Question

How is this transformation linear?
\[T(r,\theta)=(r,\theta+\phi)\]

Answer 1

\[T(r_1,\theta_1)+T(r_2,\theta_2)=(r_1+r_2,\theta_1+\theta_2+2\phi)\]
but
\[T(r_1+r_2,\theta_1+\theta_2)=(r_1+r_2,\theta_1+\theta_2+\phi)\]
what is wrong with this

Answer 2

r and theta are polar coordinates. You can't just add r's and theta's together. That transformation is simply a rotation by the angle phi.

Answer 3

How vectors are added in polar coordinates

Answer 4

It's considerably more complicated. I would advise just recognizing that that transformation can be represented by the matrix
\[\left[\begin{matrix}\cos(\phi) &-\sin(\phi) \\\sin(\phi) &\cos(\phi)\end{matrix}\right]\]

Answer 5

and invoking the knowledge that all matrices are linear transformations.

Answer 6

But if you must verify it, then you must add them component-wise. The vector with polar coordinates (r,theta) is, in component form,
\[ < r\cos(\theta), r\sin(\theta) > \]

Answer 7

I got it thanks

Answer 8

so applying the transformation yields
\[ < r\cos(\theta + \phi), r\sin(\theta + \phi) >

Answer 9

oops..
\[ < r\cos(\theta + \phi), r\sin(\theta + \phi) > \]