The differential world, i.e. the world of derivatives, was invented / discovered in the 17th century, almost at the same time that Modern Science (then called Natural Philosophy) was born. Newton formulated Mechanics in terms of ODEs, focusing on the motion of particles. The main magic formula is In the 18th century, PDEs appear in Jean Le Rond D'Alember's work on string oscillations: a group of particles move together due to elastic forces, but each of the infinitely rigid elements has a different motion, u=u(x,t).
This is one of the first cases of continuous collective dynamics. PDEs are the way of expressing such a CCD. Johann and Daniel Bernoulli and then Leonhard Euler laid the foundations of Ideal Fluid Mechanics (1730 to 1750), in Basel and St.Petersburg. The system is non-linear; it does not fit into one of the 3 types that we know today (elliptical, parabolic, hyperbolic); the primary pure mathematics problem is still unsolved (the existence of classical solutions for good data; . Clay problems, years). 2000).
Origins of PDEs
PDEs in the XIX Century
The new century is facing revolutions in the conception of heat and energy, electricity and magnetism, and what space is. All these fields mathematically end in PDE:. i) heat leads to the heat equation,ut= ∆u, and the credit goes to kJ. ii) electricity leads to the Coulomb equation in Laplace-Poisson form:. A vector potential satisfies a wave equation, the same as D'Alembert's, but has a vector value and is in more dimensions. v) Real fluids are represented by the Navier Stokes equations.
PDEs cont. XIX Century
Mathematics, Physics and PDE Modern times. iv) Geometry was transformed from the tradition of Euclid plus Cartesian Algebra in the spirit of PDEs by G. After these men, especially Riemann, reality is mostly continuous and its essence lies in physical law, which is a law about field or a series of fields.
XXth century. Summing Up
The main (technical) task of a mathematician working in mathematical physics is to understand the world of partial differential equations, linear and non-linear. The combination of functional analysis, PDES and ODE, geometry, physics and stochastic calculus is one of the great machines of today's research, a child of the 20th century. The combination of functional analysis, PDES and ODE, geometry, physics and stochastic calculus is one of the great machines of today's research, a child of the 20th century.
Outline
Georg Friedrich Bernhard Riemann (1826-1866)
First, to preserve the symmetry, there must be two real functions of two real variables:.
Complex Variables II
Complex Variables: the PDE code, called CRE
They are one of the most important examples of a PDE system with extraordinary geometric and analytical consequences.
Complex Variables, analysis and geometry
If the 2-2 function f is CR, it defines a conformal transformation of the part of the plane where f0(z)6=0. Riemann's geometric theory of one CV is based on this idea.♠. If the 2-2 function f is CR, it defines a conformal transformation of the part of the plane where f0(z)6=0. Riemann's geometric theory of one CV is based on this idea.♠. Solutions of this equation are harmonic functions, and they count as the most beautiful C∞ functions in analysis and the most important in physics, where solving ∆u=−ρ means finding the potential of ρ.
Givenuwe is found in, its conjugate pair, by integrating the differential form. dv=Pdx+Qdy, with P=vx=−uy,Q=vy=ux; it's a precise differential thanks to the CR. Go to the scalar potential of the vector field: is accurate due to irrotation). Take the harmonic conjugate Ψ and define the complex current potential as F= Φ +iΨ, a chf.
Some Pictures of 2D glory
Summary
THE BIG PICTURE IN 2D
The complex derivative of the complex potential is just the conjugate of the velocity field. These are singular points, which in physics are called stagnation points. Many things can happen at a singularity, essentially one thing can happen at a fixed point (⇒the implicit function theorem). Riemann was an expert on singular points.
From 2D to 3D
What is Geometry according to BR
In 1854 Riemann presented his ideas on geometry for the official postgraduate qualification in Gottingen; the older Gauss was an examiner and was extremely impressed. Riemann argued that the fundamental ingredients for geometry are a space of points (now called a manifold) and a way to measure the distance between curves in space. He argued that space need not be an ordinary Euclidean space and that it could have any dimension (he even considered spaces of infinite dimension).
Nor is it necessary to draw the surface as a whole in three-dimensional space. Riemann seems to have been led to these ideas in part by his dislike of the (Newtonian) concept of action at a distance in contemporary physics and by his desire to endow space with the ability to transmit forces such as electromagnetism and gravitation . It is local as it operates on local entities, tangent vectors. Forget Pythagoras but remember2 =dθ2+sin2θdφ2 on the sphere.
Note that although Christoffel symbols have three indices, they are not tensors.
Curvatures at the center of geometry
The Laplacian operator in such geometries
This is minus the contraction of the second covariant derivative tensor (∇2u)ij=∂iju−Γkij∂ku.
Yamabe problem. Ricci flow
General relativity. Einstein equation and tensor
Riemann’s interest in Physics
Die Partiellen Differentialgleichungen der Mathematischen Physik.Nach Riemann's Vorlesungen in vierter Auflage neu bearbeitet von Heinrich Weber, Professor der Mathematik an der Universit ˆat Strassburg. Riemann's lectures on the partial differential equations of mathematical physics and their application to heat conduction, elasticity and hydrodynamics were published after his death by his former student, Hattendorff. Three editions appeared, the last in 1882; and few books have proved so useful to the student of theoretical physics. The purpose of Riemann's lectures was twofold: first, to formulate the differential equations which are based on the results of physical experiments or hypotheses ; secondly, to integrate these equations and explain their limitations and their application in particular cases.
One-dimensional isentropic gas flow is a mathematical abstraction described by the system of differential equations. In the application module, x is interpreted as the length along a pipe, the transverse dimensions of which are assumed to be irrelevant, and the velocity and density of the fluid particles are interpreted. The finding of this caused B. Riemann great concern, as he says at the beginning of his article.
Hyperbolic systems
To continue we do linear algebra, calculating the eigenvectors and eigenvalues of the matrix A. Note that λ(u, ρ) therefore varies with (x,t) depending on the flow you choose at this time. Now we get a map from (x,t) to (u, ρ), with two nice directions for linearizing the evolution equation.
If you are Riemann, this allows you to construct some magic coordinates where the flow is not complicated.
Riemann invariants
Now Riemann tells us to find the characteristic lines: If we think the solution is known, then solve the ODE systems. He then tells you to find functions F1, F2, called the Riemann invariants, which are independent and constant along the corresponding. Since these functions are constant on the properties, they allow you to see what the properties are doing, and this tells you what the flow is doing at any given time.
Shocks
The theory developed by BR enables the solution of the system in a classical way, if the characteristics of the same type for different points do not cross. Rankine and Hugoniot completed Riemann's work when the gas is not isentropic and the system is three-dimensional. The story of how discontinuous functions can be correct solutions to the partial differential equations of mathematical physics, and even more so, how important what happens at the point where classical analysis breaks down, is one of the deepest and most beautiful aspects of PDE in the 20th century. Isentropy solution keyword, I also worked on this topic.
New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2006. Temple, “Shock Wave Interactions in General Relativity,” Springer Science + Business Media, LLC, New York, NY USA, 2007. The story of how discontinuous functions can be correct partial differential solutions equation of mathematical physics, and more importantly what happens at the point where classical analysis breaks down, is one of the most profound and beautiful aspects of PDE in the 20th century.
Joel Smoller, “Shock Waves and Reaction-Diffusion Equations”, Springer, 2e druk, 1994.* Jeffrey Groah, Joel Smoller, B.
Some shock waves in Nature
This nebula is an expanding blast wave from a stellar cataclysm, a supernova explosion, which occurred about 15,000 years ago. The supernova blast wave, which is moving from left to right across the image, has recently hit a cloud of denser-than-average interstellar gas. This collision causes shock waves in the cloud that heat the interstellar gas, making it glow.
Sandia releases new version of Shock Wave Physics program:. http://composite.about.com/library/PR/2001/blsandia1.htm. This nebula is an expanding explosion wave from a stellar cataclysm, a supernova explosion, that occurred about 15,000 years ago. The supernova explosion wave, which moves from left to right across the image, recently hit a cloud of denser-than-average interstellar gas.
This collision drives shock waves into the cloud that heat the interstellar gas, causing it to glow.
P D End
Danke sch ¨ on, Herr Riemann!
