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March 8th, 2015, 11:08 PM   #1
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Question Difference between vector differentiation and parametric differentiation

This might seem like a naive question to ask, but a full explanation of why these two concepts are different would be welcome. I am confused because parametric equations are $\displaystyle y = 8t^2$ and $\displaystyle x = 5t$, but at the same time, these two equations can describe the $\displaystyle x$ and $\displaystyle y$ components of a vector. Parametric differentiation is defined as $\displaystyle \frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\frac{\mathrm{d} y}{\mathrm{d} t}}{\frac{\mathrm{d} x}{\mathrm{d} t}}$, while vector differentiation is defined as $\displaystyle \frac{\mathrm{d} \vec{r}}{\mathrm{d} t} = \left \langle \frac{\mathrm{d} x}{\mathrm{d} t}, \frac{\mathrm{d} y}{\mathrm{d} t} \right \rangle$
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March 9th, 2015, 04:50 AM   #2
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Note the "denominators" of your differential operators on the left hand side. The first relates how$y$ varies with $x$ along the path of a "particle" (independently of time), the second shows how the position of that "particle" changes with time (i.e. it's velocity).
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