Why f is no longer a function of x and y ?

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Discussion Overview

The discussion revolves around the nature of a function of two variables, specifically questioning why a function denoted as ##f(x,y)## appears to lose its status as a function of ##x## and ##y## when restricted to a defined rectangular region. Participants explore the implications of changing variable names and the effects of restrictions on derivatives.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that ##f(x,y)## remains a function of ##x## and ##y## even when restricted to a rectangular region, arguing that the variable names do not change the function's nature.
  • Another participant questions the reasoning behind a specific application of the Biot-Savart Law, suggesting that the relationship between the variables may not be clear.
  • Concerns are raised about the interpretation of the derivative of a function with respect to different variables, particularly in the context of dummy integration variables.
  • Some participants discuss the implications of treating different variables as interchangeable, particularly in the context of integration and differentiation.
  • There is a mention of the potential misunderstanding regarding the relationship between the variables ##x'## and ##x##, with some arguing that ##x'## should not be considered a function of ##x##.

Areas of Agreement / Disagreement

Participants express differing views on whether the function retains its status as a function of ##x## and ##y## under restrictions, leading to an unresolved debate on the implications of variable naming and the nature of derivatives in this context.

Contextual Notes

There are unresolved issues regarding the treatment of dummy variables in integrals and how they relate to the variables of differentiation. The discussion highlights the subtleties involved in mathematical reasoning when changing variable names and applying derivatives.

  • #31
PeroK said:
The same difference as differentiating with respect to ##y_1## and ##y_2## in the above examples.
That solves my whole problem. Thank you so much.
 
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  • #32
Adesh said:
So, what’s the difference between differentiating with respect to ##x## and ##x’## ?
In that response I was trying to illustrate to you the way we think of axes, but:
Define ##\vec a=f(x)\vec e_1=(f(x), 0)## and ##\vec c=g(x')\vec e_1=(g(x'),0)##. You know how to do differentiation on vectors as you are reading Griffith's book, so what do you think?
Adesh said:
That solves my whole problem. Thank you so much.
I think the main point that you need to remember is to think of the axes as the set of real numbers. The function you have of both ##x## and ##x'## who are on the same "axis" can be seen as ##\vec a = f(x, x')\vec e##.
 
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  • #33
On #4
\nabla\times J=[\frac{\partial J_y(x',y',z')}{\partial z}-\frac{\partial J_z(x',y',z')}{\partial y}]\hat x + ...
is zero.
 
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