# Find the direction of fastest variation in temperature.

The temperature field in a body varies according to the equation $$T(x,y)=x^{3}+4xy$$.Find the direction of  fastest variation in temperature at the point $$(1,0)$$.

Asked on 25th October 2019 in
• 1 Answer(s)

The fastest variation of temperature will be given as $\frac{\partial \phi}{\partial x}i+\frac{\partial\phi }{\partial y}j$.

Here $$\phi$$ is any function in terms of $$x$$ and $$y$$.

$\frac{\partial \phi}{\partial x}=\frac{\partial }{\partial x}\left ( x^{3}+4xy\right ) \\=3x^{2}+4y$

$\frac{\partial \phi }{\partial y}=\frac{\partial }{\partial y}\left ( x^{3} +4xy\right ) \\=4x$

At point $$(1,0)$$ its value will be $3x^{2}+4y=3(1^{2})+4(0)=3$

and $4x=4(1)=4$

$=3i+4j$

At this point its directional unit vector will be $\frac{3i+4j}{\sqrt{3^{2}+4^{2}}} \\=\frac{3i+4j}{\sqrt{25}} \\=\frac{3i+4j}{5}$

$=0.6i+0.8j$

Answered on 25th October 2019.
• ### Your Answer

By posting your answer, you agree to the privacy policy and terms of service.