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MathGroup Archive 2007

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Re: Derivative recurrence

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72717] Re: Derivative recurrence
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 16 Jan 2007 03:23:39 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <eofikq$gc1$1@smc.vnet.net>

In article <eofikq$gc1$1 at smc.vnet.net>,
 "KFUPM" <hussain.alqahtani at gmail.com> wrote:

> I have the following
> 
> u1(x,x,z,t) = z f1(x,y,t)
> u2(x,y,z,t) = z f2(x,y,t)
> u3(x,y,z,t) = z f3(x,y,t)

I would implement these rules as

  u[1][{x_,y_,z_},t_] = z f[1][t][x,y]
  u[2][{x_,y_,z_},t_] = z f[2][t][x,y]
  u[3][{x_,y_,z_},t_] = z f[3][t][x,y]
 
> which is basically the displacement fied. I want to compute the strain
> which is given in indecial notation as:
> 
> Eij = 1/2( dui/dxj + duj/dxj ) 

I assume you mean Eij = 1/2( dui/dxj + duj/dxi ) ?

> i, j, k go from 1 to 3,,, x1=x, x2 =y, x3=z,

but why is k required. Here is one implementation:

  r = {x,y,z};
  
  e[i_,j_] := 1/2 (D[u[i][r,t], r[[j]] ] + D[u[j][r,t], r[[i]] ])

> I want mathematica to calculate the strain Eij autmoatically,
> 
> for example
> 
> when i = 1 and j= 2
> 
> E12 =1/2 (du1/dx2+du2/dx1 )     but x1 =x and x2 =y
> 
> therefore
> 
> E12 =1/2 (du1/dy + du2/dx) = 1/2 ( z df1/dx + z df2/dy),,,   where f1,
> f2 and f3 are arbitrary functions.

Now you can compute these quantities:

  e[1,2]

  (z Derivative[0, 1][f[1][t]][x, y] + 
   z Derivative[1, 0][f[2][t]][x, y])/2

This output looks better in StandardForm -- or TraditionalForm. 

> Eij should generate 3 by 3 matrix containing the strain components.

Use Table:

 Table[e[i,j], {i, 3}, {j, 3}]

Also, if required, the arguments [x, y] can be suppressed using 
pattern-matching:

 % /.  f_[x, y] :> f

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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