Math S21a: Multivariable calculus Oliver Knill, Summer 2011Lecture 9: Partial derivativesIf f(x, y) is a function of two variables, then∂∂xf(x, y) is defined as the derivativeof the function g(x) = f(x, y), where y is considered a constant. It is called partialderivative of f with respect to x. The partial derivative with respect to y is definedsimilarly.One also uses the short hand notation fx(x, y) =∂∂xf(x, y). For iterated deriva t ives, the notationis similar: for example fxy=∂∂x∂∂yf.The notation for partial derivatives ∂xf, ∂yf were introduced by Carl Gustav Jacobi. Josef La-grange had used the term ”partial differences”. Partial derivatives fxand fymeasure the ra t eof change of the function in the x or y directions. For functions of more variables, the partialderivatives are defined in a similar way.1 For f (x, y) = x4− 6x2y2+ y4, we have fx(x, y) = 4x3− 12xy2, fxx= 12x2− 12y2, fy(x, y) =−12x2y + 4y3, fyy= −12x2+ 12y2and see that fxx+ fyy= 0. A function which satisfies thisequation is also called harmonic. The equation fxx+ fyy= 0 is an example of a partialdifferential equation: it is an equation for an unknown function f(x, y) which involvespartial derivatives with respect to mor e than one variables.Clairot’s theorem If fxyand fyxare both continuous, then fxy= fyx.Pro of: we look at the equations without taking limits first. We extend the definition and say thata background Planck constant h is positive, then fx(x, y) = [f(x + h, y) − f( x, y)]/h. Fo r h = 0we define fxas before. Compare the two sides for fixed h > 0:hfx(x, y) = f(x + h, y) − f (x, y)h2fxy(x, y) = f(x + h, y + h) − f(x + h, y +h) − (f(x + h, y) − f(x, y))dyfy(x, y) = f(x, y + h) − f (x, y).h2fyx(x, y) = f (x + h, y + h) − f(x + h, y) −(f(x, y + h) − f (x, y))We have not taken any limits in this proof but established an identity which ho lds for all h > 0, thediscrete derivatives fx, fysatisfy the relation fxy= fyx. We could fancy the identity obtained inthe proof as a ”quantum Clairot” theorem. If the classical derivatives fxy, fyxare both continuous,we can take the limit h → 0 t o get the classical Clairot’s theorem as a ”classical limit”. Notethat the quantum Clairot theorem shown first in this proof ho lds for any functions f(x, y) of twovariables. We do not even need the functions to be continuous.2 Find fxxxxxyxxxxxfor f (x) = sin(x) + x6y10cos(y). Answer: Do not compute, but think.3 The continuity assumption for fxyis necessary. The examplef(x, y) =x3y − xy3x2+ y2contradicts Clairaut’s theorem:fx(x, y) = (3x2y − y3)/(x2+ y2) − 2x(x3y −xy3)/(x2+y2)2, fx(0, y) = −y, fxy(0, 0) = −1,fy(x, y) = (x3− 3xy2)/(x2+ y2) − 2y(x3y −xy3)/(x2+ y2)2, fy(x, 0) = x, fy,x(0, 0) = 1.An equation for an unknown function f(x, y) which involves partial derivatives withrespect to at least two different varia bles is called a partial differential equation.If only t he derivative with respect to one variable appears, it is called an ordinarydifferential equation.Here are some examples of partial differential equations. You should know the first 4 well.4 The wave equation ftt(t, x) = fxx(t, x) governs the mot ion of lig ht or sound. The f unctionf(t, x) = sin(x − t) + sin(x + t) satisfies the wave equation.5 The heat equation ft(t, x) = fxx(t, x) describes diffusion of heat or spread of an epi-demic. The functionf(t, x) =1√te−x2/(4t)satisfies the heat equation.6 The Laplace equation fxx+ fyy= 0 determines the shape of a membrane. The functionf(x, y) = x3− 3xy2is an example satisfying the Laplace equation.7 The advection equation ft= fxis used to model transport in a wire. The functionf(t, x) = e−(x+t)2satisfy the advection equation.8 The eiconal equation f2x+ f2y= 1is used to see t he evolution of wave fronts in optics.The function f (x, y) = cos(x) + sin(y) satisfies the eiconal equation.9 The Burgers equation ft+ f fx= fxxdescribes waves at the beach which break. Thefunctionf(t, x) =xt√1te−x2/(4t)1+√1te−x2/(4t)satisfies the Burgers equation.10 The KdV equation ft+ 6f fx+ fxxx= 0 models water waves in a narrow channel.The functionf(t, x) =a22cosh−2(a2(x − a2t))satisfies the KdV equation.11 The Schr¨odinger equation ft=i¯h2mfxxis used to describe a quantum particle of massm. The function f(t, x) = ei(kx−¯h2mk2t)solves the Schr¨odinger equation. [Here i2= −1 isthe imaginary i and ¯h is the Planck constant ¯h ∼ 10−34Js.]Here are the graphs of the solutions of the equations. Can you match them with the PDE’s?Notice that in all these examples, we have just given one possible solution to the partial differen-tial equation. There are in general many solutions and only additional conditions like initial orboundary conditions determine the solution uniquely. If we know f (0, x) for the Burgers equation,then the solution f(t, x) is determined. A course on par tial differential equations would show youhow to get the solution.Paul Dirac once said: ”A great deal of my work is just playing with equations and seeingwhat they give. I do n’t suppose that applies so much to other physicists; I think it’s a peculiarityof myself that I like to play about with equations, just looking for beautiful mathematicalrelations which maybe don’t have any physical meaning at all. Sometimes they do.” Diracdiscovered a PDE describing the electron which is consistent both with quantum theory and specialrelativity. This won him the No bel Prize in 1933. Dirac’s equation could have two solutions, onefor an electron with positive energy, and one for an electron with negative energy. Dirac interpretedthe later as an antiparticle: the existence of antiparticles was later confirmed. We will not learnhere to find solutions to par tial differential equations. But you should be able to verify that agiven function is a solution of the equation.Homework1 Verify that f(t, x) = sin(cos(t + x)) is a solution of the transport equation ft(t, x) =fx(t, x).2 Verify that f(x, y) = 3y2+ x3satisfies the Euler-Tricomi partial differential equationuxx= xuyy. This PDE is useful in describing transonic flow. Can you find an othersolution which is not a multiple of the solution given in this problem?3 Verify that f(x, t) = e−rtsin(x + ct) satisfies the driven transpo rt equation ft(x, t) =cfx(x, t) − rf(x, t) It is sometimes also called t he advection equation.4 The pa r tial differential equation fxx+fyy= fttis called the wave equation in two
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