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Handbook of mathematics for engineers and scienteis61ts part

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Tham khảo tài liệu 'handbook of mathematics for engineers and scienteis61ts part', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 388 Differential Geometry 9.2.1-3. Tangent line to surface. A straight line is said to be tangent to a surface if it is tangent to a curve lying in this surface. Suppose that a surface is given in vector form 9.2.1.1 and a curve lying on it is parametrized by the parameter t. Then to each parameter value there corresponds a point of the curve and the position of this point on the surface is specified by some values of the curvilinear coordinates u and v. Thus the curvilinear coordinates of points of a curve lying on a surface are functions of the parameter t. The system of equations u u t v v t 9.2.1.3 is called the intrinsic equations of the curve on the surface. The intrinsic equations completely characterize the curve if the vector equation of the surface is given since the substitution of 9.2.1.3 into 9.2.1.1 results in the equation r r u t v t which is called the parametric equation of the curve. The differential of the position vector is equal to dr r du r dv 9.2.1.4 9.2.1.5 where du u t t dt and dv v t t dt. The vectors r and r are called the coordinate vectors corresponding to the point whose curvilinear coordinates have been used in the computations. The coordinate vectors are tangent vectors to coordinate curves Fig. 9.21 . Figure 9.21. Coordinate vectors. 9.2. Theory of Surfaces 389 Formula 9.2.1.5 shows that the direction vector of any tangent line to a surface at a given point is a linear combination of the coordinate vectors corresponding to this point i.e. the tangent to a curve lies in the plane spanned by the vectors r and r at this point. The direction of the tangent to a curve on a surface at a point M is completely characterized by the ratio dv du of differentials taken along this curve. 9.2.1-4. Tangent plane and normal. If all possible curves are drawn on a surface through a given regular point M0 r0 M0 x0 y0 z0 M u0 v0 of the surface then their tangents at M0 lie in the same plane which is called the tangent plane to the surface at M0. The .