4. Extra Credit. (10 Points.) Consider the problem of solving for functions u(x, y), defined...
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Calculus
4. Extra Credit. (10 Points.) Consider the problem of solving for functions u(x, y), defined on x + y ? 1,such that(i) Au = 0, and(ii) u = 0 at all points in the boundary x + y = 1.Show that the only solution to this problem is the zero function: u(x, y) = 0.Hint: Consider the vector field F = uvu.Can you make the same conclusion if we replace replace condition (ii) with "Vun = 0 at all points onthe boundary" (so the function becomes "flat" as you approach the boundary). If not, what is a conclusionabout u you can make?Remark: The two "boundary conditions" in this problem are two very common ones you'll find in differen-tial equations.
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