MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms P Partial Differential Equations An important application of the higher partial derivatives is that they are used in partial differential equations t o express some laws of physics which are basic t o most science and engineering subjects In this section we will give examples of a few such equations The reason is partly cultural so you meet these equations early and learn t o recognize them and partly technical to give you a little more practice with the chain rule and computing higher derivatives A partial differential equation PDE for short is an equation involving some unknown function of several variables and one or more of its partial derivatives For example is such an equation Evidently here the unknown function is a function of two variables we infer this from the equation since only x and y occur in it as independent variables In general a solution of a partial differential equation is a differentiable function that satisfies it In the above example the functions w xnyn any n all are solutions to the equation In general PDE s have many solutions far too many t o find all of them The problem is always t o find the one solution satisfying some extra conditions usually called either boundary conditions or initial conditions depending on their nature Our first important PDE is the Laplace equation in three dimensions Any steady state temperature distribution in three space 2 W T x Y T temperature a t the point x y z satisfies Laplace s equation Here steady state means that it is unchanging over time here reflected in the fact that T is not a function of time For example imagine a solid object made of some uniform heat conducting material say a solid metal ball and imagine a steady temperature distribution on its surface is maintained somehow say with some arrangement of wires and thermostats Then after a while the temperature a t each point inside the ball will come t o equilibrium reach a steady state and the resulting temperature function 2 inside the ball will then satisfy Laplace s equation As another example the gravitational potential P PARTIAL DIFFERENTIAL EQUATIONS 1 resulting from some arrangement of masses in space satisfies Laplace s equation in any region R of space not containing masses The same is true of the electrostatic potential resulting from some collection of electric charges in space 1 is satisfied in any region which is free of charge This potential function measures the work done against the field carrying a unit test mass or charge from a fixed reference point to the point x y z in the gravitational or electrostatic field Knowing 4 the field itself can be recovered as its negative gradient All of this is just to stress the fundamental character of Laplace s equation we live our lives surrounded by its solutions The two dimensional Laplace equation is similar you just drop the term involving z The steady state temperature distribution in a flat metal plate would satisfy the twodimensional Laplace equation if the faces of the plate were kept insulated and a steady state temperature distribution maintained around the edges of the plate If in the temperature model we include also heat sources and sinks in the region unchanging over time the temperature function satisfies the closely related Poisson equation where f is some given function related to the sources and sinks Another important PDE is the wave equation given below are the one dimensional and two dimensional versions the three dimensional version would add a similar term in z to the left Here x y are the space variables t is the time and c is the velocity with which the wave travels this depends on the medium and the type of wave light sound etc A solution respectively W w x t W w x Y gives for each moment to of time the shape w x to w x y to of the wave The third PDE goes by two names depending on the context heat equation or diffusion equation The one and two dimensional versions are respectively It looks a lot like the wave equation 4 but the right hand side this time involves only the first derivative which gives it mathematically and physically an entirely different character When it is called the one dimensional heat equation a solution w x t represents a time varying temperature distribution in say a uniform conducting metal rod with insulated sides In the same way w x y t would be the time varying temperature distribution in a flat metal plate with insulated faces For each moment to in time w x y to gives the temperature distribution at that moment For example if we assume the distribution is steady state i e not changing with time then aw O steady state condition at 2 18 02 NOTES and the two dimensional heat equation would turn into the two dimensional Laplace equation 1 When 5 is referred to as the dzfluszon equation say in one dimension then w x t represents the concentration of a dissolved substance diffusing along a uniform tube filled with liquid or of a gas diffusing down a uniform pipe Notice that all of these PDE s are second order that is involve derivatives no higher than the second There is an important fourth order PDE in elasticity theory the bilaplacian equation but by and large the general rule seems to be either that Nature is content with laws that only require second partial derivatives or that these are the only laws that humans are intelligent enough to formulate Exercises Section 2K
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