tailieunhanh - Ebook Handbook of mathematics (5/E): Part 2

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 Basic Notions of the Theory of Vector Fields Vector Functions of a Scalar Variable 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 : () 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. ). This space curve is called the hodograph of the vector function ~a(t). P1 r1 0 P P2 r2 r r3 P3 Figure 0 dr dt Dr r+Dr Figure Derivative of a Vector Function 30 25 20 15 Figure The derivative of () 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 : () 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. ). 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 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 () (' is a scalar function of t) () () 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 .

• Toán học
• Handbook of mathematics
• Vector analysis
• Function theory
• Integral transformations
• Probability theory
• Numerical analysis
• Computer algebra systems
• Journal of Computer Science and Cybernetics
• On the real world semantics interpretability
• Fuzzy rule based systems under fuzzy set approach
• Hedge algebra approach
• The RWS interpretability
• teaching numerical analysis
• techniques
• analysis of results
• mathematics
• partial differential equations
• systems of nonlinear equations
• matrix algebra
• Engineering mathematics
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• Computer numbering systems
• Further algebra
• Partial fractions
• Simple equations
• Simultaneous equations
• Computer organization
• Design fundamentals
• Digital signals
• Numbering systems
• Logic functions
• Boolean algebra
• Standard boolean expression formats
• Karnaugh maps
• Vietnam Journal of Mechanics
• Balancing conditions of planar mechanisms with multi degree of freedom
• Multi degree of freedom
• Planar five bar linkage
• The widely accessible computer algebra systems
• digitals signals
• Signed Representations
• Computer Architecture
• information systems department
• Digital Representation
• Digital processing
• axioms of boolean algebra