Why the Inconsistent Use of Δ in Physics?

[Bullwinckle]

Hello,

Sometimes in introductory physics, I see this:

After defining P as the momentum, we show: dP/dt = F

Then, later, I see this

ΔL = mv₁ - mv₂

So my question is really simple: WHY is there no consistency? Why do we sometimes use the Δ symbol? Is there something about how the definition is being applied (small change but not infinitesimally small), that makes the books switch back to Δ

I see it here, too (where U is the potential energy in a non-dispersive system):

ΔK.E. + ΔU = 0

Is there an APPLIED (as, say, in engineering) advantage to using the above, and not: K.E. + U = constant which, as I see it, is more theoretically precise.

 

[axmls]

The d denotes the derivative. In a less rigorous sense, it usually denotes an infinitesimally small difference, as opposed to delta, which is any finite difference. The deltas are used in the conservation of energy formula because energy itself has not inherent meaning--it is the change of energy that's important. It makes no sense to say K.E + U = 0. Using the deltas tells us that there is no change in the total energy of the system (some of it might have converted between forms, but the total change is 0).

 

[Bullwinckle]

The deltas are used in the conservation of energy formula because energy itself has not inherent meaning--it is the change of energy that's important. It makes no sense to say K.E + U = 0. Using the deltas tells us that there is no change in the total energy of the system (some of it might have converted between forms, but the total change is 0).

Thank you

 

[Bullwinckle]

The d denotes the derivative. In a less rigorous sense, it usually denotes an infinitesimally small difference, as opposed to delta, which is any finite difference. The deltas are used in the conservation of energy formula because energy itself has not inherent meaning--it is the change of energy that's important. It makes no sense to say K.E + U = 0. Using the deltas tells us that there is no change in the total energy of the system (some of it might have converted between forms, but the total change is 0).

Oh, one more thing...

Are you saying that books use this:

ΔL = mv₁ - mv₂

Simply to prepare students to solve problems over a finite time difference that is not infinitesimally small?

So, here, the Δ is being used for a PRACTICAL purpose?

 

[axmls]

We're often concerned with the change in quantities--not necessarily infinitesimal changes. The change in energy would be the obvious one. These aren't (usually infinitesimal). For instance, if we slide a block down a frictionless incline, we may be interested in the change in kinetic energy between the time at which the block is on top and the time at which it's on the bottom.

So really, different problems require different approaches. It's sometimes the case that we end up taking a limit as these differences go to 0, in which case we end up with a derivative or integral (somewhat non-rigorously), but again, this all depends on context.