Calculation of true thickness of the bed

 

29) A scalar potential field in 3D space is expressed as U(x,y,z) = $x^{2}+yz^{2}$. The magnitude of the maximum rate of change in U(x,y,z) at a point (1,1,2) is____

 

(Thanks to Chandrasekhar, ANU)

Solution:

 

Given the function U(x,y,z) = $x^{2}+yz^{2}$

 

First calculate the first derivatives of the function

 

$\frac{\partial U}{\partial x}=2x$

 

$\frac{\partial U}{\partial y}=z^{2}$

 

$\frac{\partial U}{\partial z}=2yz$

 

Then

 

$\triangledown U(x,y,z) = < \frac{\partial U}{\partial x} , \frac{\partial U}{\partial y} , \frac{\partial U}{\partial z}>$

 

$\triangledown U(x,y,z) = < 2x,z^{2},2yz>$

 

$\triangledown U(1,1,2) = < 2(1),(2)^{2},2(1)(2)>$

 

$\triangledown U(1,1,2) = < 2,4,4>$

 

   For calculating the magnitude of the maximum rate of change is

 

$\mid\triangledown U(1,1,2)\mid=\sqrt{2^{2}+4^{2}+4^{2}}$

 

$\mid\triangledown U(1,1,2)\mid=\sqrt{4+16+16}$

 

$\mid\triangledown U(1,1,2)\mid=\sqrt{36}$

 

$\mid\triangledown U(1,1,2)\mid=6$

 

 

 The maximum rate of change in U(x,y,z) at a point (1,1,2) for the function $x^{2}+yz^{2}$ is 6