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 V7 Laplace s Equation and Harmonic Functions In this section we will show how Green s theorem is closely connected with solutions to Laplace s partial differential equation in two dimensions where w x y is some unknown function of two variables assumed to be twice differentiable Equation 1 models a variety of physical situations as we discussed in Section P of these notes and shall briefly review 1 The Laplace operator and harmonic functions The two dimensional Laplace operator or laplacian as it is often called is denoted by V2 or lap and defined by The notation V2 comes from thinking of the operator as a sort of symbolic scalar product In terms of this operator Laplace s equation 1 reads simply Notice that the laplacian is a linear operator that is it satisfies the two rules 3 v2 u v v2u v2v v2 cu c v2u for any two twice differentiable functions u x y and v x y and any constant c Definition A function w x y which has continuous second partial derivatives and solves Laplace s equation 1 is called a harmonic function In the sequel we will use the Greek letters q5 and to denote harmonic functions functions which aren t assumed to be harmonic will be denoted by Roman letters f g u v etc According to the definition 4 4 x y is harmonic H v2q5 0 By combining 4 with the rules 3 for using Laplace operator we see 5 q5 and harmonic q5 and cq5 are harmonic c constant Examples of harmonic functions Here are some examples of harmonic functions The verifications are left to the Exercises V VECTOR INTEGRAL CALCLUS 2 A Harmonic homogeneous polynomials1 in two variables Degree 0 all constants c are harmonic Degree 1 all linear polynomials ax by are harmonic Degree 2 the quadratic polynomials x2 y 2 and xy are harmonic all other harmonic homogeneous quadratic polynomials are linear combinations of these q5 x y a x2 Y2 bxy a b constants Degree n the real and imaginary parts of the complex polynomial x Check this against the above when n 2 are harmonic d m B Functions with radial symmetry Letting r the function given by r In r is harmonic and its constant multiples c l n r are the only harmonic functions with radial symmetry i e of the form f r C Exponentially growing or decaying oscillations ekx sin ky and ekx cos ky are harmonic For all k the functions In general harmonic functions cannot be written down explicitly in terms of elementary functions Nevertheless we will be able to prove things about them by using Green s theorem 2 Harmonic functions and vector fields The relation between harmonic functions and vector fields rests on the simple identity 6 div Vf v2f which is easily verified since its truth is suggested symbolically by There is an important connection between harmonic functions and conservative fields which follows immediately from 6 Let F V f 7 Then div F 0 f is harmonic Another way to put this is to say in a simply connected region curl F 0 and div F 0 7 F Vq5 where q5 is harmonic This is just 7 combined with the criterion for gradient fields Section V5 X In other words from the vector field viewpoint the theory of harmonic functions and Laplace s equation is the same as the theory of conservative vector fields with zero divergence Where do such functions and fields occur A homogeneous polynomial in several variables is one in which all the terms have the same total degree like x 2 y 2y3 or x5 6xZy3 4xy4 V7 LAPLACE S EQUATION AND HARMONIC FUNCTIONS One place is in heat flow problems Imagine a thin uniform metal plate which is insulated on the faces so no heat can enter or escape on the faces and imagine that some temperature distribution is maintained along the edge of the plate Then when the temperature distribution on the plate has reached steady state it will be given by a harmonic function x y namely it must satisfy the heat a2gt but gt 0 since the equation see Section P of these notes q5 temperature is not changing with time by assumption Harmonic functions also occur as the potential functions for two dimensional gravitational electrostatic and electromagnetic fields in regions of space which are respectively free of mass static charge or moving charges Here twodimensional means not that the fields lie in the xy plane but rather that as fields in three space the vectors all lie in horizontal planes and the field looks the same no matter what horizontal plane it is viewed in A typical example would be the field arising from a uniform mass or charge distribution on a set of vertical wires or from uniform currents on vertical wires 3 Boundary value problems As the example given above of a temperature distribution on a uniform insulated metal plate suggests the typical problem in solving Laplace s equation would be to find a harmonic function satisfying given boundary conditions That is we are given a region R of the xy plane bounded by a simple closed curve C The problem is to find a function g x y which is defined and harmonic on R and which takes on prescribed boundary values along the curve C The boundary values are commonly given in one of two ways i as the values of q5 along C 84 of q5 along C ii as the values of the normal derivative drl To explain this last the normal derivative is just the directional derivative in the direction of the outward pointing unit normal vector n 2In g n normal derivative drl The tangential derivative is defined similarly using the unit tangent vector t instead of n 84 For heat flow problems boundary values of the first type i would be most common you are maintaining a definite temperature distribution q5 along C and want to know what the temperature will look like in R For conservative force field problems with F Vq5 one could also get boundary values of the second type ii For example if you were given the field vector F at each point of C then you would know Vq5 n and Vq5 t the normal derivative and the tangential derivative at each point of C Knowing the tangential derivative however is equivalent to knowing g5 itself on C for ds s arclength along C and therefore s can be obtained by integrating the tangential derivative So to prescribe F on the boundary is equivalent to prescribing both i and ii above for its potential function 4 V VECTOR INTEGRAL CALCLUS The basic problems are now these A Existence Does there exist a x y harmonic in some region containing C and its
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