Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 58', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Chapter 9 Differential Geometry . Theory of Curves . Plane Curves . Regular points of plane curve. A plane curve r in a Cartesian coordinate system can be defined by equations in the following form Explicitly y f x . Implicitly F x y 0. Parametrically x x t y y t . In vector form r r t where r t x t i y t j . In a polar coordinate system the curve is usually given by the equation r r where the relationship between Cartesian and polar coordinates is given by formulas x r cos y and y r sin . Remark. The explicit equation can be obtained from the parametric equations if the abscissa is taken for the parameter x t y f t . A point M x t y t is said to be regular if the functions x t and y t have continuous first derivatives not simultaneously equal to zero in a sufficiently small neighborhood of this point. For implicitly defined functions a point M x y is said to be regular if grad F VF 0 at this point. If a curve is given parametrically then the positive sense is defined on this curve . the direction in which the point M x t y t of the curve moves as the parameter t increases. If the curve is given explicitly by then the positive sense corresponds to the direction in which the abscissa increases . moves from left to right . In a polar coordinate system the positive sense corresponds to the direction in which the angle p increases . the positive sense is counterclockwise . If s is the curve length from some constant point M0 to M then the infinitesimal length increment of the arc M0 M is approximately expressed by the formula for the arc length 367 368 Differential Geometry differential ds . the following formulas hold As ds y 1 yx 2 dx As ds y xt 2 y t 2 di As ds y r2 r v 2 dp if the curve is given explicitly if the curve is given parametrically for a curve in the polar coordinate system. Example 1. The arc length differential of the curve y cos x has the form ds V1 sin x dx. Example 2. For the
