Calculating the metric in $\mathbb R^2$ in polar coordinates

Question

On this lecture the metric in polar coordinates in $\mathbb R^2$ is defined as

$$g = dx \otimes dx + dy \otimes dy = dr \otimes dr +r^2 d\theta \otimes d\theta$$

with

$$dx = \cos\theta dr -r\sin\theta d\theta$$

and

$$dy = \sin \theta dr + r \cos\theta d\theta$$

How is the first equation calculated?

I would have calculated, for example

$$dx \otimes dx$$

as

$$dx \otimes dx=(\cos\theta -r\sin\theta) (\cos\theta -r\sin\theta).$$

And $$dy\otimes dy$$

as

$$dy\otimes dy=(\sin \theta + r \cos\theta) (\sin \theta + r \cos\theta)$$

Answer 1

$$dx \otimes dx=(\cos\theta dr -r\sin\theta d\theta) \otimes(\cos\theta dr -r\sin\theta d\theta)$$

hence $$\big(\cos^2\theta \big)dr\otimes dr+\big(r^2\sin^2\theta\big)d\theta\otimes d\theta- r\cos\theta\sin\theta\big(dr\otimes d\theta+d\theta\otimes dr\big).$$

Similarly

$$dy \otimes dy=(\sin\theta dr -r\cos\theta d\theta)\otimes(\sin\theta dr -r\cos\theta d\theta)$$

hence

$$\big(\sin^2\theta \big)dr\otimes dr+\big(r^2\cos^2\theta\big)d\theta\otimes d\theta+ r\cos\theta\sin\theta\big(dr\otimes d\theta+d\theta\otimes dr\big).$$

Finally: $$dx\otimes dx+dy\otimes dy=\big(\cos^2\theta \big)dr\otimes dr+\big(r^2\sin^2\theta\big)d\theta\otimes d\theta+\\+\big(\sin^2\theta \big)dr\otimes dr+\big(r^2\cos^2\theta\big)d\theta\otimes d\theta$$

that is $$dx\otimes dx+dy\otimes dy=dr\otimes dr+r^2d\theta\otimes d\theta$$ in polar coordinates.