Difference between small delta t, big delta t, and dt?

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Discussion Overview

The discussion revolves around the differences between the notations Δf, δf, and df, particularly in the context of solid state physics and equations of motion in semiconductor crystals. Participants explore the meanings and implications of these symbols as they relate to changes in variables and calculus concepts.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that Δf represents the difference between two values, while df and δf refer to infinitesimal changes, although the distinction between df and δf is unclear.
  • Another participant clarifies that Δx indicates a change in f for specific changes in the variables f depends on, while df is defined as the differential in calculus, which may not strictly be considered infinitesimal.
  • A third participant notes that δf has various meanings and should be defined in the context of the text being used.
  • A participant expresses confusion regarding why the differential is not considered infinitesimal, questioning the transition from δt to dt in a specific equation from their textbook.
  • The same participant seeks clarification on the reasoning behind changing symbols from δk to dk and δt to dt, wondering if they are essentially the same.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of the symbols Δf, δf, and df, indicating that the discussion remains unresolved with multiple competing interpretations.

Contextual Notes

There are limitations in the definitions provided, as the meanings of δf can vary based on context, and the relationship between infinitesimals and differentials is not universally agreed upon.

schrodingerscat11
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Greetings! I am confused with the difference between Δf, δf, and df. I think Δf is a difference between two values, while df and δf refer to infinitesimal change (but I do not know the difference between the two.) Can anybody explain the difference? I am studying solid state physics (I am using Kittel) and the explanation of the equations of motion in semiconductor crystals is confusing without understanding these notations. Thank you very much.
 
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Yes, "[itex]\Delta x[/itex]" is the change in f for specific changes in the variables f depends on. "df" is the differential as defined in Calculus. It is, strictly speaking, not an "infinitesal" (which, outside of some very deep logical texts) is at best a loose concept). "[itex]\delta x[/itex]" may have a number of different meanings and should be defined in the text.
 
The last symbolism has a rather well-established meaning from the calculus of variations, but as HallsofIvy said, this is by no means a dominant-bordering-on-universal meaning.
 
Okay. I am kinda confused. How come the differential is not infinitesimal? Isn't it a very small increment to the function? But anyway, I'll just post the transition (from the book) of δt to dt. This is what actually confused me:

δk = -(eE/h)δt
(h)(dk/dt) = -eE

Note: h here actually means h/2*pi or h-bar. I just wrote h to simplify stuff.

Why change the δk to dk and δt and dt? Are the two just the same? Why change symbols all of a sudden? Thanks again.
 

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