Answer:
![D_uT(3,1)=-\frac{44}{9}*\frac{1}{\sqrt{2} } \approx-3.46](https://tex.z-dn.net/?f=D_uT%283%2C1%29%3D-%5Cfrac%7B44%7D%7B9%7D%2A%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D%20%5Capprox-3.46)
Step-by-step explanation:
To find the rate of change of temperature with respect to distance at the point (3, 1) in the x-direction and the y-direction we need to find the Directional Derivative of T(x,y). The definition of the directional derivative is given by:
![D_uT(x,y)=T_x(x,y)i+T_y(x,y)j](https://tex.z-dn.net/?f=D_uT%28x%2Cy%29%3DT_x%28x%2Cy%29i%2BT_y%28x%2Cy%29j)
Where i and j are the rectangular components of a unit vector. In this case, the problem don't give us additional information, so let's asume:
![i=\frac{1}{\sqrt{2} }](https://tex.z-dn.net/?f=i%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D)
![j=\frac{1}{\sqrt{2} }](https://tex.z-dn.net/?f=j%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D)
So, we need to find the partial derivative with respect to x and y:
In order to do the things easier let's make the next substitution:
![u=2+x^2+y^2](https://tex.z-dn.net/?f=u%3D2%2Bx%5E2%2By%5E2)
and express T(x,y) as:
![T(x,y)=88*u^{-1}](https://tex.z-dn.net/?f=T%28x%2Cy%29%3D88%2Au%5E%7B-1%7D)
The partial derivative with respect to x is:
Using the chain rule:
![\frac{\partial u}{\partial x}=2x](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x%7D%3D2x)
Hence:
![T_x(x,y)=88*(u^{-2})*\frac{\partial u}{\partial x}](https://tex.z-dn.net/?f=T_x%28x%2Cy%29%3D88%2A%28u%5E%7B-2%7D%29%2A%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x%7D)
Symplying the expression and replacing the value of u:
![T_x(x,y)=\frac{-176x}{(2+x^2+y^2)^2}](https://tex.z-dn.net/?f=T_x%28x%2Cy%29%3D%5Cfrac%7B-176x%7D%7B%282%2Bx%5E2%2By%5E2%29%5E2%7D)
The partial derivative with respect to y is:
Using the chain rule:
![\frac{\partial u}{\partial y}=2y](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%3D2y)
Hence:
![T_y(x,y)=88*(u^{-2})*\frac{\partial u}{\partial y}](https://tex.z-dn.net/?f=T_y%28x%2Cy%29%3D88%2A%28u%5E%7B-2%7D%29%2A%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D)
Symplying the expression and replacing the value of u:
![T_y(x,y)=\frac{-176y}{(2+x^2+y^2)^2}](https://tex.z-dn.net/?f=T_y%28x%2Cy%29%3D%5Cfrac%7B-176y%7D%7B%282%2Bx%5E2%2By%5E2%29%5E2%7D)
Therefore:
![D_uT(x,y)=(\frac{1}{\sqrt{2} } )*(\frac{-176x}{(2+x^2+y^2)^2} -\frac{176y}{(2+x^2+y^2)^2})](https://tex.z-dn.net/?f=D_uT%28x%2Cy%29%3D%28%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D%20%29%2A%28%5Cfrac%7B-176x%7D%7B%282%2Bx%5E2%2By%5E2%29%5E2%7D%20-%5Cfrac%7B176y%7D%7B%282%2Bx%5E2%2By%5E2%29%5E2%7D%29)
Evaluating the point (3,1)
![D_uT(3,1)=(\frac{1}{\sqrt{2} } )*(\frac{-176(3)-176(1)}{(2+3^2+1^2)^2})=(\frac{1}{\sqrt{2} })* (-\frac{704}{144})=(\frac{1}{\sqrt{2} }) ( - \frac{44}{9})\approx -3.46](https://tex.z-dn.net/?f=D_uT%283%2C1%29%3D%28%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D%20%29%2A%28%5Cfrac%7B-176%283%29-176%281%29%7D%7B%282%2B3%5E2%2B1%5E2%29%5E2%7D%29%3D%28%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D%29%2A%20%28-%5Cfrac%7B704%7D%7B144%7D%29%3D%28%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D%29%20%28%20-%20%5Cfrac%7B44%7D%7B9%7D%29%5Capprox%20-3.46)