Q. 7. - Calculus

Question

Explain why Theorem $12.33$ is a special case of Theorem

$12.34$ with $n=2$ and $m=2$

Step-by-Step Solution

Answer

The required answer is

$\frac{\partial z}{\partial t_2}=\frac{\partial z}{\partial x_1}\frac{\partial x_1}{\partial t_2}+\frac{\partial z}{\partial x_2}\frac{\partial x_2}{\partial t_2}$

Step 1: Given information

The complete version of the chain rule is for a given function $z=f\left(x_1,x_2,\dots ,x_n\right)$and $x_i=u_i\left(t_1,t_2,\dots ,t_m\right)$for$1\le i\le n$or all values of $t_1,t_2,\dots ,t_m$at that $u_i$each  is differentiable and if $f$  is differentiable at

$(x_1,x_2,....x_n)$ then

$\frac{dz}{d{t}_j}=\frac{dz}{d{x}_1}\frac{d{x}_1}{d{t}_j}+\frac{dz}{d{x}_2}\frac{d{x}_2}{d{t}_j}+....+\frac{dz}{d{x}_n}\frac{d{x}_n}{d{t}_j}......\left(1\right)$

Where $1\le j\le m$

The chain rule version three states that for a given function $z=f\left(x_1,x_2\right)$

and $x_i=u_i\left(t_1,t_2\right)$for $1\le i\le 2$for all values of $t_1,t_2$at that each $u_i$

is differentiable and if $f$  is differentiable at $\left(x_1,x_2\right)$then

$$\begin{gathered} \frac{\partial z}{\partial t_1}=\frac{\partial z}{\partial x_1}\frac{\partial x_1}{\partial t_1}+\frac{\partial z}{\partial x_2}\frac{\partial x_2}{\partial t_1} \\ And, \frac{\partial z}{\partial t_2}=\frac{\partial z}{\partial x_1}\frac{\partial x_1}{\partial t_2}+\frac{\partial z}{\partial x_2}\frac{\partial x_2}{\partial t_2} \end{gathered}$$

Step 2: The objective is to show when n = 2   and m = 2   then the complete version of the chain rule gives the chain rule version three.

When $n=m=2$then the complete version of the chain rule is as follows. For a given function $z=f\left(x_1,x_2\right)$and $x_i=u_i\left(t_1,t_2\right)$for $1\le i\le 2$

For the values of $t_1 and t_2$ at which $u_1 and u_2$ are differentiable and if

$f$ is differentiable at $x_1 and x_2$ then

The first put $n=2$and $m=1$in the equation $\left(1\right)$

$\frac{\partial z}{\partial t_1}=\frac{\partial z}{\partial x_1}\frac{\partial x_1}{\partial t_t}+\frac{\partial z}{\partial x_2}\frac{\partial x_2}{\partial t_1}$

Again put  $n=2$and $m=2$in the equation $\left(1\right)$

$\frac{\partial z}{\partial t_2}=\frac{\partial z}{\partial x_1}\frac{\partial x_1}{\partial t_2}+\frac{\partial z}{\partial x_2}\frac{\partial x_2}{\partial t_2}$