Can someone give me an intuitive definition for differentials?

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Discussion Overview

The discussion revolves around the concept of differentials, particularly seeking an intuitive understanding of them in the context of calculus and their application in thermodynamics. Participants express a desire for practical insights rather than theoretical depth, as they prepare for upcoming coursework.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant asks for an intuitive definition of differentials, suggesting they may be "infinitely small differences."
  • Another participant references a resource (Tenenbaum and Pollard) that may provide clarity on the topic.
  • A participant shares a link to a visual representation of differentials, explaining that it relates to the change in linear approximation for a function.
  • It is noted that the differential of a function can be understood through the family of tangent lines to a smooth curve, indicating that the differential represents these linear functions.
  • Another participant suggests reading Loomis and Sternberg for further insights, indicating that preliminary reading may be necessary to grasp the concept fully.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and seek different resources, indicating that there is no consensus on a singular intuitive definition of differentials. Multiple viewpoints and references are presented without resolution.

Contextual Notes

Some participants emphasize the need for practical understanding over theoretical explanations, which may limit the depth of discussion on the foundational aspects of differentials.

Who May Find This Useful

Students preparing for calculus and thermodynamics who are looking for intuitive insights into the concept of differentials.

Howers
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Can someone give me an intuitive definition for differentials? My prof said to brush up on them because we'll be seeing them lots in thermo. I don't need all the theory because I'll be seeing them in november in calc. Right now I just have to work with them. Are they just infinitely small differences?
 
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read tenenbaum and pollard, ode, first few pages.
 
This picture show what differential is for a function f(x)
http://www.bymath.com/studyguide/ana/sec/ana4a.gif
Basically, it is the change in the linear approximation for a function for a change in x, dx.

dy/dx = f '(x) -> differential dy = f '(x) dx
When dx is small dy is a good approximation for f(x + dx) - f(x);
 
consider for a smooth curve y=f(x), its family of tangent lines. the differential of f, df, is the family of linear functions whose graphs are those tangent lines.

so that picture depicted the graph of df(x0), the graph of one of the linear functions making up the differential.
 
Howers said:
Can someone give me an intuitive definition for differentials? My prof said to brush up on them because we'll be seeing them lots in thermo. I don't need all the theory because I'll be seeing them in november in calc. Right now I just have to work with them. Are they just infinitely small differences?

Loomis and Sternberg, chapter 3 section 5 and following sections.
You'll have to do some preliminary reading in order to get to this point. Their text is freely available.

Even if you don't do more than look at it now, it'll serve you well in November.
 

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