Crear matriz de derivadas

Question

Dadas las siguientes definiciones,

Clear["Global`*"]
Remove["Global`*"]
n = 4;
wb = Table[Sum[qw[i, j] x^j, {j, n}], {i, 2}]
ub = Table[Sum[qu[i, j] x^j, {j, n}], {i, 2}];  

um = Sum[0.5*ub[[i]], {i, 2}]
wm = 1/h (wb[[1]] - wb[[2]])

¿Alguien puede ayudar a construir la matriz en la imagen de abajo?

Intenté hacerlo con el siguiente código:

nx = 4;
ny = 3;
    u = Sum[{qx[i, j], qy[i, j]} x^i y^j, {i, 1, nx}, {j, 0, ny}]
    gradu = D[u, {{x, y}}]
    \[CurlyEpsilon] = (gradu + Transpose[gradu])/2

pero no pude hacerlo funcionar.

Answer 1

mat = {{D[um, x], 0.5 D[um, x] + D[wm, x]}, {0.5 D[um, x] + D[wm, x], D[wm, x]}};
mat // FullSimplify

da:

 {{
 0.5 qu[1, 1] + 0.5 qu[2, 1] + x (1. qu[1, 2] +
 1. qu[2, 2] + x (1.5 qu[1, 3] + 2. x qu[1, 4] + 
 1.5 qu[2, 3] + 2. x qu[2, 4])), 0.25 (qu[1, 1] + 
 x (2 qu[1, 2] + x (3 qu[1, 3] + 4 x qu[1, 4]))) +
 0.25 (qu[2, 1] + x (2 qu[2, 2] + x (3 qu[2, 3] + 
 4 x qu[2, 4]))) + (1/h)(qw[1, 1] - qw[2, 1] + x
 (2 qw[1, 2] - 2 qw[2, 2] + x (3 qw[1, 3] + 4 x 
 qw[1, 4] - 3 qw[2, 3] - 4 x qw[2, 4])))
 }, 
 {
  0.25 (qu[1, 1] + x (2 qu[1, 2] + x (3 qu[1, 3] +      
 4 x qu[1, 4]))) + 0.25 (qu[2, 1] + x (2 qu[2, 2] 
 + x (3 qu[2, 3] + 4 x qu[2, 4]))) + (1/h)(qw[1,  
 1] - qw[2, 1] + x (2 qw[1, 2] - 2 qw[2, 2] + x (3 
 qw[1, 3] + 4 x qw[1, 4] - 3 qw[2, 3] - 4 x qw[2, 
 4]))), (1/h)(qw[1, 1] - qw[2, 1] + x (2 qw[1, 2] 
 - 2 qw[2, 2] + x (3 qw[1, 3] + 4 x qw[1, 4] - 3 
 qw[2, 3] - 4 x qw[2, 4])))
 }}

Answer 2

No es la mejor forma pero funciona así:

\[CurlyEpsilon] = ( {
    {D[um, {x}], 0.5*(D[um, {x}] + D[wm, {x}])},
    {0.5*(D[um, {x}] + D[wm, {x}]), D[wm, {x}]}
   } );