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Chapter 12: Q. 16 (page 944)

Assume that g(x, y, z) is a function of three variables with partial derivatives of every order. Assume also the order in which the partial derivatives are taken is significant.

(a) How many different second-order partial derivatives does g have?

(b) How many different third-order partial derivatives does g have?

(c) How many different nth-order partial derivatives does g have?

Short Answer

Expert verified

Part (a) The number of the second-order partial derivative is 9.

Part (b)The number of third order partial derivative is 27.

Part (c) The number of n-th order partial derivatives is3n.

Step by step solution

01

Given information

We are given a function g of three variable x ,y, z

02

Part (a) Step 1: Explanation 

As g is a function of three variables x, y, z.

The second-order partial derivatives can be given as,

fxx,fxy,fxz,fyx,fyz,fyy,fzx,fzy,fzz.

It has 9 second-order partial derivatives.

03

Part (b) Step 1: Explanation 

As g is a function of three variables x, y, z.

It will have 27 third-order partial derivatives.

04

Part (c) Step 1: Explanation. 

As g is a function of x, y, z The number of n-th partial derivatives is 3n.

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