Exercise 4.2.4. Suppose we require the geometric condition that every level curve f(x, y) = 0 be a line. What types of m
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- Exercise 4 2 4 Suppose We Require The Geometric Condition That Every Level Curve F X Y 0 Be A Line What Types Of M 1 (32.55 KiB) Viewed 29 times
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Exercise 4.2.4. Suppose we require the geometric condition that every level curve f(x, y) = 0 be a line. What types of minimal surfaces do we now obtain? Hints:
K • In general, if a curve is given implicitly as f(x, y) = c, then the curvature is given by - fxf7 +2fx fyfry - fyy.tz VP Show this by invoking the implicit function theorem and writing f(x, 8(x)) = c with para- metrization a(x) = (x,8(x), c). Then KE 18" (1 +82) =- = = Show g' = -by using f.a' = 0. Now find g" by implicit differentiation • Show H = 0 + fu + fjs = \S3 + 5311 • Now K = 0 (why?), so firx + fyy = 0. • Show that a solution to Laplace's equation above with the given geometric constraint is f(x, y) = A arctan y - yo X-XO +B.