lecture36-09

lecture36-09 - 18.02 Multivariable Calculus(Spring 2009...

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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 36 On Laplace and Maxwell’s equations. May 8 Reading Material: From From Lecture Notes : V7 and V15 Last time: Stokes’ Theorem Today: On Laplace and Maxwell’s 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 = Δ u = ∂ 2 x u + ∂ 2 y u + ∂ 2 z u. Let’s now consider two (so called) boundary values problem: Consider the solid D bounded by a closed “nice” surface S : The Laplace equation Δ u = 0 (we say that u is harmonic) with condition on S given by u = φ on S (2.1) is the so called Dirichelet Problem. 1 On the other hand if we replace (2.1) with ∂ ∂η u = ~ ∇ u · ˆ n = ψ on S (2.2) where ∂ ∂η u denotes the directional derivative of u in the direction of the normal vector ˆ 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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This note was uploaded on 03/08/2010 for the course MATH 18.022 taught by Professor Hartleyrogers during the Spring '06 term at MIT.

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lecture36-09 - 18.02 Multivariable Calculus(Spring 2009...

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