BENG 221: Mathematical Methods in Bioengineering Lecture 11 Review of Vector Calculus References http://en.wikipedia.org/wiki/Gradient http://en.wikipedia.org/wiki/Gradient_theorem http://en.wikipedia.org/wiki/Divergence http://en.wikipedia.org/wiki/Divergence_theorem http://en.wikipedia.org/wiki/Curl_(mathematics)http://en.wikipedia.org/wiki/Stokes%27_theorem http://en.wikipedia.org/wiki/Spherical_coordinates http://mathworld.wolfram.com/SphericalCoordinates.htmlï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 0x (units R)J0((h0i)1/2l)ï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 1x (units R)J1((h1i)1/2l)ï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 2x (units R)J2((h2i)1/2l)ï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 3x (units R)J3((h3i)1/2l)ï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 4x (units R)J4((h4i)1/2l)h01 = 5.783/R2h02 = 30.47/R2h03 = 74.89/R2h04 = 139/R2h05 = 222.9/R2h11 = 14.68/R2h12 = 49.22/R2h13 = 103.5/R2h14 = 177.5/R2h15 = 271.3/R2h21 = 26.37/R2h22 = 70.85/R2h23 = 135/R2h24 = 218.9/R2h25 = 322.6/R2h31 = 40.71/R2h32 = 95.28/R2h33 = 169.4/R2h34 = 263.2/R2h35 = 376.7/R2h41 = 57.58/R2h42 = 122.4/R2h43 = 206.6/R2h44 = 310.3/R2Cylindrical Value Boundary Conditionsh45 = 433.8/R2h11 = 19.74/L2h12 = 49.35/L2h13 = 98.7/L2h14 = 167.8/L2h15 = 256.6/L2h21 = 49.35/L2h22 = 78.96/L2h23 = 128.3/L2h24 = 197.4/L2h25 = 286.2/L2h31 = 98.7/L2h32 = 128.3/L2h33 = 177.7/L2h34 = 246.7/L2h35 = 335.6/L2h41 = 167.8/L2h42 = 197.4/L2h43 = 246.7/L2h44 = 315.8/L2h45 = 404.7/L2h51 = 256.6/L2h52 = 286.2/L2h53 = 335.6/L2h54 = 404.7/L2Cartesian Box Value Boundary Conditionsh55 = 493.5/L2ï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 0x (units R)J0((h0i)1/2l)ï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 1x (units R)J1((h1i)1/2l)ï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 2x (units R)J2((h2i)1/2l)ï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 3x (units R)J3((h3i)1/2l)ï1 ï0.8 ï0.6 ï0.4 ï0.2 0 0.2 0.4 0.6 0.8 1ï1ï0.8ï0.6ï0.4ï0.200.20.40.60.81n = 4x (units R)J4((h4i)1/2l)h01 = 0/R2h02 = 14.68/R2h03 = 49.22/R2h04 = 103.5/R2h05 = 177.5/R2h11 = 3.39/R2h12 = 28.42/R2h13 = 72.87/R2h14 = 137/R2h15 = 220.9/R2h21 = 9.328/R2h22 = 44.97/R2h23 = 99.39/R2h24 = 173.5/R2h25 = 267.2/R2h31 = 17.65/R2h32 = 64.24/R2h33 = 128.7/R2h34 = 212.7/R2h35 = 316.4/R2h41 = 28.28/R2h42 = 86.16/R2h43 = 160.8/R2h44 = 254.9/R2Cylindrical Flux Boundary Conditionsh45 = 368.5/R2h00 = 0/L2h01 = 9.87/L2h02 = 39.48/L2h03 = 88.83/L2h04 = 157.9/L2h10 = 9.87/L2h11 = 19.74/L2h12 = 49.35/L2h13 = 98.7/L2h14 = 167.8/L2h20 = 39.48/L2h21 = 49.35/L2h22 = 78.96/L2h23 = 128.3/L2h24 = 197.4/L2h30 = 88.83/L2h31 = 98.7/L2h32 = 128.3/L2h33 = 177.7/L2h34 = 246.7/L2h40 = 157.9/L2h41 = 167.8/L2h42 = 197.4/L2h43 = 246.7/L2Cartesian Box Flux Boundary Conditionsh44 =