Any point on the plane can be specified with an ##r## and a ##\theta##, where ##\mathbf{r} = r \hat{\mathbf{r}}(\theta)##. From this, my book derives ##\displaystyle \frac{d \mathbf{r}}{dt}## by making the substitution ##\hat{\mathbf{r}}(\theta) = \cos \theta \hat{\mathbf{i}} + \sin \theta \hat{\mathbf{j}}##, and then deriving it from there, concluding that ##\mathbf{v} = \dot{r} \hat{\mathbf{r}} + r \dot{\theta} \hat{\mathbf{\theta}}##.(adsbygoogle = window.adsbygoogle || []).push({});

My question is, is is possible to make the derivation without referring to a different coordinate system, the Cartesian system, when the substitution ##\hat{\mathbf{r}}(\theta) = \cos \theta \hat{\mathbf{i}} + \sin \theta \hat{\mathbf{j}}## is made?

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# I Polar Coordinates

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