Answer:
Therefore the rate change of temperature at the point P(2,-1,2) in the direction toward the point (3,-3,3) is
°C/m.
Step-by-step explanation:
Given that, the temperature at a point (x,y,z) is
.
Rate change of temperature at the point P(2,-1,2) in the direction toward the point Q (3,-3,3) is ![D_uT(2,-1,2)](https://tex.z-dn.net/?f=D_uT%282%2C-1%2C2%29)
![T_x= 200(-2x)e^{-x^2-3y^2-9z^2}](https://tex.z-dn.net/?f=T_x%3D%20200%28-2x%29e%5E%7B-x%5E2-3y%5E2-9z%5E2%7D)
![=-400x200.2xe^{-x^2-3y^2-9z^2}](https://tex.z-dn.net/?f=%3D-400x200.2xe%5E%7B-x%5E2-3y%5E2-9z%5E2%7D)
![T_y = 200.(-6y)e^{-x^2-3y^2-9z^2}](https://tex.z-dn.net/?f=T_y%20%3D%20200.%28-6y%29e%5E%7B-x%5E2-3y%5E2-9z%5E2%7D)
![=-1200ye^{-x^2-3y^2-9z^2}](https://tex.z-dn.net/?f=%3D-1200ye%5E%7B-x%5E2-3y%5E2-9z%5E2%7D)
![T_z=200(-18z)e^{-x^2-3y^2-9z^2}](https://tex.z-dn.net/?f=T_z%3D200%28-18z%29e%5E%7B-x%5E2-3y%5E2-9z%5E2%7D)
![=-3600ze^{-x^2-3y^2-9z^2}](https://tex.z-dn.net/?f=%3D-3600ze%5E%7B-x%5E2-3y%5E2-9z%5E2%7D)
The gradient of the temperature
![=-400e^{-x^2-3y^2-9z^2}(x,3y,9z)](https://tex.z-dn.net/?f=%3D-400e%5E%7B-x%5E2-3y%5E2-9z%5E2%7D%28x%2C3y%2C9z%29)
![\bigtriangledown T(2,-1,2) = -400e^{-2^2-3(-1)^2-9.2^2}(2,3.(-1),9.2)](https://tex.z-dn.net/?f=%5Cbigtriangledown%20T%282%2C-1%2C2%29%20%3D%20-400e%5E%7B-2%5E2-3%28-1%29%5E2-9.2%5E2%7D%282%2C3.%28-1%29%2C9.2%29)
![=-400 e^{-43}(2,-3,18)](https://tex.z-dn.net/?f=%3D-400%20e%5E%7B-43%7D%282%2C-3%2C18%29)
V=
=(3,-3,3)-(2,-1,2)=(1,-2,1)
The unit vector of V is ![\frac{(1,-2,1)}{\sqrt{1^2+(-2)^2+1^2}}](https://tex.z-dn.net/?f=%5Cfrac%7B%281%2C-2%2C1%29%7D%7B%5Csqrt%7B1%5E2%2B%28-2%29%5E2%2B1%5E2%7D%7D)
![=\frac{1}{\sqrt6}(1,-2,1)](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B%5Csqrt6%7D%281%2C-2%2C1%29)
Therefore,
![D_uT(2,-1,2) = \bigtriangledown T(2,-1,2).\frac{1}{\sqrt6}(1,-2,1)](https://tex.z-dn.net/?f=D_uT%282%2C-1%2C2%29%20%3D%20%5Cbigtriangledown%20T%282%2C-1%2C2%29.%5Cfrac%7B1%7D%7B%5Csqrt6%7D%281%2C-2%2C1%29)
![=-400 e^{-43}(2,-3,18).\frac{1}{\sqrt6}(1,-2,1)](https://tex.z-dn.net/?f=%3D-400%20e%5E%7B-43%7D%282%2C-3%2C18%29.%5Cfrac%7B1%7D%7B%5Csqrt6%7D%281%2C-2%2C1%29)
![=-\frac{400}{\sqrt6}e^{-43}(2.1+(-3).(-2)+18.1)](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B400%7D%7B%5Csqrt6%7De%5E%7B-43%7D%282.1%2B%28-3%29.%28-2%29%2B18.1%29)
°C/m
Therefore the rate change of temperature at the point P(2,-1,2) in the direction toward the point (3,-3,3) is
°C/m.