Differentials multiplied by differentials

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Homework Help Overview

The discussion revolves around the manipulation of differentials in mathematical expressions, particularly in the context of calculus and differential geometry. Participants are examining whether certain terms involving differentials can be disregarded under specific conditions.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the validity of dropping terms like dAdZ and (dT)^2 in their expressions. Some participants question the meaning and implications of manipulating differentials, while others suggest that in certain contexts, such as limits approaching zero, these terms may be considered negligible.

Discussion Status

The discussion is exploring various interpretations of the treatment of differentials, with some participants providing insights into when terms can be dropped based on their relative size in limits. There is no explicit consensus, but productive lines of reasoning are being examined.

Contextual Notes

Participants note that the treatment of differentials may vary depending on the context, such as in differential geometry or when considering integrals and derivatives. There is an acknowledgment of potential ambiguity in the definitions and representations of differentials across different coordinate systems.

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

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