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Let's consider the formula for curvature

$κ$=$|\frac{dT}{dS}|$

This is the rate of change of $Tangent$ $vector$ as we walk along the curve.

Now a more useful way to represent this is multiplying and dividing by $dt$ ;

$κ$=$|\frac{\frac{dT}{dt}}{\frac{ds}{dt}}|$.

This seems logical to me as we can further simplify the expression as

$κ$=$\frac{1}{|\vec{v}|}$$|\frac{dT}{dt}|$.

This is a useful expression.

But from the perspective of a Mathematician, why does he abhor the idea of multiplying and dividing a differential ,what is the correct mathematical way to approach this?

Harry Case
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1 Answers1

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The fact is that it seems you are dividing differentials but you can interpret it as a change of variables.

For example, suppose $\color{blue}{\dfrac{dt}{ds}=\alpha(x,t)}$ and $\color{green}{\dfrac{dx}{ds}=\beta(x,t)}\implies \dfrac{\frac{dx}{ds}}{\frac{dt}{ds}}"="\dfrac{dx}{dt}=\dfrac{\beta(x,t)}{\alpha(x,t)}$... Why?
Since $x=x(t(s))$, we have $$\color{green}{\dfrac{dx}{ds}}=\dfrac{d}{ds}x(t(s))=\underbrace{\dfrac{d}{dt}x(t(s))\cdot\color{blue}{\dfrac{d}{ds} t(s)}}_{\text{chain rule}}=\dfrac{d}{dt}x\cdot\color{blue}{\alpha(x,t)}\implies\color{green}{\beta(x,t)}=\dfrac{dx}{dt}\cdot\color{blue}{\alpha(x,t)}\\\dfrac{dx}{dt}=\dfrac{\beta(x,t)}{\alpha(x,t)}.$$

So when you see calculations in which it seems that differentials are treated as fractions you should see if the variables are implicitly dependent.

Vajra
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