The Laplacian of a function f of n variables ¤1, x2, · ·· ¤n, denoted V² ƒ is defined by V² f (#1, ®2, · · • , xn):= a² f _ в ƒ Now assume that f depends only on r wherer = (x² + æ; + · .. + x%)³, i.e. ƒ (x1, æ2,·.,æn) = 9 (r), for some function g. Show that, for ¤1, ¤2, · ·' , ¤n #0, v°f = ",'d (-) + s" ()

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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The Laplacian of a function f of n variables x1, x2,·· xn, denoted V2 f is defined by
V² f (x1, x2, · · , xn) :=
+….+
Now assume that f depends only on r wherer =
x1, x2,, n + 0,
(x + x + ...+ x%)? , i.e. f (x1, x2,· , xn) = g (r), for some function g. Show that, for
n - 1
V² f =
-g (r) + g" (r)
Transcribed Image Text:The Laplacian of a function f of n variables x1, x2,·· xn, denoted V2 f is defined by V² f (x1, x2, · · , xn) := +….+ Now assume that f depends only on r wherer = x1, x2,, n + 0, (x + x + ...+ x%)? , i.e. f (x1, x2,· , xn) = g (r), for some function g. Show that, for n - 1 V² f = -g (r) + g" (r)
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