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# If $u=\sqrt{x^{2}+y^{2}},$ Show that $x\large\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=u$

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com

Toolbox:
• Euler's Theorem: If $f(x,y)$ is a homogeneous function of degree n, then $x\large\frac{\partial f}{\partial x}$$+y\large\frac{\partial f}{\partial y}$$=nf$ This can be extended to several variables
$u=\sqrt {x^2+y^2}$
Step 1:
u is a homogeneous function in x,y of degree 1.
$\therefore$ by Euler's Theorem ,
$x\large\frac{\partial u}{\partial x}$$+y\large\frac{\partial u}{\partial y}$$=1u$
Step 2:
$x\large\frac{\partial u}{\partial x}$$+y \large \frac{\partial u}{\partial y}$$=u$