# Mathematical Physics by Eugene Butkov: A Classic Textbook You Can Read for Free

## Mathematical Physics by Eugene Butkov: A Classic Textbook for Students and Researchers

Mathematical physics is a branch of physics that applies mathematical methods and techniques to solve physical problems and understand natural phenomena. It is an interdisciplinary field that connects physics with mathematics, as well as other sciences such as chemistry, biology, engineering, and computer science.

## Mathematical Physics Butkov Pdf Download

Mathematical physics is essential for developing new theories and models of physical reality, as well as testing and verifying existing ones. It also provides powerful tools for analyzing complex systems, such as fluids, plasmas, solids, quantum mechanics, relativity, cosmology, and more.

One of the most comprehensive and authoritative textbooks on mathematical physics is Mathematical Physics by Eugene Butkov. This book was first published in 1968 by Addison-Wesley Publishing Company, and has been widely used by students and researchers around the world ever since.

In this article, we will review the main topics covered by Butkov's book, and show you how to download the PDF version of it for free. We will also discuss the benefits of having a digital copy of the book, as well as the legal and ethical issues of downloading copyrighted material.

## What is mathematical physics and why is it important?

As we mentioned earlier, mathematical physics is a branch of physics that uses mathematics to describe and explain physical phenomena. It is not a separate discipline from physics, but rather a way of applying mathematics to physics.

Mathematics is often called the language of physics, because it allows us to express physical laws and concepts in precise and concise terms. Mathematics also helps us to generalize and abstract physical situations, so that we can find common patterns and principles that apply to different cases.

Moreover, mathematics enables us to perform calculations and simulations that would be impossible or impractical to do experimentally. For example, we can use mathematics to predict the behavior of atoms, molecules, stars, galaxies, black holes, etc., without having to observe them directly.

However, mathematical physics is not just about applying mathematics to physics. It is also about developing new mathematics that are inspired by physical problems. For instance, some of the most important branches of modern mathematics, such as calculus, differential equations, linear algebra, complex analysis, topology, geometry, algebraic structures, etc., were originally motivated by physical questions.

Therefore, mathematical physics is a two-way street between physics and mathematics. It enriches both fields with new ideas, methods, techniques, results, and applications. It also fosters collaboration and communication among physicists and mathematicians from different backgrounds and perspectives.

## The main topics covered by Butkov's book

Butkov's book covers a wide range of topics in mathematical physics that are relevant for both undergraduate and graduate students of physics and related fields. The book consists of 12 chapters, each with several sections and subsections. The chapters are:

Chapter 1: Introduction

Chapter 2: Partial Differential Equations and Boundary Value Problems

Chapter 3: Special Functions and Orthogonal Expansions

Chapter 4: Green's Functions and Integral Equations

Chapter 5: Variational Methods and Calculus of Variations

Chapter 6: Tensor Analysis and Differential Geometry

Chapter 7: Group Theory and Its Applications

Chapter 8: Complex Variables and Analytic Functions

Chapter 9: Integral Transforms

Chapter 10: Asymptotic Methods

Chapter 11: Perturbation Theory

Chapter 12: Nonlinear Problems and Chaos

In the following sections, we will briefly summarize the main contents of each chapter, and provide some examples of how they are used in physics.

### Partial Differential Equations and Boundary Value Problems

A partial differential equation (PDE) is an equation that involves partial derivatives of an unknown function with respect to more than one independent variable. For example, the heat equation, the wave equation, and the Laplace equation are PDEs that describe the diffusion of heat, the propagation of waves, and the potential of a static electric field, respectively.

A boundary value problem (BVP) is a problem that consists of finding a solution to a PDE that satisfies certain conditions on the boundary of the domain where the function is defined. For example, the Dirichlet problem is a BVP that requires the solution to have a given value on the boundary, while the Neumann problem requires the normal derivative of the solution to have a given value on the boundary.

In this chapter, Butkov introduces the basic concepts and methods for solving PDEs and BVPs, such as separation of variables, Fourier series, eigenvalue problems, Sturm-Liouville theory, etc. He also discusses some important types of PDEs and BVPs that arise in physics, such as the heat equation, the wave equation, the Laplace equation, the Poisson equation, the Helmholtz equation, etc.

#### Example: The Heat Equation

The heat equation is a PDE that describes how the temperature of a body changes over time due to heat conduction. It can be written as:

$$\frac\partial u\partial t = k \nabla^2 u$$ where $u(x,y,z,t)$ is the temperature at point $(x,y,z)$ and time $t$, and $k$ is a constant that depends on the thermal conductivity of the material.

To solve this equation, we need to specify some initial and boundary conditions. For example, suppose we have a thin metal rod of length $L$ with insulated ends. The initial temperature distribution along the rod is given by:

$$u(x,0) = f(x)$$ where $f(x)$ is some known function. The boundary conditions are:

$$u(0,t) = u(L,t) = 0$$ which means that the ends of the rod are kept at zero temperature.

To solve this BVP, we can use the method of separation of variables. We assume that the solution has the form:

$$u(x,t) = X(x)T(t)$$ where $X(x)$ and $T(t)$ are functions of $x$ and $t$ only. Substituting this into the heat equation, we get:

$$X(x)T'(t) = k X''(x)T(t)$$ where $'$ denotes differentiation. Dividing both sides by $kXT$, we obtain:

$$\fracT'(t)kT(t) = \fracX''(x)X(x) = -\lambda$$ where $\lambda$ is a constant. This implies that both sides of the equation are equal to a constant, which we call $-\lambda$. Therefore, we have two ordinary differential equations (ODEs):

$$T'(t) + k\lambda T(t) = 0 \quad \textand \quad X''(x) + \lambda X(x) = 0$$ The first ODE can be solved by using an exponential function:

$$T(t) = A e^-k\lambda t$$ where $A$ is an arbitrary constant. The second ODE can be solved by using trigonometric functions:

$$X(x) = B \cos(\sqrt\lambda x) + C \sin(\sqrt\lambda x)$$ where $B$ and $C$ are arbitrary constants.

, B, and C, we need to apply the initial and boundary conditions. The boundary conditions imply that:

$$X(0) = X(L) = 0$$ which means that:

$$B = 0 \quad \textand \quad C \sin(\sqrt\lambda L) = 0$$ The second equation has two possible solutions: either $C = 0$ or $\sin(\sqrt\lambda L) = 0$. The first solution gives a trivial solution $X(x) = 0$, which is not interesting. The second solution gives:

$$\sqrt\lambda L = n\pi \quad \textfor some integer n$$ Therefore, we have:

$$\lambda = \left(\fracn\piL\right)^2 \quad \textand \quad X(x) = C \sin\left(\fracn\pi xL\right)$$ where $n$ can be any positive integer. This means that we have an infinite number of possible solutions, each corresponding to a different value of $n$. These solutions are called eigenfunctions, and the corresponding values of $\lambda$ are called eigenvalues.

To find the general solution, we can use the principle of superposition, which states that any linear combination of solutions is also a solution. Therefore, we can write:

$$u(x,t) = \sum_n=1^\infty A_n e^-k\left(\fracn\piL\right)^2 t \sin\left(\fracn\pi xL\right)$$ where $A_n$ are arbitrary constants. To determine these constants, we need to use the initial condition:

$$u(x,0) = f(x) = \sum_n=1^\infty A_n \sin\left(\fracn\pi xL\right)$$ This is a Fourier sine series of the function $f(x)$ on the interval $[0,L]$. The coefficients $A_n$ can be found by using the orthogonality property of the sine functions:

$$A_n = \frac2L \int_0^L f(x) \sin\left(\fracn\pi xL\right) dx$$ Thus, we have found the complete solution to the heat equation for this BVP. It shows how the temperature distribution along the rod evolves over time, depending on the initial temperature and the thermal conductivity.

### Special Functions and Orthogonal Expansions

Special functions are functions that arise frequently in mathematical physics and have special properties and applications. Some examples of special functions are Bessel functions, Legendre polynomials, Hermite polynomials, Laguerre polynomials, etc.

Orthogonal expansions are methods of representing a function as an infinite series of orthogonal functions, such as Fourier series, Fourier-Bessel series, Fourier-Legendre series, etc. Orthogonal functions are functions that satisfy an orthogonality relation, such as:

$$\int_a^b f(x) g(x) w(x) dx = 0$$ where $w(x)$ is a weight function. Orthogonal expansions are useful for solving PDEs and BVPs by using the method of separation of variables.

In this chapter, Butkov introduces some of the most important special functions and orthogonal expansions in mathematical physics, such as Bessel functions and Fourier-Bessel series, Legendre polynomials and Fourier-Legendre series, Hermite polynomials and Fourier-Hermite series, Laguerre polynomials and Fourier-Laguerre series, etc. He also discusses some of their properties and applications in physics.

#### Example: Bessel Functions and Fourier-Bessel Series

Bessel functions are solutions to the Bessel differential equation:

$$x^2 y''(x) + x y'(x) + (x^2 - n^2) y(x) = 0$$ where $n$ is a constant. This equation arises when solving PDEs in cylindrical or spherical coordinates, such as the wave equation or the Helmholtz equation.

The Bessel functions of the first kind are denoted by $J_n(x)$, and the Bessel functions of the second kind are denoted by $Y_n(x)$. They can be defined by using power series, integral representations, or recurrence relations. They have many properties, such as:

$J_n(x)$ is an even function if $n$ is even, and an odd function if $n$ is odd.

$Y_n(x)$ is singular at $x = 0$, and behaves like $\ln(x)$ as $x \to 0$.

$J_n(x)$ and $Y_n(x)$ are linearly independent for any fixed $n$.

$J_n(x)$ and $J_-n(x)$ are linearly dependent if $n$ is an integer, and linearly independent if $n$ is not an integer.

$J_n(x)$ has infinitely many zeros on the positive real axis, denoted by $j_n,k$, where $k$ is the index of the zero. The zeros are simple and interlace with the zeros of $J_n+1(x)$.

$J_n(x)$ satisfies the orthogonality relation:

$$\int_0^a J_n(\alpha x) J_n(\beta x) x dx = \fraca^22 J_n+1^2(\alpha a) \delta_\alpha \beta$$ where $\alpha$ and $\beta$ are positive zeros of $J_n(x)$, and $\delta_\alpha \beta$ is the Kronecker delta.

A Fourier-Bessel series is a series of the form:

$$f(x) = \sum_k=1^\infty c_k J_n(\alpha_k x)$$ where $\alpha_k$ are positive zeros of $J_n(x)$, and $c_k$ are coefficients given by:

$$c_k = \frac2a^2 J_n+1^2(\alpha_k a) \int_0^a f(x) J_n(\alpha_k x) x dx$$ A Fourier-Bessel series can be used to represent any piecewise continuous function on the interval $[0,a]$, and to solve BVPs involving PDEs in cylindrical or spherical coordinates.

### Green's Functions and Integral Equations

A Green's function is a function that satisfies a differential equation with a delta function as the source term. For example, a Green's function for the Laplace equation is a function that satisfies:

$$\nabla^2 G(\mathbfx,\mathbfx') = -\delta(\mathbfx-\mathbfx')$$ where $\mathbfx$ and $\mathbfx'$ are points in space, $\nabla^2$ is the Laplacian operator, and $\delta(\mathbfx-\mathbfx')$ is the Dirac delta function. A Green's function can be used to construct the solution to a differential equation with any given source term, by using the principle of superposition. For example, if we have:

$$\nabla^2 u(\mathbfx) = f(\mathbfx)$$ where $f(\mathbfx)$ is any given function, then we can write:

$$u(\mathbfx) = \int G(\mathbfx,\mathbfx') f(\mathbfx') d\mathbfx'$$ where the integral is taken over the domain of interest. This is called a Green's representation formula.

An integral equation is an equation that involves an unknown function under an integral sign. For example, a Fredholm integral equation of the second kind is an equation of the form:

$$u(x) = f(x) + \lambda \int_a^b K(x,y) u(y) dy$$ where $f(x)$ and $K(x,y)$ are known functions, $\lambda$ is a constant, and $u(x)$ is the unknown function. An integral equation can be used to reformulate a differential equation or a BVP in a different way, or to model physical phenomena that involve interactions among different points or regions.

function methods, Fredholm theory, etc.

#### Example: Green's Function for the Laplace Equation

The Laplace equation is a PDE that describes the potential of a static electric field, or the steady-state temperature distribution in a homogeneous medium. It can be written as:

$$\nabla^2 u(\mathbfx) = 0$$ where $\mathbfx$ is a point in space, and $u(\mathbfx)$ is the unknown function.

A Green's function for the Laplace equation is a function that satisfies:

$$\nabla^2 G(\mathbfx,\mathbfx') = -\delta(\mathbfx-\mathbfx')$$ where $\mathbfx'$ is a fixed point in space, and $\delta(\mathbfx-\mathbfx')$ is the Dirac delta function. A Green's function can be used to construct the solution to the Laplace equation with any given source term, by using the Green's representation formula:

$$u(\mathbfx) = \int G(\mathbfx,\mathbfx') f(\mathbfx') d\mathbfx'$$ where $f(\mathbfx')$ is any given function.

The form of the Green's function depends on the dimension and geometry of the domain where the Laplace equation is defined. For example, if we consider a three-dimensional domain that extends to infinity in all directions, then the Green's function is given by:

$$G(\mathbfx,\mathbfx') = \frac1\mathbfx-\mathbfx'$$ where $\mathbfx-\mathbfx'$ is the distance between $\mathbfx$ and $\mathbfx'$. This Green's function can be derived by using the method of images, or by using Fourier transform techniques.

### Variational Methods and Calculus of Variations

Variational methods are methods that involve finding the extrema (minimum or maximum) of a functional. A functional is a function that maps a function to a number. For example, the length of a curve is a functional that maps a curve to its length.

Calculus of variations is a branch of mathematics that deals with finding the extrema of functionals, and studying their properties and applications. It can be considered as a generalization of calculus, where instead of dealing with functions of one or more variables, we deal with functions of functions.

Variational methods and calculus of variations are useful for solving differential equations and BVPs that arise from physical principles, such as the principle of least action, the principle of minimum potential energy, the principle of maximum entropy, etc. They are also useful for finding optimal solutions to problems involving geometry, mechanics, optics, economics, etc.

In this chapter, Butkov introduces some of the basic concepts and methods of variational methods and calculus of variations, such as functionals, Euler-Lagrange equation, Hamilton's principle, Lagrangian and Hamiltonian mechanics, variational principles in physics, etc.

#### Example: Euler-Lagrange Equation

The Euler-Lagrange equation is a differential equation that gives the necessary condition for a functional to have an extremum. For example, suppose we have a functional of the form:

$$F[y] = \int_a^b f(x,y,y') dx$$ where $y$ is an unknown function of $x$, and $y'$ is its derivative. To find the extrema of this functional, we need to find the function $y$ that satisfies:

$$\fracddx \left( \frac\partial f\partial y' \right) - \frac\partial f\partial y = 0$$ This is called the Euler-Lagrange equation. It can be derived by using the method of variations, which involves perturbing the function $y$ by a small amount $\epsilon \eta$, where $\eta$ is an arbitrary function that vanishes at the endpoints $a$ and $b$. Then we expand the functional $F[y+\epsilon \eta]$ in a Taylor series around $\epsilon = 0$, and set the first-order term to zero. This gives the Euler-Lagrange equation.

The Euler-Lagrange equation can be used to solve various problems in physics and mathematics, such as finding the shortest path between two points, finding the shape of a hanging chain, finding the equation of motion of a particle or a system, etc.

## How to download the PDF version of Butkov's book for free?

Butkov's book is a classic textbook on mathematical physics that has been out of print for a long time. However, thanks to the Internet, it is possible to find and download the PDF version of the book for free from various sources online. However, before doing so, one should be aware of the benefits and drawbacks of having a digital copy of the book, as well as the legal and ethical issues of downloading copyrighted material.

### The benefits of having a digital copy of the book

There are several advantages of having a digital copy of Butkov's book, such as:

It is convenient and portable. You can access the book from any device that can read PDF files, such as a computer, a tablet, or a smartphone. You can also store the book in a cloud service or a flash drive, and carry it with you wherever you go.

It is searchable and editable. You can use the search function to find any word or phrase in the book, or use the highlight and annotation tools to mark important parts or add your own notes. You can also copy and paste text or images from the book to other documents or applications.

It is cheaper and eco-friendly. You do not have to pay for the book or its shipping costs, nor do you have to worry about its availability or condition. You also save paper and ink by not printing the book.

### The drawbacks of having a digital copy of the book

There are also some disadvantages of having a digital copy of Butkov's book, such as:

It is less comfortable and enjoyable. Reading a PDF file on a screen can cause eye strain, headache, or fatigue, especially if you read for a long time or in poor lighting conditions. You may also miss the feeling and smell of holding a physical book in your hands, or flipping through its pages.

It is less reliable and secure. You may lose access to the book if your device breaks down, gets lost, stolen, or hacked, or if your file gets corrupted, deleted, or overwritten. You may also face technical issues such as compatibility problems, formatting errors, or broken links.

It is less interactive and social. You may have difficulty sharing the book with others, such as your classmates, friends, or teachers. You may also miss the opportunity to borrow or lend the bo