Chapter 13.6, Problem 82ES

(a)

To determine

To calculate: The rate of change of temperature with respect to distance at $\left(1,1,1\right)$ in the direction of the origin if the temperature at the point, $\left(x,y,z\right)$ , in a metal solid is $T\left(x,y,z\right)=\frac{xyz}{1+x^2+y^2+z^2}$ .

(b)

To determine

To calculate: The direction in which the temperature rises most rapidly at the point, $\left(1,1,1\right)$ , if the temperature at a point $\left(x,y,z\right)$ in a metal solid is $T\left(x,y,z\right)=\frac{xyz}{1+x^2+y^2+z^2}$ .

(c)

To determine

To calculate: The rate at which the temperature rises moving from $\left(1,1,1\right)$ in the direction obtained in part (b) if the temperature at a point $\left(x,y,z\right)$ in a metal solid is $T\left(x,y,z\right)=\frac{xyz}{1+x^2+y^2+z^2}$ .