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Jxs63J
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Homework Statement
Show the principal curvatures on x sin z - y cos z = 0 are +-1/(1 + x^2 + y^2)
Actually I have devoted hours and hours to this problem. It is in Chapter 9 of the old Schaum's book. I have tried to put this in a form u*e1 + v*e2 + f(u,v)*e3 with little success. I have used other hints from this chapter such as determinant [dx, fx, Dfx] = 0 and I can get some simplifications but then I get lost in overly extensive equations. I have tried graphing this function as z = arctan(y/x) but without Mathematica or Maple the graph gets hazy in my mind. I know it would have several sheets but try to stick with the "main sheet". So I feel I am missing something crucial, but at the same time feel it should have been easy and that it is a really cool problem.
Differential geometry is a branch of mathematics that studies the properties of curves and surfaces in space. It uses tools from calculus, linear algebra, and topology to understand the geometric properties of these objects.
Principle curvatures are the maximum and minimum curvatures of a surface at a given point. They represent the amount of bending or curvature in different directions on the surface.
Gaussian curvature is the product of the two principle curvatures at a given point. It is a measure of how much a surface curves in all directions, and it is an important concept in differential geometry.
One example of a surface with constant principle curvatures is a sphere. At every point on a sphere, the principle curvatures are equal, resulting in a constant Gaussian curvature.
Differential geometry has many practical applications, such as in computer graphics, robotics, and physics. It is used to model and analyze complex shapes and surfaces, and to understand the behavior of objects in motion.