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Calculus (Curvature) Question
When deriving the expression for curvature, I can follow the example given here.However, between steps 13 and 14, they replace the dummy variable (t), since we know y=f(x), we can greatly simplify the expression.
What is this technique called, and where could I find a straightforward example of it?
I should probably know this, so try not to make me feel too dumb with your replies.
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Re: Calculus (Curvature) Question
The technique is "reparametrization"; we don't care what exactly the functions x(t) and y(t) are, as long as they produce the curve of interest; there will generally be more than one way to do so. So by changing the rate at which we trace along the curve, we can get different but equivalent functions (i.e., different functions which define the same curve). Thus, in particular, if we know that y is just some function f of x, then we can, without loss of generality, assume that we're tracing out the curve with the parametric definition x(t) = t and y(t) = f(t) [thus, just making t another name for x].
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