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Introduction to Linear Algebra with Mathematica

Glossary

Preface

Augustin-Louis Cauchy.
Baron Augustin-Louis Cauchy (1789--1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra.

Cauchy's father was active in the education of young Augustin-Louis. Laplace and Lagrange were visitors at the Cauchy family home and Lagrange in particular seems to have taken an interest in young Cauchy's mathematical education. Lagrange advised Cauchy's father that his son should obtain a good grounding in languages before starting a serious study of mathematics. In 1802 Augustin-Louis entered the École Centrale du Panthéon where he spent two years studying classical languages.

From 1804 Cauchy attended classes in mathematics and he took the entrance examination for the École Polytechnique in 1805. He was examined by Biot and placed second. At the École Polytechnique he attended courses by Lacroix, de Prony and Hachette while his analysis tutor was Ampère. In 1807 he graduated from the École Polytechnique and entered the engineering school École des Ponts et Chaussées. He was an outstanding student and for his practical work he was assigned to the Ourcq Canal project where he worked under Pierre Girard.

Sofya Kovalevskaya.
Sofya Vasilyevna Kovalevskaya (Russian: Софья Васильевна Ковалевская), born Korvin-Krukovskaya (1850 – 1891), was a Russian mathematician who made noteworthy contributions to analysis, partial differential equations and mechanics. She was a pioneer for women in mathematics around the world – the first woman to obtain a doctorate (in the modern sense) in mathematics, the first woman appointed to a full professorship in northern Europe and one of the first women to work for a scientific journal as an editor.

n 1868 Kovalevskaya entered into a marriage of convenience with a young paleontologist, Vladimir Kovalevsky, in order to leave Russia and continue her studies. The pair traveled together to Austria and then to Germany, where in 1869 she studied at the University of Heidelberg under the mathematicians Leo Königsberger and Paul du Bois-Reymond and the physicist Hermann von Helmholtz. The following year she moved to Berlin, where, having been refused admission to the university on account of her gender, she studied privately with the mathematician Karl Weierstrass. In 1874 she presented three papers—on partial differential equations, on Saturn’s rings, and on elliptic integrals—to the University of Göttingen as her doctoral dissertation and was awarded the degree, summa cum laude, in absentia. Her paper on partial differential equations, the most important of the three papers, won her valuable recognition within the European mathematical community. It contains what is now commonly known as the Cauchy-Kovalevskaya theorem, which gives conditions for the existence of solutions to a certain class of partial differential equations. Having gained her degree, she returned to Russia, where her daughter was born in 1878. She separated permanently from her husband in 1881.

The Cauchy--Kovalevskaya theorem on existence and uniqueness of solutions to systes of PDEs. This theorem establishes sufficient conditions on the individual PDEs of the system, as well as the initial conditions, called Cauchy data.

Suppose u1, u2, … , uM are dependent variables of time t and spacial variables x1, x2, … , xN. The general expression for a PDE of variable um (1 ≤ m ≤ M) is

\begin{equation} \label{EqCK.1} \frac{\partial^{k_m} u_m}{\partial t^{k_m}} = F_m \left( t, x_1 , x_2 , \ldots , x_M, u_1 , u_2 , \ldots , u_M , \ldots , \frac{\partial^j u_i}{\partial^{j_0} t \,\partial^{j_1} x_1 \,\cdots \cdot \partial^{j_N} x_N} , \ldots \right) , \end{equation}
where j0 + j1 + ··· + jN = j ≤ km and j0 < km. The notation reflects the requirement that PDE for an arbitrary dependent variable um be such that the km-th partial derivative of um with respect to t may be usilated on the left-hand side of Eq.\eqref{EqCK.1}. The right-hand side of the equation is a function Fm of the independent variables t, x1, x2, … , xN, the dependent variables u1, u2, … , uM, and the partial derivatives of the dependent variables, with order less than or equal to the order km. The order of any partial derivative of um with respect to any spacial variable xi is a nonnegative integer ji. Finally, the order j0 of the partial derivative of um with respect to t must be an integer strictly less than the order km.

Recall from experience with ODEs that a first order equation

\[ \frac{{\text d}y}{{\text d}t} = F(t,y) \]
requires an initial condition y(t0) = y0 for determing the unique particular solution. The second order ODE
\[ \frac{{\text d}^2 y}{{\text d}t^2} = F \left( t, y, \frac{{\text d}y}{{\text d}t} \right) \]
must have initial values for both y and its derivative prescribed for the particular solution to be found.

In similar way, initial data is needed in the case of the system of PDEs for a particular solution to be determined. Equation \eqref{EqCK.1} is an expression for the km-th partial derivative of dependent avriable um with respect to time t. Consequently, initial functions for all partial derivatives of order zero to km −1 of each dependent variable must be specified. The general form of such data is

\begin{equation} \label{EqCK.2} \frac{\partial^{j} u_i}{\partial t^{j}} \left( t_0 , x_1, \ldots , x_M \right) = \phi_{i,j} \left( x_1 , x_2 , \ldots , x_M \right) , \end{equation}
where ϕi,j(x1, x2, … , xN) is a given function for 0 ≤ j≤ km −1 and 1 ≤ i ≤ M.

Before the Cauchy--Kovalevskaya statement is given, it make sense to refresh your knowledge about analytic or holomorphic functions. For that purpose, suppose f is defined in terms of N variables x1, x2, … , xN. The point x0 = (x01, … , x0N) is said to be interior to the domain Ω of f if there exists a positive real number ϵ such that the N-dimensional "ball" of radius ϵ (often is refderred to as the ϵ-neighborhood of x0) is contained entirely within Ω. The function f is said to be holomorphic or analytic (however, generally speaking, definition of analytic functions is more general and include holomorphic functions) at x0 if f can be expressed as a power series about x0 such as

\[ f(x_1 , x_2 , \ldots , x_N ) = \sum_{k\ge 0} \left( \sum_{k_1 + \cdots + k_N = k} a_{k_1 k_2 \cdots k_N} \left( x_1 - x_1^0 \right)^{k_1} \cdots \left( x_N - x_N^0 \right)^{k_N} \right) , \]
that is valid for all x in the ϵ-neighborhood of x0. The poers ki are nonnegative integers.

Cauchy--Kovalevskaya theorem: Suppose that the system of PDEs with eqch PDE of the form as in Eq.\eqref{EqCK.1} is such that each Fm is holomorphic in a neighborhood of the point
\[ \left\{ t_0 , x_1^{0} , x_2^{0} , \ldots , x_N^{0} , \ldots , \left[ \frac{\partial^{j_m - j_0} u_i}{\partial^{j_1} x_1 \partial^{j_2} x_2 \cdots \partial^{j_N} x_N} \right]_{x_1^{0} , x_2^{0} , \ldots , x_N^{0}} , \ldots \right\} . \]
Further, suppose initial data, expressed as functions as shown in Eq.\eqref{EqCK.2}, is such that each ϕi,j is holomorphic on a neighborhood of x0 = (x01, … , x0N). Then there exists a unique holomorphic solution to the system of PDEs in a neighborhood of (t0, x01, … , x0N).