Heidelberg University Publishing

This introductory text is based on lectures within a three-semester cycle on "Real Analysis" given by the author at Heidelberg University. The present first part is devoted to the classical calculus of differentiation and integration for functions of one real variable. Content and presentation are particularly oriented towards the needs of the application in the theory of differential equations, in Mathematical Physics and in Numerical Analysis.
For supporting self-study each chapter contains exercises with solutions collected in the appendix.

This introductory text is based on lectures within a three-semester course on "Real Analysis" given by the author at Heidelberg University. The present second part is devoted to the classical calculus of differentiation and integration for functions of several real variables. Content and presentation are particularly oriented towards the needs of the application in the theory of differential equations, in Mathematical Physics and in Numerical Analysis. The understanding of the contents requires besides the material of the preceding part of this series, "Analysis 1 (Differential- und Integralrechnung fur Funktionen einer reellen Veränderlichen)", only some basic prior knowledge of Linear Algebra.
For supporting self-study each chapter contains exercises with solutions collected in the appendix.

This introductory text is based on lectures within a three-semester course on "Real Analysis", given by the author at Heidelberg University. The present third part treats the Riemann integral over lines and surfaces and the integral formulas of Gauß and Stokes. Further, the Lebesgue integral and the corresponding function spaces are introduced. Then, applications are discussed in the theory of Fourier integrals, for basic problems in the calculus of variations and in the theory of partial differential equations. Contents and
presentation are particularly oriented towards the needs of applications in the theory of differential equations, in Mathematical Physics and in Numerical Analysis. The understanding of the contents requires besides the material of the preceding parts of this series, "Analysis 1 (Differential- und Integralrechnung für Funktionen einer reellen Veränderlichen)" and "Analysis 2 (Differential- und Integralrechnung für Funktionen mehrerer reeller Veränderlichen)", only some basic prior knowledge of Linear Algebra. For supporting
self-study each chapter contains exercises with solutions collected in the appendix.

The lecture notes give an overview of modern cosmology: After introducing the necessary concepts from general relativity, the FLRW-class of cosmological models is discussed, with emphasis on dark energy. Cosmic structure formation, the necessity of dark matter and the interplay between statistics and nonlinear fluid mechanics are treated in detail. The physics behind cosmological observations that have led to the standard model of cosmology is explained, in particular supernovae, the cosmic microwave background and gravitational lensing.

Einstein‘s theory of general relativity is still the valid theory of gravity and has been confirmed by numerous tests and measurements. It is built upon simple principles and relates the geometry of space-time to its matter-energy content. These lecture notes begin by introducing the physical principles and by preparing the necessary mathematical tools taken from differential geometry. Beginning with Einstein’s field equations, which are introduced in two different ways in the lecture, the motion of test particles in a gravitational field is then discussed, and it is shown how the properties of weak gravitational fields follow from the field equations. Solutions for compact objects and black holes are derived and discussed as well as cosmological models. Two applications of general relativity to astrophysics conclude the lecture notes.

General relativity is the theory of the structure and dynamics of spacetime. These lecture notes provide an introduction into the concepts of differential geometry, in particular pseudo-Riemannian geometry, and discusses the ideas behind the construction of a gravitational field equation. Exact solutions to the field equation for highly symmetric spacetimes, i.e. black holes, FLRW cosmologies and gravitational waves, are worked out. Advanced topics that are covered include Lie derivatives and the Killing equation, the derivation of the field equations from variational principles, and the formulation of field theories on curved spacetimes.

This introductory text is based on courses within a multi-semester cycle on “Numerical Mathematics” given by the author at the Universities in Saarbrücken and Heidelberg. In the present part basic concepts of numerical methods are presented for solving linear optimization problems (so-called “Linear Programming”). This includes besides the classical ”Simplex method“ also modern ”Interior-point methods“. As natural extensions methods for convex nonlinear, especially quadratic, optimization problems are discussed. Theoretical as well as practical aspects are considered. As prerequisite only that prior knowledge is required, which is usually taught in the introductory Analysis, Linear Algebra, and Numerics courses. For facilitating self-learning the book contains theoretical and practical exercises with solutions collected in the appendix.

This introductory text is based on courses within a multi-semester cycle on “Numerical Mathematics” given by the author at the Universities in Saarbrücken and Heidelberg.

In the present part basic concepts of numerical methods are presented for solving linear optimization problems (so-called “Linear Programming”). This includes besides the classical ”Simplex method“ also modern ”Interior-point methods“. As natural extensions methods for convex nonlinear, especially quadratic, optimization problems are discussed.

Theoretical as well as practical aspects are considered. As prerequisite only that prior knowledge is required, which is usually taught in the introductory Analysis, Linear Algebra, and Numerics courses. For facilitating self-learning the book contains theoretical and practical exercises with solutions collected in the appendix.

This introductory text is based on courses on “Numerical Mathematics” given by the author at Heidelberg University within a multi-semester cycle over a period of 25 years. The present first part covers fundamental concepts of numerical methods for solving ordinary analysis and linear algebra problems. Both theoretical and practical aspects are considered. The understanding of the contents requires only that prior knowledge as is usually provided in the basic Analysis and Linear Algebra courses. For facilitating self-learning the book contains theoretical and practical exercises with solutions.

This introductory text is based on courses within a multi-semester cycle on “Numerical Mathematics” given by the author at Heidelberg University over a period of 25 years. The present second part treats numerical methods for solving ordinary differential equations. Again theoretical as well as practical aspects are considered. The last chapter provides an outlook on numerical methods for partial differential equations. The  understanding of the contents requires besides the material of the first volume in this series “Numerik 0” only that prior knowledge as is usually provided in the basic Analysis and Linear Algebra courses. For facilitating self-learning the book contains theoretical and practical exercises with solutions.

This introductory text is based on courses within a multi-semester cycle on “Numerical Mathematics” given by the author at Heidelberg University over a period of 25 years. The present third part is devoted to numerical methods for solving partial differential equations. Again theoretical as well as practical aspects are considered.

The understanding of the contents requires besides the material of the first two parts of this series, "Numerik 0 (Einführung in die Numerische Mathematik)" and "Numerik 1 - (Numerik gewöhnlicher  Differentialgleichungen)", only that prior knowledge as is usually provided in the basic Analysis and Linear Algebra courses. For facilitating self-learning the book contains theoretical exercises with solutions.

This introductory text is based on courses within a multi-semester cycle on “Numerical Mathematics” given by the author at Heidelberg University over a period of 25 years. The present fourth part is devoted to problems in Continuum Mechanics, especially in Structural and Fluid Mechanics, and their numerical solution by finite element methods. Again theoretical as well as practical aspects are considered. As basis of an appropriate numerical approximation the mathematical models are systematically derived from fundamental physical principles. The understanding of the contents requires besides the material of the preceding parts of this series, “Numerik 0 (Einführung in die Numerische Mathematik)”, “Numerik 1 (Numerik gewöhnlicher Differentialgleichungen)”, and “Numerik 2 (Numerik partieller Differentialgleichungen)” only that prior knowledge as is usually provided in the basic Analysis and Linear Algebra courses.

What is Palliative Medicine / Palliative Care? What stresses are patients and their relatives exposed to in incurable and advanced disease situations? How can appropriate comprehensive treatment and support be managed? How can a decision to limit or continue therapeutic measures at the end of life be justified?

This script is intended to provide medical students and all interested parties with an insight into the necessities and possibilities of comprehensive palliative medical support and to be of assistance for reference and preparation for the palliative medicine examinations in QB 13 and the state examination.

Statistical physics provides the microscopic theory for thermodynamic macroscopic properties of a physical system. These lecture notes introduce the necessary concepts of statistics and analytical mechanics for equilibrium thermodynamics with partition functions. They cover classical and quantum statistics, and treat advanced topics such as Langevin dynamics, the Fokker-Planck equation and phase transitions. Many systems like ideal classical and relativistic gases are worked out in detail.

Understanding astronomical objects requires knowledge and methods from different branches of theoretical physics: we diagnose these objects mostly by the light we receive; the observed phenomena often have to do with the flow of fluids, sometimes ionised, sometimes magnetised; and the measured velocities reflect dynamics driven by gravitational fields. Courses in theoretical physics lay the foundation in classical and quantum mechanics, electrodynamics, and thermodynamics, but a gap remains between this foundation and its application to astrophysics. These lecture notes build upon the core courses in theoretical physics and provides the methods for understanding astrophysics theoretically.

Theoretical physics is commonly taught in separate lectures, illustrating the physics behind the great constants of Nature: Electrodynamics and the speed of light, quantum mechanics and Planck’s constant, thermodynamics and Boltzmann’s constant, and finally relativity with Newton’s constant as well as the cosmological constant. In these lecture notes, the concepts of theoretical physics are illustrated with their commonalities, and phenomena are traced back to their origin in fundamental concepts.