Multivariable Calculus and Classical Physics problems

Description

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).

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