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Lecture 36: 8 December 2010Lecture 37: 8 December 2010 Discussed the impact of a different kind of algebraic structure on the solution of linear PDEs: symmetry. Looked at the specific examples of mirror symmetries and translational symmetry. Showed that a symmetry corresponds to a symmetry operator that commutes with  and preserves the boundary conditions, and allows us to find simultaneous eigenfunctions of  and the symmetry operator. For mirror symmetries, this leads to even/odd solutions, and for translational symmetry this leads to separable solutions with exp(ikz) exponentials in the invariant directions. Discussed projection operators onto the symmetry eigenfunctions, and showed that the projections also commute with Â. Hence, showed that the symmetry eigenvalue is "conserved": if you have an initial u(x) that is even/odd, or a source that is even/odd, then the resulting solutions u at later times will also be even/odd. Similarly for exp(ikz) and translational symmetry: k is conserved, and this leads e.g. to Snell's law. Further reading: The general subject of symmetry and linear PDEs leads to group theory (for the symmetry group) and group representation theory (to generalize the symmetry "eigenfunctions" to non-commutative groups). For a simple introduction similar to the one in class but applied to Maxwell's equations, see e.g. chapter 3 of our book. For a more complete treatment, see any book on the applications of group theory to physics; my favorite is this book by Inui but it is out of print; a classic with cheap reprints is this book by TinkhamMIT OpenCourseWarehttp://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and NumericsFall 2010 For information about citing these materials or our Terms of Use, visit:


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