Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
Part 2 book “Handbook of mathematics” has contents: Vector analysis and vector fields, function theory, integral transformations, probability theory and mathematical statistics, dynamical systems and chaos, optimization, numerical analysis, computer algebra systems, tables, bibliography. | 642 13. Vector Analysis and Vector Fields 13 VectorAnalysisandVectorFields 13.1 Basic Notions of the Theory of Vector Fields 13.1.1 Vector Functions of a Scalar Variable 13.1.1.1 De nitions 1. Vector Function of a Scalar Variable t A vector function of a scalar variable is a vector ~a whose components are real functions of t: ~a = ~a(t) = ax(t)~ex + ay (t)~ey + az (t)~ez : (13.1) The notions of limit, continuity, di erentiability are de ned componentwise for the vector ~a(t). 2. Hodograph of a Vector Function If we consider the vector function ~a(t) as a position or radius vector ~r = ~r(t) of a point P , then this function describes a space curve while t varies (Fig. 13.1). This space curve is called the hodograph of the vector function ~a(t). P1 r1 0 P P2 r2 r r3 P3 Figure 13.1 0 dr dt Dr r+Dr Figure 13.2 13.1.1.2 Derivative of a Vector Function 30 25 20 15 Figure 13.3 The derivative of (13.1) with respect to t is also a vector function of t: d~a = lim ~a(t + %t) ; ~a(t) = dax(t)~e + day (t)~e + daz (t)~e : (13.2) dt t!0 %t dt x dt y dt z r The geometric representation of the derivative d~ dt of the radius vector is a vector pointing in the direction of the tangent of the hodograph at the point P (Fig. 13.2). Its length depends on the choice of the parameter t. If t is the time, then the vector ~r(t) describes the motion of a point P in space (the r space curve is its path), and d~ motion. If t = s dt has the direction and magnitude of the velocity of this d~ r is the arclength of this space curve, measured from a certain point, then obviously = 1. ds 13.1.1.3 Rules of Di erentiation for Vectors d dt d dt d dt ~ (~a ~b ~c) = d~a db d~c dt dt dt d~a ('~a) = d' dt ~a + ' dt a ~ d~b (~a~b) = d~ dt b + ~a dt (13.3a) (' is a scalar function of t) (13.3b) (13.3c) 13.1 Basic Notions of the Theory of Vector Fields 643 d d~b ~ d~a ~ dt (~a b) = dt b + ~a dt d d~a d' dt ~a '(t)] = d' dt (the factors must not be .