transformation between cartesian coordinates and polar coordinates

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

¯i j ∂¯xi T = T ∂xj

(1)

we get the transformation matrix from the partial derivatives

i Aij = ∂¯x-- ∂xj

(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 (x,y) as the bar coordinates, then the transformation components from polar coordinates (r,𝜃) to cartesian coordinates (x,y) is calculted as

1 A11 = ∂-¯x- = ∂x- = cos𝜃 ∂x1 ∂r

∂¯x1 ∂x A12 = ----= ---= − r sin 𝜃 ∂x2 ∂𝜃

∂ ¯x2 ∂y A21 = ∂x1- = ∂r- = sin𝜃

∂¯x2- ∂y- A22 = ∂x2 = ∂𝜃 = r cos𝜃

The components from cartesian coordinates to polar coordinates transform the same way, but now the polar coordinates have the bar

∂¯x1 ∂r x B11 = --1-= ---= ∘--2----2- ∂x ∂x x + y

∂¯x1 ∂r y B12 = ∂x2-= ∂y-= ∘--2----2- x + y

∂¯x2 ∂𝜃 y B21 = ----= ---= − -------- ∂x1 ∂x x2 + y2

2 B22 = ∂¯x--= ∂𝜃-= ---x---- ∂x2 ∂y x2 + y2

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

[ ] [ ][ ] x cos𝜃 − r sin 𝜃 r y = sin𝜃 r cos𝜃 𝜃

[ ] [ √--x--- √--y--- ][ ] r = x2+y2 x2+y2 x 𝜃 − x2y+y2 x2x+y2 y

"transformation between cartesian coordinates and polar coordinates" is owned by bloftin.
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