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General Vector Taylor Series Expansion: Measure of deviation

2. Arfken, p. 342, 5th Ed. (p. 359, 6th Ed.), ) Prob. 5.6.7. Use the General Vector Taylor Series Expansion For a General Function, cI) (r) = (I) (x, y, z) , Of a Three-Dimensional Vector Coordinate, Expressed In Cartesian Coordinates, which is Expanded About the Origin, r = 0 Or x = y = z = 0 , Where 0(0 = (1)(x', y', z') 1 ir, vy, (0) + (D (0, 0, 0) , n=0 n! n=0 11! ax ay az an an an With the Notation that — cl) (0, 0, 0) , (13 (0, 0, 0), (13 (0, 0, 0) , Are the nth ax ae

Order Derivatives Of the Function, c (x, y, z) , Evaluated At the Origin, (x, y, z) (0, 0, 0) , Which Are Constants. In Addition, In This Notation, the Laplacian Of the Function Evaluated At the Origin Is Denoted As
v cp(r )
( 2 52 a2 = urqx,y,z) r=0 •!‘._ .2 ay2 az2 GX
• A2 A2 A2 = u (NO 0 0)+'±---' -CD(0,0,0)-F--CD(0,0,0) axe ay2 aZ2
1 Expand the Function,

(1)(r), To Second Order (x2 , y2 , z2 Terms), Using the General Taylor Series Expansion, As an Approximation To the Function, Near the Origin, Labeled As °approx. (0 • Calculate the Average Value Of the Function, Labeled As (1)(0) , About the Origin, By Integrating this Approximation Of the Function, cDapprox. (r) , Over a Cube Of Side Lengths a , Where the Average Value Of the Function At the Origin Is
1 di (0) = ---73- fff dxdydzcl)approx. (r) . From the Average Value Integral, you a -a/2-a/2-a/2
a/2 a/2 a/2

should be able to Isolate the Laplacian Of the Function, V2(1:1(r) , Evaluated At the r=0
Origin. As a Result of this, Express the Laplacian Of the Function, V2c1)(r)1, r=0 Evaluated At the Origin, In Terms Of a Constant Factor Times the Difference Between the Average Value Of the Function, 45(0) , And the Function, 1 (0) , all Evaluated At the Origin. Note That this Expression Becomes Exact In the Limit That the Size Of the Cube Length Goes To Zero, a —> 0 . The Objective Of this Exercise Is To Demonstrate the Fact That the Laplacian Of a Function At Any Point Can Be Interpreted As Being Directly Related To the Difference Of the Average Value Of a Function Around the Point And the Function Itself At that Point. Note: Make Sure To Write Out All the Terms In your Second Order Vector Taylor Series Expansion Of Function, Before Performing the Averaging Integral.

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Solution Summary

With good explanations and calculations, the problem is solved.