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 \((1,0)\).
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 \((1,0)\).
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 \[\frac{3i+4j}{\sqrt{3^{2}+4^{2}}}
\\=\frac{3i+4j}{\sqrt{25}}
\\=\frac{3i+4j}{5}\]
\[=0.6i+0.8j\]
