# Polar coordinates

Dear All,

How do you derive both equations below. Let r be the position vector (rcos(θ), rsin(θ)), with r and θ depending on time t.

These equations can be found in wiki under polar coordinates.

jgens
Gold Member
This looks just like a staight-forward application of the chain and product rules . . .

Proof: $$\mathbf{r} = [rcos(\theta), rsin(\theta)]$$

$$\mathbf{\hat{r}} = [cos(\theta),sin(\theta)]$$

$$\mathbf{\hat{\theta}} = [-sin(\theta),cos(\theta)]$$

$$\mathbf{r} = rcos(\theta)\mathbf{e_1} + rsin(\theta)\mathbf{e_2}$$

$$\frac{d\mathbf{r}}{dt} = [\dot{r}cos(\theta) - rsin(\theta)\dot{\theta}]\mathbf{e_1} + [\dot{r}sin(\theta) + rcos(\theta)\dot{\theta}]\mathbf{e_2}$$

Expressing this in terms of our previously defined unit vectors we have that.

$$\frac{d\mathbf{r}}{dt} = \dot{r}\mathbf{\hat{r}} + r\dot{\theta}\mathbf{\hat{\theta}}$$

As desired. A similar method could probably be used to get the second result. This is probably a bit sloppy but I'm just learning vector calculus.

I'm new to vector calculus too. What does e1 and e2 mean?

jgens
Gold Member
They're unit vectors.

I'm new to vector calculus too. What does e1 and e2 mean?

The notation $$e_i$$ is often used for the i-th vector in the standard basis. So $$e_1$$ is the vector that points in the positive x direction, and $$e_2$$ to the positive y direction, etc.

Here's a good link I found on deriving those equations: http://mathworld.wolfram.com/PolarCoordinates.html" [Broken]

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jgens
Gold Member
Which is pretty much exactly what I showed!