## Related questions with answers

Question

Experiments show that in a temperature field, heat flows in the direction of maximum decrease of temperature T. Find this direction in general and at the given point P. Sketch that direction at P as an arrow. T=x²+y²+4z², P:(2, -1, 2)

Solution

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1 of 3$T=x^2+y^2+4z^2\qquad,\qquad P(2\ ,\ -1\ ,\ 2)$

Direction of heat flow :$\quad -\nabla T$

$\nabla T=2x\ \hat i+2y\ \hat j+8z\ \hat k$

Direction of flow ,$\qquad -\nabla =-2x\ \hat i-2y\ \hat j-8z\ \hat k$

Direction of flow at$\qquad P(2\ ,\ -1\ ,\ 2)$ is

$-2(2)\ \hat i-2(-1)\ \hat j-8(2)\ \hat k\quad$ i.e $\quad-4\ \hat i+2\ \hat j-16\ \hat k$

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