TAILIEUCHUNG - Ebook Math advanced calculus: Part 2

(BQ) Part 2 book "Math advanced calculus" has contents: The derivative of a vector valued function of a vector variable, nonlinear functions, transformation of integrals, line and surface integrals, infinite series,.and other contents. | The Derivative of a Vector-Valued Function of a Vector Variable 9 Definition of the Derivative We have seen in Chapter 3 that the derivative f a of a real-valued function 2 - E 2 c E of a real variable at an accumulation point a G 2 of 2 may be characterized as follows f is differentiable at a and f à is the derivative if and only if there exists a linear function la E - E with the property that for every 0 there is a 5 e 0 such that l - - ta x - a x - a for all X G Nẳ a n 3 and f à h la h for all h G E. See Theorem . In generalizing the concept of a derivative to vector-valued functions of a vector variable from 2 s E into Em we proceed in a straightforward manner replacing the absolute value by the norm and la E - E by a linear function ỉ E - E 1. However we shall restrict our discussion to interior points of 2 rather than consider the more general case where a G 2 is merely an accumulation point of 2 . Otherwise we would not obtain a unique derivative when n 1. Definition 86. J If f 2 - Ew 2 E and if a is an interior point of 2 then is differentiable at a if and only if there exists a linear function I E - Em with the property that for every 0 there is a ô è 0 where Nẵ e a 2 such that x -f a - l x - a fi x - all for all X G Nổ j a . The function is called the derivative of at a and is denoted by a . Note that a is for given a a linear function from E into Em that is a A k a ỉ a fc for all h k G E and a A Ấ a ỉ for all h G E and any Ấ G R. 312 Definition of the Derivative 313 Example 1 Let E - Em be defined by c Z x where c e Em and where E - Em is a linear function. Then for any a e E a I x - a - x - a II x - a - x - a II 0 for all X e E . This is a generalization of the well-known result that the derivative of E - E given by x a bx is z . Example 2 Let E2 - E be defined by i 2 2 and let a a2 . We propose to show that E2 - E is the linear function that is defined by a fl 2a1co1 co2 for all u e E2. We have x - fl f a x - a 2 - a - a2 - .

• Toán học
• Math advanced calculus
• The derivative
• A vector valued function
• A vector variable
• Nonlinear functions
• Transformation of integrals
• Surface integrals
• Infinite series
• The riemann integral
• The euclidean n space
• Vector valued functions
• Sequences of functions
• Linear functions
• Borel sets
• Baire classification of functions
• Porosity of sets
• Porosity character of sets
• Infinite series of real terms
• advanced mathematics
• Convergence Test
• Taylor
• Maclaurin series
• The Binomial Series
• Differential Equations
• Fourier Series
• Trigonometric expressions
• Binomial series
• Hypergeometric series
• Trigonometric functions
• Binomial coefficients
• báo cáo khoa học
• báo cáo hóa học
• công trình nghiên cứu về hóa học
• tài liệu về hóa học
• cách trình bày báo cáo
• Functional Analysis
• Linear algebra
• Discrete mathematics
• Integral calculus
• Linear integral equations
• Early transcendentals
• Mathematical modeling
• Polar curves
• Conic sections
• Advanced calculus
• Multiple integrals
• Line integrals
• Improper integrals
• Functions of a complex variable
• Ebook Calculus
• Topics in differentiation
• Three dimensional space
• Inverse Operations
• Negative Numbers
• Fractional Numbers
• Imaginary Numbers
• Exponential Function
• INFINITE SEQUENCES AND SERIES