<Li> Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose . </Li> <P> So the cartesian unit vectors are related to the spherical unit vectors by: </P> <Dl> <Dd> (x ^ y ^ z ^) = (sin ⁡ θ cos ⁡ φ cos ⁡ θ cos ⁡ φ − sin ⁡ φ sin ⁡ θ sin ⁡ φ cos ⁡ θ sin ⁡ φ cos ⁡ φ cos ⁡ θ − sin ⁡ θ 0) (ρ ^ θ ^ φ ^) (\ displaystyle (\ begin (bmatrix) \ mathbf (\ hat (x)) \ \ \ mathbf (\ hat (y)) \ \ \ mathbf (\ hat (z)) \ end (bmatrix)) = (\ begin (bmatrix) \ sin \ theta \ cos \ phi & \ cos \ theta \ cos \ phi & - \ sin \ phi \ \ \ sin \ theta \ sin \ phi & \ cos \ theta \ sin \ phi & \ cos \ phi \ \ \ cos \ theta & - \ sin \ theta &0 \ end (bmatrix)) (\ begin (bmatrix) (\ boldsymbol (\ hat (\ rho))) \ \ (\ boldsymbol (\ hat (\ theta))) \ \ (\ boldsymbol (\ hat (\ phi))) \ end (bmatrix))) </Dd> </Dl> <Dd> (x ^ y ^ z ^) = (sin ⁡ θ cos ⁡ φ cos ⁡ θ cos ⁡ φ − sin ⁡ φ sin ⁡ θ sin ⁡ φ cos ⁡ θ sin ⁡ φ cos ⁡ φ cos ⁡ θ − sin ⁡ θ 0) (ρ ^ θ ^ φ ^) (\ displaystyle (\ begin (bmatrix) \ mathbf (\ hat (x)) \ \ \ mathbf (\ hat (y)) \ \ \ mathbf (\ hat (z)) \ end (bmatrix)) = (\ begin (bmatrix) \ sin \ theta \ cos \ phi & \ cos \ theta \ cos \ phi & - \ sin \ phi \ \ \ sin \ theta \ sin \ phi & \ cos \ theta \ sin \ phi & \ cos \ phi \ \ \ cos \ theta & - \ sin \ theta &0 \ end (bmatrix)) (\ begin (bmatrix) (\ boldsymbol (\ hat (\ rho))) \ \ (\ boldsymbol (\ hat (\ theta))) \ \ (\ boldsymbol (\ hat (\ phi))) \ end (bmatrix))) </Dd>

Relation between unit vectors in cartesian and spherical coordinates