# lec13-1 - Laplace equation in Cartesian coordinates • The...

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Unformatted text preview: Laplace equation in Cartesian coordinates • The Laplace equation is written ∇ 2 φ = 0 • For example, let us work in two dimensions so we have to find φ ( x , y ) from, ∂ 2 φ ∂ x 2 + ∂ 2 φ ∂ y 2 = 0 • We use the method of separation of variables and write φ ( x , y ) = X ( x ) Y ( y ) X 00 X + Y 00 Y = 0 Laplace equation in Cartesian coordinates, continued • Again we have two terms that only depend on one independent variable, so Y 00 Y =- k 2 • This is called a Helmholtz equation (we’ve seen in before), and we can write it Y 00 + k 2 Y = 0 • The we have another equation to solve, X 00- k 2 X = 0 • Here k is real and k ≥ • Could we have done this a different way? Yes! Laplace equation in Cartesian coordiates, continued • We could have a different sign for the constant, and then Y 00- k 2 Y = 0 • The we have another equation to solve, X 00 + k 2 X = 0 • We will see that the choice will determine the nature of the solutions, which in turn will depend on the boundary conditions Steady-state temperature in a semi-infinite plate • Imagine a metal plate bounded at y = 0 but extending to infinity...
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lec13-1 - Laplace equation in Cartesian coordinates • The...

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