General Description
Suppose one has a function u which describes the temperature at a given location (x, y, z). This function will change over time as heat spreads throughout space. The heat equation is used to determine the change in the function u over time. The image to the right is animated and describes the way heat changes in time along a metal bar. One of the interesting properties of the heat equation is the maximum principle which says that the maximum value of u is either earlier in time than the region of concern or on the edge of the region of concern. This is essentially saying that temperature comes either from some source or from earlier in time because heat permeates but is not created from nothingness. This is a property of parabolic partial differential equations and is not difficult to prove mathematically (see below).
Another interesting property is that even if u has a discontinuity at an initial time t = t0, the temperature becomes smooth as soon as t > t0. For example, if a bar of metal has temperature 0 and another has temperature 100 and they are stuck together end to end, then very quickly the temperature at the point of connection will become 50 and the graph of the temperature will run smoothly from 0 to 100.
The heat equation is used in probability and describes random walks. It is also applied in financial mathematics for this reason.
It is also important in Riemannian geometry and thus topology: it was adapted by Richard Hamilton when he defined the Ricci flow that was later used by Grigori Perelman to solve the topological Poincaré conjecture.
Read more about this topic: Heat Equation
Famous quotes containing the words general and/or description:
“I have never looked at foreign countries or gone there but with the purpose of getting to know the general human qualities that are spread all over the earth in very different forms, and then to find these qualities again in my own country and to recognize and to further them.”
—Johann Wolfgang Von Goethe (1749–1832)
“Whose are the truly labored sentences? From the weak and flimsy periods of the politician and literary man, we are glad to turn even to the description of work, the simple record of the month’s labor in the farmer’s almanac, to restore our tone and spirits.”
—Henry David Thoreau (1817–1862)