Infinitesimal step in different coordinates

Question

I was wondering why an infinitesimal step in different coordinates are not the same size unless specifically stated. For example, dx and dy are both infinitesimal steps in spatial coordinates, but dx/dy is not equal to 1. Is this related to how there are different 'size' infinities?

Answer 1

$\frac{dy}{dx}$ is not a fraction, but a notation. In the same way, $dx$ and $dy$ are not actual nonzero quantities. You may instead interpret them as limits of differences $\Delta x$ and $\Delta y$, as they tend to zero.

For example, for a differentiable function $y(x)$, the notation means: $$\frac{d y}{ dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x},$$ where $\Delta y = y(x+\Delta x)-y(x)$ is a shorthand notation of a difference as well.

Answer 2

Firstly, consider a function $y = f(x)$ where $x$ and $y$ are not independent, and we are interested in the effects of a small change in $x$, denoted as $dx$. Correspondingly, $dy$ represents the small change in $y$ resulting from this adjustment in $x$.

The concept of a differential, often expressed as $dy$, is a measure of how the value of $y$ changes as we transition from one point to another in $x$. In mathematical terms, $dy = f'(x) \cdot dx$, where $f'(x)$ denotes the derivative of $f$ with respect to $x$. This expression encapsulates the rate of change of $y$ concerning $x$ and quantifies the effect of a small $dx$ on $y$.

you are thinking $dy/dx$ as a ratio of independent variables and thats against the basic idea of differentials, differential doesnt really make sense without having a relation between $x$ and $y$. While some may argue against interpreting it as a fraction, asserting that it defies traditional fraction rules, but i disagree diffential is just a fraction

Consider a small change in $x$, denoted as $x_2 - x_1$. If $y = f(x)$, the corresponding change in $y$ is given by $f(x_2) - f(x_1)$. Notably, these two quantities are generally not equal. Why? Because you cant ignore relationship between $x$ and $y$ , you are getting value of y by knowing how it is related to x . The rate of change of $y$ concerning $x$ is contingent on the nature of the function $f(x)$.

The assertion that $dy/dx$ is akin to a fraction is rooted in the idea that it represents the "fractional" change in $y$ concerning the change in $x$. While unconventional, this perspective provides an insightful intuition into the dynamics of the function. The ratio $dy/dx$ signifies how much of the change in $y$ is attributed to the change in $x$, offering a quantitative understanding of the relationship between these variables.

i hope we can make sense of this now that its the ratio of change in both y and x when change in one variable is too small $$\frac{d y}{ dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}, $$ hope it helps

Answer 3

In the context of a modern theory of infinitesimals such as nonstandard analysis, $\frac{dy}{dx}$ can indeed be defined as a fraction of infinitesimals. Just as standard numbers can be of different sizes, infinitesimal numbers can also be of different sizes, entailing no contradiction at all.