Question

if phi = 2x3y2z4 find div.grad phi​

Answer 1

SOLUTION

GIVEN

$\sf{ \phi = 2 {x}^{3} {y}^{2} {z}^{4} }$

FORMULA TO BE IMPLEMENTED

$\displaystyle \sf{ div \: grad \: \phi = \nabla \: . \nabla \: \phi = \frac{{ \partial}^{2} \phi}{ \partial {x}^{2} } + \frac{{ \partial}^{2} \phi}{ \partial {y}^{2} } + \frac{{ \partial}^{2} \phi}{ \partial {z}^{2} } }$

TO DETERMINE

$\sf{ div \: grad \: \phi }$

EVALUATION

Here it is given that

$\sf{ \phi = 2 {x}^{3} {y}^{2} {z}^{4} }$

Now

$\displaystyle \sf{ \frac{ \partial \phi}{ \partial x} = 6 {x}^{2} {y}^{2} {z}^{4} }$

$\displaystyle \sf{ \frac{{ \partial}^{2} \phi}{ \partial {x}^{2} } = 12 x {y}^{2} {z}^{4} }$

$\displaystyle \sf{ \frac{ \partial \phi}{ \partial y} = 4 {x}^{3} y {z}^{4} }$

$\displaystyle \sf{ \frac{{ \partial}^{2} \phi}{ \partial {y}^{2} } = 4 {x}^{3} {z}^{4} }$

$\displaystyle \sf{ \frac{ \partial \phi}{ \partial z} = 8 {x}^{3} {y}^{2} {z}^{3} }$

$\displaystyle \sf{ \frac{{ \partial}^{2} \phi}{ \partial {z}^{2} } = 24 {x}^{3} {y}^{2} {z}^{2} }$

Hence

$\sf{ div \: grad \: \phi }$

$\sf{ = \nabla \: . \nabla \: \phi }$

$\displaystyle \sf{ = \frac{{ \partial}^{2} \phi}{ \partial {x}^{2} } + \frac{{ \partial}^{2} \phi}{ \partial {y}^{2} } + \frac{{ \partial}^{2} \phi}{ \partial {z}^{2} } }$

$\displaystyle \sf{ = 12 x {y}^{2} {z}^{4} + 4 {x}^{3} {z}^{4} + 24 {x}^{3} {y}^{2} {z}^{2} }$

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