[理学]线性代数III-chapter25.pptVIP

Solution Notations * Prof Liubiyu Directional derivative 方向导数 Gradient 梯度 Electrostatic field 静电场 Electric potential 电势 Electric field intensity 电场强度 Permittivity 介电常数，电容率 Potential function 势函数 New Words Contents §2.2 Limits and continuity of multivariable functions §2.3 Partial derivatives and higher order partial derivatives §2.1 Basic concept of multivariable functions §2.4 Total differentials and their applications §2.6 Differentiation of multivariable composite functions §2.7 Differentiation of multivariable implicit functions §2.8 Application of partial derivatives in geometry §2.5 Directional derivatives and gradient §2.9 Extreme value of multivariable functions §2.5 Directional derivatives and the gradient In this section we generalize the notion of a partial derivative to that of a directional derivative. Then we introduce a vector, called “ the gradient”, to provide a short formula for the directional derivative. The gradient will have other uses later in the following study. 1、The directional derivative We know that partial derivatives are the rates of a function of several variables changing along positive direction of coordinate axis. in the section, we shall consider the rates of a function of several variables changing along any direction. Definition 1 ( a directional derivative) Notations 2、Geometric interpretation of the directional derivative Example 1 Solution Notations Using the definition to compute the directional derivative is not convenient. The following theorem gives a sufficient condition for the existence of directional derivative and a computational formula. Theorem 1 Proof Notations Example 2 Solution Example 3 Solution Example 4 Solution The study of the direction of the maximum rate of change of a function and the maximum value of this rate is a very important problem in the natural science and in many applied problem. 3、Gradient Definition 2 Notations (6