transformation between cartesian basis vectors and polar basis vectors

From the definition of a covariant vector (covariant tensor of rank 1)

¯ ∂xj- Ti = Tj ∂¯xi

(1)

the corresponding transformation matrix is

j Aij = ∂x-- ∂¯xi

(2)

In order to calculate the transformation matrix, we need the equations relating the two coordinates systems. For cartesian to polar, we have

∘ -------- r = x2 + y2

( ) 𝜃 = tan −1 y- x

and for polar to cartesian

x = rcos 𝜃

y = rsin𝜃

So if we designate (ê_x,ê_y) as the bar coordinates, then the transformation components from a polar basis vector (ê_r,ê_𝜃) to a cartesian basis vector (ê_x,ê_y) is calculted as

1 A = ∂x--= ∂r-= ∘---x----- 11 ∂¯x1 ∂x x2 + y2

∂x2- ∂𝜃- ---y---- A12 = ∂¯x1 = ∂x = − x2 + y2

∂x1 ∂r y A21 = ∂¯x2-= ∂y-= ∘--2----2- x + y

∂x2 ∂𝜃 x A22 = ----= ---= -------- ∂¯x2 ∂y x2 + y2

The components of cartesian basis vectors to polar basis vectors transform the same way, but now the polar coordinates have the bar

∂x1- ∂x- B11 = ∂ ¯x1 = ∂r = cos𝜃

∂x2 ∂y B12 = ∂-¯x1 = ∂r- = sin𝜃

∂x1- ∂x- B21 = ∂¯x2 = ∂𝜃 = − r sin 𝜃

∂x2- ∂y- B22 = ∂¯x2 = ∂𝜃 = r cos𝜃

In summary, the components of covariant basis vectors in cartesian coordinates and polar coordinates transform between each other according to

⌊ ⌋ [ ] √--x--- − -2y-2 [ ] eˆx = ⌈ x2+y2 x +y ⌉ ˆer eˆy √--y2--2 x2x+y2 ˆe𝜃 x +y

[ ] [ ] [ ] ˆer = cos𝜃 sin 𝜃 ˆex ˆe𝜃 − rsin 𝜃 rcos𝜃 ˆey

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