Question

1, 19: + 'fax + Son + 4x + xy + yafya + xzbu where subscripts denote the cube. The first te of x, y, and z is zero, as is th ubscripts denote the derivatives evaluated at (0.0.0). Let's average these terms over . The first term is constant, and thus equals its average. By symmetry, the average d is zero, as is the average of the cross-terms xy, yz, and xz. The average of x- is given by 1 102 0/2 0/2 a J-a/2 J-s/2J- * dx dy dz - 12 and similarly for yo and z. Thus the difference between 10,0,0) and its average value throughout the small cube is given approximately by 10,0,0) - fax = Sau + fu+fa) - 12 500,0,0) We alluded to this property of the laplacian in section 3.6. EXERCISES 1. Verify that the zeroth, first, and second partial derivatives of f and its second-degree Taylor polynomial agree at the base point Ro. 2. Work out the second-order Taylor polynomials for the function ety + z2 expanded around (0,0,0) and (1,1,1). 3. Work out the third-order Taylor polynomial for a function of two variables, f(x,y). 4. The Taylor (Maclauren) series for ek is 1 + x + 2! 37 .. Use this to interpret the expression e suf; what do you get? 5. Show that the difference between f(0,0,0) and its average over a small sphere of radius R is given approximately by --(R2/10)V2/(0,0,0). (Hint: The average value of x2 over the sphere is R2/5.) 6. Another way of deriving the Taylor polynomials for f(x,y,z) is as follows. Regard y and z as fixed and expand f in powers of (x - Xo). The coefficients in this expansion will be functions of y and z. Regard z as fixed and expand each coefficient in powers of (y - yo). Now the coefficients are functions of z. Expand them in powers of (z - zo). Carry out this program to second order and verify that the results agree with the display in eq. (3.26).

Question 5

Answer 1

414.4,2) = 10,0,0) + fx x + ty uttzz. ny foxy + y2 tyz tez tez au partial derivation are point 20,90). Taking avorge of these values over sphere of Radius R. (rmall). now, Volume = 45 R3. . - J sva dni -o; x is symmetric V S55 dy orrs vilnphere) =) Youg = zaug zo. 10 = say av .; again ay 1 sss lav i symmetric over vi =) (xylaug = (y2) avg = (m2) avg=0 . = (ny) avg

(x²) avg = J J Ju xZ dv = R². =) (7²) ang=(2²) ong (x²) avg - Til ar. = R2 - 5 f(x,y; 2) – +10,0,0) + role (fou + tay + tar)" -) 610,0,0) - fang =-2 (fun+ try too) aug. = + 10,0,0) - tava = - R2 V2F10,0,0) 10