[理学]线性代数III-chapter26.ppt

This property is called the invariance of the first order total differential form or simply the invariance of the total differential form Solution Example 10 Solution Example 11 By the invariance of the total differential form, we have Solution Example 12 * Prof Liubiyu 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 §2.6 Differentiation of multivariable composite functions We have seen that the chain rule has played a very important role in calculating derivatives of composite functions of one variable. In this section, we will extend the chain rule to multivariable functions and discuss the differentiation of composite functions of several variables. 1、The chain rule for multivariable composite functions Similarly, we have Theorem 1 ( The chain rule 1 ) Proof The proof of the next chain rule is almost identical to the proof of theorem 1. Notations Theorem 2 ( The chain rule 2) (3) Notations (2) The following diagram is useful in using the chain rule of Theorem 2 According to the characteristics of the chain rule we can extend the chain rule to a composite function with m intermediate variables and n independent variables. There are various different cases for multivariable composite functions and you should learn to analyze the composite relations among these functions, clarify the the intermediate variables and independent variables. For example: They are different Solution Example 1 In the next example there are two middle variables and two bottom variables. Solution Example 2 In the next example there