whole, from planetary systems to galaxies and black holes. This field heavily relies on advanced classical mechanics, particularly celestial mechanics, to understand orbital dynamics, gravitational interactions, and the evolution of cosmic structures. The Multivariable Calculus and Classical Physics problems course provides an exceptional foundation for an astrophysicist. The course's comprehensive coverage of the action principle, Lagrangian, and Hamiltonian formalism is fundamental to celestial mechanics and general relativity, allowing for the precise modeling of astronomical systems. Concepts like non-inertial frames of reference are crucial for understanding observations from Earth or spacecraft, while the kinematics and dynamics of rigid bodies are applicable to planetary rotation and spacecraft dynamics. The rigorous mathematical approach and the advanced problem-solving techniques taught are perfectly aligned with the demands of astrophysical research. A PhD is almost always required for this highly specialized and research-intensive field.

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Robotics Engineer

Robotics engineers design, build, and program robotic systems that automate tasks across various industries, from manufacturing to healthcare. A core aspect of this profession involves understanding and controlling the intricate motion of multi-link mechanical structures. The Multivariable Calculus and Classical Physics problems course is exceptionally suited for those aspiring to become a robotics engineer. It provides comprehensive coverage of the kinematics and dynamics of rigid bodies, which directly translates to analyzing robot arm movements and ensuring stable operation. Learners will delve into calculating angular velocity, kinetic energy, and the inertia matrix for complex systems, skills directly applicable to robot design and control. The course’s exploration of the Lagrangian formalism offers a powerful method for deriving the equations of motion for multi-joint robotic systems, while problems like the 4-bar linkage provide practical insights into mechanical linkages. An advanced degree is frequently pursued for specialized research or complex system design within robotics.

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Applied Mathematician

Applied mathematicians develop and apply mathematical theories and techniques to solve practical problems in engineering, science, business, and other fields. Their work often involves creating models, analyzing data, and developing algorithms for complex systems. The Multivariable Calculus and Classical Physics problems course offers an excellent toolkit for an applied mathematician. The first part of the course focuses profoundly on understanding the key concepts of multivariable calculus, including partial differentiation, multiple integration, Jacobian matrices, and vector calculus operations (gradient, divergence, curl). The rigorous approach to mathematical reasoning, including the proofs of Gauss and Stokes theorems, is invaluable for developing new analytical methods. Furthermore, the classical mechanics problems, tackled with advanced formalisms like Lagrangian and Hamiltonian mechanics, exemplify how mathematical principles are used to model and solve complex physical phenomena, a core skill for this profession. An advanced degree is very common, and often required for research and development roles.

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Control Systems Engineer

Control systems engineers design, implement, and maintain systems that regulate the behavior of other dynamic systems, from robotic arms and automatic pilots to manufacturing processes and power grids. This field relies heavily on modeling system behavior, predicting responses, and designing feedback mechanisms. The Multivariable Calculus and Classical Physics problems course is exceptionally relevant for an aspiring control systems engineer. It offers a deep understanding of how to mathematically analyze the dynamics of complex systems, crucial for deriving equations of motion for controllers. The course's detailed exploration of the Lagrangian formalism provides a powerful method for modeling multi-degree-of-freedom systems, while the study of kinematics and dynamics of rigid bodies directly applies to designing robust control strategies for mechanical systems. Furthermore, the analysis of small oscillations is fundamental for understanding system stability and designing effective feedback loops. The inclusion of the 4-bar linkage engineering problem provides a direct application of these principles in a control systems context. An advanced degree is often beneficial for advanced design roles.

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Computational Scientist

Computational scientists leverage high-performance computing and advanced mathematical algorithms to model, simulate, and analyze complex scientific and engineering problems. Their work often involves translating physical laws into numerical methods and developing software to gain insights into phenomena that are difficult or impossible to study experimentally. The Multivariable Calculus and Classical Physics problems course is highly relevant for an aspiring computational scientist. The course fosters a deep understanding of multivariable calculus concepts like partial differentiation, multiple integration, and the Jacobian matrix, which are essential for numerical methods and scientific computing. Furthermore, the rigorous treatment of classical mechanics problems, including the derivation of Lagrangians and Hamiltonians, provides the necessary theoretical foundation for developing physics-based simulations. The inclusion of a numerically solved 4-bar linkage problem directly showcases the application of these principles in computational contexts, making it particularly useful for this career path. An advanced degree is often required, particularly for research and advanced modeling roles.

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Biomechanics Engineer

Biomechanics engineers apply principles of mechanics to biological systems, such as the human body, animals, or plants. Their work involves analyzing movement, designing prosthetic devices, optimizing sports performance, or understanding the forces on tissues and organs. The Multivariable Calculus and Classical Physics problems course is highly relevant for an aspiring biomechanics engineer. It provides a deep understanding of the kinematics and dynamics of rigid bodies, which directly translates to analyzing the motion of limbs, joints, and entire body segments. Learners will gain expertise in calculating angular velocity, kinetic energy, and the inertia matrix for complex biological systems. The Lagrangian formalism is particularly useful for deriving equations of motion for multi-joint human movements, and the course’s problems on small oscillations can be applied to gait analysis or the study of vibrations in anatomical structures. The rigorous problem-solving approach helps build the analytical skills necessary for innovation in this interdisciplinary field. An advanced degree is often pursued for specialized roles.

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Physics Engine Developer

A physics engine developer specializes in creating software that simulates physical phenomena in virtual environments, such as video games, virtual reality, or industrial simulations. This role demands a strong grasp of classical mechanics to accurately model interactions, movements, and forces within a digital world. The Multivariable Calculus and Classical Physics problems course is an excellent resource for a physics engine developer. It provides a comprehensive understanding of the kinematics and dynamics of rigid bodies, which is directly applicable to simulating objects in virtual space. Learners will gain expertise in calculating angular velocity, kinetic energy, and the inertia matrix for various shapes, skills crucial for realistic object behavior. While real-time engines often simplify dynamics, the underlying Lagrangian and Hamiltonian formalisms offer a deep theoretical foundation for robust and accurate physics implementations. The course's approach to solving complex mechanical problems helps build the analytical skills necessary to tackle diverse simulation challenges.

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Geophysicist

Geophysicists study the Earth's physical processes and phenomena, including earthquakes, volcanoes, plate tectonics, and the Earth's magnetic field. Their work often involves complex modeling of subterranean structures, fluid dynamics, and seismic wave propagation, all requiring strong foundations in physics and advanced mathematics. The Multivariable Calculus and Classical Physics problems course provides a valuable background for a geophysicist. The course’s detailed treatment of multivariable calculus, including concepts like partial differentiation and multiple integration, is crucial for modeling Earth's internal structure and processes. The study of non-inertial frames of reference, specifically problems related to the Earth's rotation and the Coriolis effect, is directly applicable to understanding atmospheric and oceanic currents, as well as the deflection of objects on Earth's surface. The general classical mechanics principles and advanced problem-solving skills learned are beneficial for analyzing seismic waves, gravitational anomalies, and other geophysical phenomena. An advanced degree is often required for research or specialized roles.

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Structural Engineer

Structural engineers design and analyze the integrity and stability of structures such as buildings, bridges, and tunnels, ensuring they can withstand various forces and stresses. While their daily work might not directly involve quantum mechanics, the fundamental principles of statics, dynamics, and material science that underpin their designs are deeply rooted in classical mechanics. The Multivariable Calculus and Classical Physics problems course may be helpful for a structural engineer by providing a rigorous foundation in advanced mathematical methods and the dynamics of rigid bodies. Understanding concepts like moments of inertia, kinetic energy, and how to analyze complex systems of rods or bodies under motion helps build the analytical framework necessary for structural analysis. The emphasis on intuitive understanding and mathematical reasoning helps prepare one for complex calculations and simulations of structural behavior. An advanced degree is beneficial for specialized design or research roles in this discipline.

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Data Scientist

Data scientists analyze large datasets to extract insights, build predictive models, and inform decision-making across various industries. While not directly focused on physical mechanics, the role demands strong analytical thinking and a robust mathematical foundation to understand complex algorithms and model relationships within data. The Multivariable Calculus and Classical Physics problems course may be useful for a data scientist by enhancing their foundational mathematical rigor. The course's emphasis on intuitive understanding of multivariable calculus, including partial differentiation, multiple integration, and the Jacobian matrix, is relevant for understanding optimization algorithms, dimensionality reduction techniques, and advanced statistical modeling that are common in data science. The rigorous problem-solving approach employed in classical mechanics problems also cultivates the ability to dissect complex challenges, abstract essential components, and construct logical solutions, which are highly transferable skills to data analysis and model development. An advanced degree is frequently required or preferred in this field.

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Quantitative Analyst

Quantitative analysts, often called 'quants,' develop and implement complex mathematical models for financial markets, assessing risk, pricing derivatives, and identifying trading opportunities. This highly analytical role demands exceptional mathematical rigor and the ability to understand and model dynamic systems, even if those systems are not physical in nature. The Multivariable Calculus and Classical Physics problems course may be helpful for a quantitative analyst by strengthening their advanced mathematical toolkit. The course's comprehensive coverage of multivariable calculus, including partial differentiation, Jacobian matrices, and the rigorous approach to mathematical reasoning, is highly relevant for understanding stochastic calculus, option pricing models, and optimization techniques frequently used in finance. While the classical mechanics content is less directly applicable, the underlying philosophy of abstracting complex systems into mathematical frameworks and deriving their behavior from first principles is a highly transferable and valuable skill for quantitative finance. An advanced degree, often a Master's or PhD in a quantitative field, is very common.

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Materials Scientist

Materials scientists study the properties, structure, and performance of materials, from metals and polymers to ceramics and composites, to develop new materials or improve existing ones. Their work often involves understanding the physical and mechanical behavior of substances under various conditions. The Multivariable Calculus and Classical Physics problems course may be helpful for a materials scientist by providing a strong grounding in advanced mathematics and the principles of classical mechanics. While the course’s direct focus on rigid body dynamics is not always central to materials science, the robust understanding of multivariable calculus is applicable to continuum mechanics, stress-strain analysis, and modeling material responses at various scales. The course's emphasis on careful mathematical reasoning and problem-solving for physical systems helps build an analytical mindset, which is beneficial when investigating material properties and developing new theoretical models for material behavior. A Master's or PhD is typically required for research and development roles.

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