Differentiating Matrices

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Haast
Haast 2011-10-2
I have a 2 by 2 matrix and i need to differentiate each term in this matrix by each term in another 2 by 2 matrix so that i end up with a 4 by 4 result. Eventually i will be doing the same thing with 20 by 5 matrices. Is there any way of doing this? Can anybody help please?
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Walter Roberson
Walter Roberson 2011-10-5
Pasting the code would be a good start. Commenting the code would help after that.
Haast
Haast 2011-10-10
clc
clear
%% Defining variables
%symbolic variables
syms c11 c12 c21 c22 x y
%array
c_ij = [c11 c12; c21 c22];
%dimensions
L = 20;
W = 20;
T = 0.125;
%unit loads
Nx = 1;
Ny = 1;
Nxy = 1;
%material properties
E = 72.4*10^9;
nu = 0.3;
D = (E*T^3)/(12*(1-nu.^2));
G = E/(2*(1+nu));
%% Defining and differentiating Omega
%Omega
w = c11*sin((pi*x)/L)*sin((pi*y)/W) + c12*sin((pi*x)/L)*sin((2*pi*y)/W) + c21*sin((2*pi*x)/L)*sin((pi*y)/W) + c22*sin((2*pi*x)/L)*sin((2*pi*y)/W);
%derivatives
dw_x = diff(w, x);
dw_y = diff(w, y);
dw_xy = diff(w, x, y);
%second derivatives
dw2_x = diff(dw_x, x);
dw2_y = diff(dw_y, y);
dw2_xy = diff(dw_xy, x, y);
%% FINDING KMN and PMN
F = ((dw2_x + dw2_y)^2)-2*(1-nu)*((dw2_x*dw2_y)-(dw2_xy)^2);
KMN = int(F,x,0,L);
KMN2 = D*int(KMN,y,0,W);
S = (Nx*dw_x^2+Ny*dw_y^2+2*Nxy*dw_xy^2);
PMN = int(S,0,L);
PMN2 = int(PMN,0,W);
%% Differentiation wrt cij
KMNF = diff(KMN2, c_ij);
this works up until i try and do the KMNF :(

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回答（2 个）

Walter Roberson
Walter Roberson 2011-10-3
Is this symbolic or numeric differentiation ? If it is symbolic, is the second matrix containing just one symbol per entry or does it contain expressions? Differentiating with respect to an expression is not easy.
Haast
Haast 2011-10-3
it is symbolic. Actually i realised there was an error in my code and i am trying to differentiate one expression with respect to a matrix. I haven't typed it up here because it is a very long expression but it is essentially an expression in terms of c11 c12 c21 and c22 and i want to differentiate it with respect to each of these and because i am going to eventually be doing the same with many c values i thought the neatest way to set it out would be as a matrix!
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Haast
Haast 2011-10-5
I am sorry I am very much a beginner with MATLAB and have never worked with anything to do with MuPad before. Am I just able to create a new notebook file with Mupad and enter these expressions into it??
Walter Roberson
Walter Roberson 2011-10-5
You could do that, but there are alternatives. For example, at the MATLAB level, you might be able to use
diffs_list = simple(subs('map(proc(thisvar,expression) diff(expression,thisvar) end_proc, indets(YourExpression) minus Type::ConstantIdents, YourExpression)', 'YourExpression', your_symbolic_expression));
Unfortunately I do not have the symbolic toolbox myself, so I am not certain it is possible to activate the internal toolkit map() operation in this manner.

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