Solution Set 2. y z. + j. u + j

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1 Soltion Set 2. Review of Div, Grd nd Crl. Prove:. () ( A) =, where A is ny three dimensionl vector field. i j k ( Az A = y z = i A A y A z y A ) ( y A + j z z A ) ( z Ay + k A ) y ( A) = ( Az y A ) y + ( A z y z A ) z + ( Ay z A ) y = by eqlity of mied prtils. 2. (b) =, where is ny three dimensionl sclr field (i.e. = (, y, z)). i j k = y z y z ( 2 ) ( = i zy 2 2 ) ( + j yz z 2 2 ) + k z y 2 y = 2. Orthogonlity. Do problem in Hbermn. Note tht the object of this problem is to prove tht es re orthogonl. Therefore yo mst ctlly evlte the integrls rther thn invoking orthogonlity. Hint: For the cse n = m se the identity 2 θ + cos 2 θ = nd notice tht π 2 θ dθ = π cos2 θ dθ. nπ mπ d = [ [ cos 2 = 2 π (n m) = for m n π (n m) cos π (n m) ] π (n + m) d π (n + m) π (n + m) ] For m = n: 2 mπ d = [ 2 mπ 2 d + cos 2 mπ ] d = 2 d = 2 3. Het Eqtion on Thin Circlr Wire. Do problem in Hbermn. () At eqilibrim, 2 =. Solve ODE with polynomil: = c 2 + c 2. Periodic bondry conditions c =. Since there is no fl ot of the system the finl energy mst be the sme s the initil energy ths c 2 is the verge of the initil conditions. () = c 2 = f() d. 2

2 (b) (, t) = + n= s t, (, t). Therefore n cos nπ 2 e (nπ/) kt + n= (, t) = f() d. 2 b n nπ 2 e (nπ/) kt 4. Seprtion of Vribles, plce s Eqtion. Solve problem in Hbermn. See Hbermn. 5. Minimm Principles. Solve problem in Hbermn. See pge 8 in Hbermn. Sppose the minimm occrs t point P. By the Men Vle Theorem, the vle t P mst be the verge of the points on circle rond P. This is impossible if (P ) is less thn the vle of t ll the points on the circle. Ths, by contrdiction, the minimm mst be on the bondry. 6. Wndering Bcteri. Derive prtil differentil eqtion to describe the evoltion of the concentrtion of bcteri ssming tht t every time step, ech bcteri hs eql probbility of moving to the left or moving to the right or stying pt. (Recll tht in clss we did this problem ssming eql probbility of moving left or right nd zero probbility of not moving.) Tylor Series: c + t c t +... = 3 c(, t + t) = c( +, t) + from the right ( c(, t) bcteri stying pt + c(, t) from the left c + c ) c c + ( c c ) c c t = c c 3 t 2 t = D 2 c 2 Sme s before ecept diffsion coefficient is smller (i.e. it tkes longer for the bcteri to diffse). 7. Conservtion of Mss. Do problems nd in Hbermn. D: (mss in slice) = mss flowing in mss flowing ot t (ρa ) = ρ()()a ρ( + )( + )A t ρ t = ρ()() ρ( + )( + ) = (ρ)

3 3D: ρ dv = ρ n ds = t This is only tre if the integrnd is zero. (ρ) dv If ρ is constnt: (ρ) = =. ρ t + (ρ) = 8. Flow Pst Cylinder. Do problem in Hbermn. θ = c ( ) r U + 2 θ r 2 At the srfce of the cylinder, θ = c 2U θ. When the circltion is negtive, c is positive. At the top of the cylinder (θ = π 2 ) θ = c 2U nd t the bottom θ = c + 2U. Ths the flid is moving fstest (nd bckwrds) t the top. Mtlb 2. Forier series. How good is it? For the following fnctions, plot the fnction nd the first M terms of the Forier series where M =, 2, 4, 8. (Mke one plot for ech fnction listed below with ll five crves so yo cn see how well ech pproimtion compres with the originl). () f() = { 3 on the intervl 2π (b) f() = on the intervl 2 > (c) f() = on the intervl Comment on which fnctions seem to be pproimted best by the Forier series. Which series ppers to converge to the ctl fnction fstest? Why? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Forier Series % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% y = step fnction %% figre(2) N = 4; = linspce(, 2, N); y(:n/2) = ; y(n/2:n) = ; plot(,y) hold on FS = ones(,4)/2.;

4 for i = :8 (i) =.; b(i) = (cos(i*pi) *cos(i*pi/2.))/(i*pi); FS = FS + cos(i*pi*/2.)*(i)+ (i*pi*/2.)*b(i); plot(,fs, ro ) end is([ ]) lbel( ) ylbel( y = step fnction ) title( Forier Series (First 8 Terms) ) hold off Similrly for others ecept s nd b s become: %% y = ^3() %% (i) =.; b = [ 3./4. -./4. ]; %% y = %% (i) = 2./(i^2*pi^2)*(cos(i*pi)-); b(i) =.; Forier Series (First 8 Terms).5 Forier Series (First 8 Terms) y= y = step fnction

5 Forier Series (First 8 Terms) y =.5.5 Note tht the smoother the fnction is, the better the Forier pproimtion.