(2) Supose f(x, Y, z) = g(p), where p = Vx? + y? + 2² and g is a function of one variable. Prove that 2 faæ (x, Y, z) + fyy(x, Y, z) + fzz(x, Y, z) = g" (p) + -g'(e) as long as (x, y, z) # (0,0,0). Note that g'(p) dg do %3D

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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 \( g'(\rho) = \frac{dg}{d\rho} \).
Transcribed Image Text:**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

given f(x,y,z)=g(ρ) where ρ=x2+y2+z2first we find fx=g(ρ)x=g(ρ)ρρx=g'(ρ)xx2+y2+z2again differentiate w.r.to x partiallyfxx=g"(ρ)x2x2+y2+z2+g'(ρ)x2+y2+z2-x2x2+y2+z2x2+y2+z2       =g"(ρ)x2x2+y2+z2+g'(p)y2+z2x2+y2+z232 fy=g(ρ)y=g(ρ)ρρy=g'(ρ)yx2+y2+z2fyy=g"(ρ)y2x2+y2+z2+g'(ρ)x2+y2+z2-y2x2+y2+z2x2+y2+z2      =g"(ρ)y2x2+y2+z2+g'(p)x2+z2x2+y2+z232 fz=g(ρ)z=g(ρ)ρρz=g'(ρ)zx2+y2+z2fzz=g"(ρ)z2x2+y2+z2+g'(ρ)x2+y2+z2-z2x2+y2+z2x2+y2+z2      =g"(ρ)z2x2+y2+z2+g'(p)x2+y2x2+y2+z232now add fxx+fyy+fzz=g"(ρ)x2x2+y2+z2+g'(p)y2+z2x2+y2+z232+g"(ρ)y2x2+y2+z2+g'(p)x2+z2x2+y2+z232+g"(ρ)z2x2+y2+z2+g'(p)x2+y2x2+y2+z232              =g"(ρ)x2+y2+z2x2+y2+z2+g'(ρ)2x2+y2+z2x2+y2+z232             =g"(ρ)+2x2+y2+z212g'(ρ)             =g"(ρ)+2ρg'(ρ)

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