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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 36 On Laplace and Maxwells equations. May 8 Reading Material: From From Lecture Notes : V7 and V15 Last time: Stokes Theorem Today: On Laplace and Maxwells equations. The scope of this lecture is to use many of the theorems we introduced in this class to analyze some of the most important partial differential equations proposed in physics. 2 Laplace Equation We already introduced the notation ~ ~ u = u = 2 x u + 2 y u + 2 z u. Lets now consider two (so called) boundary values problem: Consider the solid D bounded by a closed nice surface S : The Laplace equation u = 0 (we say that u is harmonic) with condition on S given by u = on S (2.1) is the so called Dirichelet Problem. 1 On the other hand if we replace (2.1) with u = ~ u n = on S (2.2) where u denotes the directional derivative of u in the direction of the normal vector n to S , we define the Neumann Problem: For both problems the solution u denotes the steady state of the temperature of the body...
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 Spring '06
 HartleyRogers
 Equations, Multivariable Calculus

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