Quiz #6 q#2 **Problem 2:**Suppose $ f(x, y, z) = g(\rho) $, where $ \rho = \sqrt{x^2 + y^2 + z^2} $ and $ g $ is a function of one variable. Prove that\[f_{xx}(x, y, z) + f_{yy}(x, y, z) + f_{zz}(x, y, z) = g''(\rho) + \frac{2}{\rho} g'(\rho)\]as long as $ (x, y, z) \neq (0, 0, 0) $. Note that $ g'(\rho) = \frac{dg}{d\rho} $.

Name: Quiz #6 q#2 **Problem 2:**Suppose $ f(x, y, z) = g(\rho) $, where $
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Description: Quiz #6 q#2 **Problem 2:**Suppose \( f(x, y, z) = g(\rho) $, where $ \rho = \sqrt{x^2 + y^2 + z^2} $ and $ g $ is a function of one variable. Prove that\[f_{xx}(x, y, z) + f_{yy}(x, y, z) + f_{zz}(x, y, z) = g''(\rho) + \frac{2}{\rho} g'(\rho)\]

given f(x,y,z)=g(ρ) where ρ=x2+y2+z2first we find fx=∂g(ρ)∂x=∂g(ρ)∂ρ∂ρ∂x=g'(ρ)xx2+y2+z2again…

Answered: (2) Supose f(x, Y, z) = g(p), where p =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Quiz #6 q#2

**Problem 2:**

Suppose $ f(x, y, z) = g(\rho) $, where $ \rho = \sqrt{x^2 + y^2 + z^2} $ and $ g $ is a function of one variable. Prove that

\[
f_{xx}(x, y, z) + f_{yy}(x, y, z) + f_{zz}(x, y, z) = g''(\rho) + \frac{2}{\rho} g'(\rho)
\]

as long as $ (x, y, z) \neq (0, 0, 0) $. Note that $ g'(\rho) = \frac{dg}{d\rho} $.

Expert Solution

Step 1

$g i v e n f (x, y, z) = g (ρ) w h e r e ρ = \sqrt{x^{2} + y^{2} + z^{2}} f i r s t w e f i n d f_{x} = \frac{\partial g (ρ)}{\partial x} = \frac{\partial g (ρ)}{\partial ρ} \frac{\partial ρ}{\partial x} = g' (ρ) \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}} a g a i n d i f f e r e n t i a t e w . r . t o x p a r t i a l l y f_{x x} = g " (ρ) \frac{x^{2}}{x^{2} + y^{2} + z^{2}} + g' (ρ) (\frac{\sqrt{x^{2} + y^{2} + z^{2}} - \frac{x^{2}}{\sqrt{x^{2} + y^{2} + z^{2}}}}{x^{2} + y^{2} + z^{2}}) = g " (ρ) \frac{x^{2}}{x^{2} + y^{2} + z^{2}} + g' (p) \frac{y^{2} + z^{2}}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} f_{y} = \frac{\partial g (ρ)}{\partial y} = \frac{\partial g (ρ)}{\partial ρ} \frac{\partial ρ}{\partial y} = g' (ρ) \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}} f_{y y} = g " (ρ) \frac{y^{2}}{x^{2} + y^{2} + z^{2}} + g' (ρ) (\frac{\sqrt{x^{2} + y^{2} + z^{2}} - \frac{y^{2}}{\sqrt{x^{2} + y^{2} + z^{2}}}}{x^{2} + y^{2} + z^{2}}) = g " (ρ) \frac{y^{2}}{x^{2} + y^{2} + z^{2}} + g' (p) \frac{x^{2} + z^{2}}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} f_{z} = \frac{\partial g (ρ)}{\partial z} = \frac{\partial g (ρ)}{\partial ρ} \frac{\partial ρ}{\partial z} = g' (ρ) \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} f_{z z} = g " (ρ) \frac{z^{2}}{x^{2} + y^{2} + z^{2}} + g' (ρ) (\frac{\sqrt{x^{2} + y^{2} + z^{2}} - \frac{z^{2}}{\sqrt{x^{2} + y^{2} + z^{2}}}}{x^{2} + y^{2} + z^{2}}) = g " (ρ) \frac{z^{2}}{x^{2} + y^{2} + z^{2}} + g' (p) \frac{x^{2} + y^{2}}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} n o w a d d f_{x x} + f_{y y} + f_{z z} = g " (ρ) \frac{x^{2}}{x^{2} + y^{2} + z^{2}} + g' (p) \frac{y^{2} + z^{2}}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} + g " (ρ) \frac{y^{2}}{x^{2} + y^{2} + z^{2}} + g' (p) \frac{x^{2} + z^{2}}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} + g " (ρ) \frac{z^{2}}{x^{2} + y^{2} + z^{2}} + g' (p) \frac{x^{2} + y^{2}}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} = g " (ρ) (\frac{x^{2} + y^{2} + z^{2}}{x^{2} + y^{2} + z^{2}}) + g' (ρ) (\frac{2 (x^{2} + y^{2} + z^{2})}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}}) = g " (ρ) + \frac{2}{{(x^{2} + y^{2} + z^{2})}^{\frac{1}{2}}} g' (ρ) = g " (ρ) + \frac{2}{ρ} g' (ρ)$

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