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"Don't panic!"
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Apologies if this isn't quite the right forum to post this in, but I was unsure between this and the calculus forum.
Something that has always bothered me since first learning calculus is how to interpret [itex]dx[/itex], essentially, what does it "mean"? I understand that it doesn't make sense to consider it as an infinitesimal change in [itex]x[/itex] (in a rigorous sense) as the idea of an infinitesimal cannot be formulated rigorously (at least in standard analysis), but can any sense of this notion be retained?
I read in Spivak's book that we can not consider quantities such as [itex]df[/itex] in the classical sense, but to overcome this we can replace this notion of infinitesimals by promoting the quantities [itex]df[/itex] to functions of infinitesimal changes along particular directions, i.e. functions of tangent vectors. In this sense the function [itex]df: T_{p}M\rightarrow\mathbb{R}[/itex] contains all information about the infinitesimal changes in the function [itex]f[/itex] as it moves in particular directions (i.e. along particular directions). Thus we can consider the functions [itex]dx^{i}: T_{p}M\rightarrow\mathbb{R}[/itex] as containing all information about the infinitesimal change in the coordinate functions in particular directions. I have paraphrased what is written in the book and tried to reformulate it in the way that I can understand it; would what I put be correct?
Also, is there a way to formulate the idea of a differential in elementary calculus (without resorting to non-standard analysis)? Is it correct to say that one can consider the rate of change in a function at a point, [itex]f'(x_{0})[/itex] which is the gradient of the tangent line [itex]y[/itex] to the function [itex]f[/itex] at this point. From this we can construct a new function [itex]df[/itex] which is dependent on the point [itex]x[/itex] and this change in its value [itex]\Delta x[/itex], such that [tex]df(x_{0},\Delta x)=f'(x_{0})\Delta x[/tex] Thus, the (finite) change in the function near a point [itex]x_{0}[/itex], [itex]\Delta f[/itex] can be expressed as the following [tex]\Delta f=f'(x_{0})\Delta x +\varepsilon =df+\varepsilon[/tex] where [itex]\varepsilon[/itex] is some error function. We note that [tex]dx(x_{0},\Delta x)=\Delta x[/tex] and so [tex]\Delta f=f'(x_{0})dx +\varepsilon =df+\varepsilon \Rightarrow df=f'(x_{0})dx[/tex] where [itex]df=f'(x_{0})dx[/itex] represents a (finite) change along the tangent line to the function [itex]f[/itex] at the point [itex]x_{0}[/itex]. I'm unsure how to proceed from here though?!
Something that has always bothered me since first learning calculus is how to interpret [itex]dx[/itex], essentially, what does it "mean"? I understand that it doesn't make sense to consider it as an infinitesimal change in [itex]x[/itex] (in a rigorous sense) as the idea of an infinitesimal cannot be formulated rigorously (at least in standard analysis), but can any sense of this notion be retained?
I read in Spivak's book that we can not consider quantities such as [itex]df[/itex] in the classical sense, but to overcome this we can replace this notion of infinitesimals by promoting the quantities [itex]df[/itex] to functions of infinitesimal changes along particular directions, i.e. functions of tangent vectors. In this sense the function [itex]df: T_{p}M\rightarrow\mathbb{R}[/itex] contains all information about the infinitesimal changes in the function [itex]f[/itex] as it moves in particular directions (i.e. along particular directions). Thus we can consider the functions [itex]dx^{i}: T_{p}M\rightarrow\mathbb{R}[/itex] as containing all information about the infinitesimal change in the coordinate functions in particular directions. I have paraphrased what is written in the book and tried to reformulate it in the way that I can understand it; would what I put be correct?
Also, is there a way to formulate the idea of a differential in elementary calculus (without resorting to non-standard analysis)? Is it correct to say that one can consider the rate of change in a function at a point, [itex]f'(x_{0})[/itex] which is the gradient of the tangent line [itex]y[/itex] to the function [itex]f[/itex] at this point. From this we can construct a new function [itex]df[/itex] which is dependent on the point [itex]x[/itex] and this change in its value [itex]\Delta x[/itex], such that [tex]df(x_{0},\Delta x)=f'(x_{0})\Delta x[/tex] Thus, the (finite) change in the function near a point [itex]x_{0}[/itex], [itex]\Delta f[/itex] can be expressed as the following [tex]\Delta f=f'(x_{0})\Delta x +\varepsilon =df+\varepsilon[/tex] where [itex]\varepsilon[/itex] is some error function. We note that [tex]dx(x_{0},\Delta x)=\Delta x[/tex] and so [tex]\Delta f=f'(x_{0})dx +\varepsilon =df+\varepsilon \Rightarrow df=f'(x_{0})dx[/tex] where [itex]df=f'(x_{0})dx[/itex] represents a (finite) change along the tangent line to the function [itex]f[/itex] at the point [itex]x_{0}[/itex]. I'm unsure how to proceed from here though?!