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 Propulsion.
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    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.
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