I am learning differential forms on my own through lectures on youtube and one of the things that I am attempting to do is to check if I can derive matrix differential rules using the rules for differential forms. i.e. Prove:
d(X+Y) = d(X) + d(Y) using the 4 differential form rules:
The 4 differential form rules being:
- d(d(w)) = 0
- df = $\sum_{1,n} \frac{\delta f}{\delta x_{i}} dx_{i}$
- d(aw + bx) = ad(w) + bd(x) where w,x are 1-form
- d(w ^ x) = d(w) ^ x + (-1)^pq w ^ d(x) where w is p-form and x is q-form
I am stuck in this proof and this probably indicates some gap in my understanding.
It is enough to use rule 3.
If $X, Y$ are the matrices of elements $(x_{ij}), (y_{ij})$, then $X+Y$ is the matrix of elements $(x_{ij}+y_{ij})$.
Now, you have to define $d$ for matrices. You do it component by component. That is, the matrix $dX$ is the matrix of components $(dx_{ij})$, by definition.
Now, using our expression for the sum of matrices: $$ d(X+Y) = d(x_{ij}+y_{ij}). $$
Using our definition of $d$ for matrices: $$ d(x_{ij}+y_{ij}) = (d(x_{ij}+y_{ij})). $$
Using rule 3: $$ (d(x_{ij}+y_{ij})) = (dx_{ij}+dy_{ij}). $$
Re-using our expression for the sum: $$ (dx_{ij}+dy_{ij}) = (dx_{ij}) + (dy_{ij}), $$
which are the matrices (using again our definition for $d$): $$ (dx_{ij}) + (dy_{ij}) = dX + dY. $$