Differentials multiplied by differentials

[Signifier]

If I have

(A + dA)dZ

= A dZ + dAdZ

Can I drop the dAdZ?

Likewise, with

(dT + 1)dT

= (dT)^2 + dT

Can I drop the (dT)^2? Is there any basis for these actions?

Are (dU)^3 and (dU)^2 equivalent?

 

[StatusX]

Those expressions don't really mean anything by themselves, but usually you use them in integrals or derivatives, and when you do you take the limit as dt, dz, etc goes to zero, so terms like dt^2 drop out.

 

[Gagle The Terrible]

I'd be carefull thought with manipulation regarding deifferentials.

I remember in a differential geometry class the teacher had "split"
ds^2 in ds*ds . But is true that in most cases, terms like dt^2 are said to be negligible comparatively to dt.

Also, terms like dA sometimes refer to the differential area for instance, and can also be represented differently depending on the coordinate system you are using.

 

[Berko]

Often in limits as t --> 0, the dt^2 is ignored because is is "much smaller" than dt.