differential forms, 1-form, 2-form

Can I express a 2-form? Can matlab diff a 1-form?

[Newton Campbell]

I would love it if you found an answer to this question.

Answer 1

[Divyam]

Hi @Thomas Starai,

You can express a 2-form using the "Symbolic Math Toolbox". However, you cannot use the traditional wedge product syntax (^) to enforce antisymmetry as this is not natively supported by MATLAB. The code below represents the following 2-form in 3D,

:

syms x y z
% Represent dx∧dy as diff(x)*diff(y) 
omega = x*y * diff(x) * diff(y) + z * diff(y) * diff(z);

You can also use MATLAB to diff a 1-form. The code below performs the diff for the following 1-form, :

syms x y dx dy real
omega = x^2*dx + y^2*dy;

% Take the exterior derivative (d_omega), mimicked by diff()
d_omega = diff(x^2, x)*diff(x)*dx + diff(y^2, y)*diff(y)*dy;

Note: As mentioned before, MATLAB can differentiate the symbolic expressions using diff. However, it doesn't handle exterior derivatives or wedge products automatically.

For more information regarding the "diff" function, refer to the following documentation: https://www.mathworks.com/help/symbolic/sym.diff.html