Coherent operations on Jacobian matrices

[Mapprehension]

Is there a notion of “coherent” operations on Jacobian matrices? By this I mean, an operation on a Jacobian matrix A that yields a new matrix A' that is itself a Jacobian matrix of some (other) system of functions. You can ascertain whether A' is coherent by integrating its partials of one variable. If those integrations yield the same function, then that variable’s contribution is coherent. Repeat for each variable, and if all are coherent, then A' is coherent.

Multiplying A by a constant matrix yields a coherent A'. Maybe there are other operations that do, but the universe of such operations seems… sparse. Are there known, non-trivial operations that retain coherency? Is this a thing? Is there any way to determine coherency without having to integrate? (I can’t imagine how…)

Thanks.
— Mapp

 

[WWGD]

Is there a notion of “coherent” operations on Jacobian matrices? By this I mean, an operation on a Jacobian matrix A that yields a new matrix A' that is itself a Jacobian matrix of some (other) system of functions. You can ascertain whether A' is coherent by integrating its partials of one variable. If those integrations yield the same function, then that variable’s contribution is coherent. Repeat for each variable, and if all are coherent, then A' is coherent.

Multiplying A by a constant matrix yields a coherent A'. Maybe there are other operations that do, but the universe of such operations seems… sparse. Are there known, non-trivial operations that retain coherency? Is this a thing? Is there any way to determine coherency without having to integrate? (I can’t imagine how…)

Thanks.
— Mapp

Isn't J(A+B)=J(A)+J(B), given $d/dx_i(f+g)=d/dx_i(f)+d/dx_i(g)$ ?

 

[Mapprehension]

Isn't J(A+B)=J(A)+J(B), given $d/dx_i(f+g)=d/dx_i(f)+d/dx_i(g)$ ?

Yes. I should have noted addition as well. I was musing over anything more elaborate.

Thanks.
— Mapp