# If f=x^3+y^3+z^3+3xy, then show that  x(partial f)/(partial x) + y (partial f)/(partial y) + z (partial f)/(partial z) = 3f?

Apr 5, 2018

$f = {x}^{3} + {y}^{3} + {z}^{3} + 3 x y z$

$\textcolor{red}{\partial \frac{f}{\mathrm{dx}}} = 3 {x}^{2} + 3 y z$

$\textcolor{b l u e}{\partial \frac{f}{\mathrm{dy}}} = 3 {y}^{2} + 3 x z$

$\textcolor{m a \ge n t a}{\partial \frac{f}{\mathrm{dz}}} = 3 {z}^{2} + 3 x y$

color(white)(dd

We need to show that $x \textcolor{red}{\partial \frac{f}{\mathrm{dx}}} + y \textcolor{b l u e}{\partial \frac{f}{\mathrm{dy}}} + z \textcolor{m a \ge n t a}{\partial \frac{f}{\mathrm{dz}}} = 3 f$

$L H S = x \left[\textcolor{red}{3 {x}^{2} + 3 y z}\right] + y \left[\textcolor{b l u e}{3 {y}^{2} + 3 x z}\right] + z \left[\textcolor{m a \ge n t a}{3 {z}^{2} + 3 x y}\right]$

color(white)(dd

$\implies \left[3 {x}^{3} + 3 x y z\right] + \left[3 {y}^{3} + 3 x y z\right] + \left[3 {z}^{3} + 3 x y z\right]$

$\implies 3 {x}^{3} + 3 {y}^{3} + 3 {z}^{3} + 9 x y z$

$\implies 3 \left({x}^{3} + {y}^{3} + {z}^{3} + 3 x y z\right) = 3 f$

Apr 5, 2018

The function:

$f = {x}^{3} + {y}^{3} + {z}^{3} + 3 x y$

as given does not satisfy the Partial Differential Equation:

$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z} = 3 f$

and the question is in error.

If however, we modify the function then we can prove a modified result. Consider the modified function:

$f = {x}^{3} + {y}^{3} + {z}^{3} + 3 x y z$

and we seek to validate that $f$ satisfies the Partial differential Equation:

$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z} = 3 f$

(In other words we are validating that a solution to the given PDE is $f$). We compute the partial derivative (by differentiating wrt to specified variable and treating all other variables as constants):

${f}_{x} = \frac{\partial f}{\partial x} = 3 {x}^{2} + 3 y z$

${f}_{y} = \frac{\partial f}{\partial y} = 3 {y}^{2} + 3 x z$

${f}_{z} = \frac{\partial f}{\partial z} = 3 {z}^{2} + 3 x y$

Next we compute the LHS of the desired expression:

$L H S = x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = x \left(3 {x}^{2} + 3 y z\right) + y \left(3 {y}^{2} + 3 x z\right) + z \left(3 {z}^{2} + 3 x y\right)$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = 3 {x}^{3} + 3 x y z + 3 {y}^{3} + 3 x y z + 3 {z}^{3} + 3 x y z$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = 3 \left({x}^{3} + x y z + {y}^{3} + x y z + {z}^{3} + x y z\right)$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = 3 \left({x}^{3} + {y}^{3} + {z}^{3} + 3 x y z\right)$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = 3 f \setminus \setminus \setminus$ QED