MAP 4306 Lecture Notes - Lecture 1: Thermal Conduction, Joseph Fourier, Thermal Conductivity

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6 Feb 2015
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If u = u(x, y, z) is a function of three variables, we de ne the laplacian of u by. If u = u(x, y) only depends on two variables, then the laplacian is de ned by. And if u = u(x) depends on a single variable, the laplacian reduces to the second derivative d2u dx2 of u. Frequently our functions will depend on one, two or three space variables, denoted by x, y, z (or x1, x2, x3) in the case of three space variables, and an extra variable t interpreted as being time. The laplacian will then be interpreted as acting only on the space variables. So if u = u(x, y, t), then we still have. A simple model (more on it below) states that u satis es an equation of the form. A model for waves traveling in space is described by the wave equation.

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