<Dl> <Dd> ∂ z ^ ∂ φ = 0 . (\ displaystyle (\ frac (\ partial \ mathbf (\ hat (z))) (\ partial \ varphi)) = \ mathbf (0).) </Dd> </Dl> <Dd> ∂ z ^ ∂ φ = 0 . (\ displaystyle (\ frac (\ partial \ mathbf (\ hat (z))) (\ partial \ varphi)) = \ mathbf (0).) </Dd> <P> The unit vectors appropriate to spherical symmetry are: r ^ (\ displaystyle \ mathbf (\ hat (r))), the direction in which the radial distance from the origin increases; φ ^ (\ displaystyle (\ boldsymbol (\ hat (\ varphi)))), the direction in which the angle in the x-y plane counterclockwise from the positive x-axis is increasing; and θ ^ (\ displaystyle (\ boldsymbol (\ hat (\ theta)))), the direction in which the angle from the positive z axis is increasing . To minimize redundancy of representations, the polar angle θ (\ displaystyle \ theta) is usually taken to lie between zero and 180 degrees . It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of φ ^ (\ displaystyle (\ boldsymbol (\ hat (\ varphi)))) and θ ^ (\ displaystyle (\ boldsymbol (\ hat (\ theta)))) are often reversed . Here, the American "physics" convention is used . This leaves the azimuthal angle φ (\ displaystyle \ varphi) defined the same as in cylindrical coordinates . The Cartesian relations are: </P> <Dl> <Dd> r ^ = sin ⁡ θ cos ⁡ φ x ^ + sin ⁡ θ sin ⁡ φ y ^ + cos ⁡ θ z ^ (\ displaystyle \ mathbf (\ hat (r)) = \ sin \ theta \ cos \ varphi \ mathbf (\ hat (x)) + \ sin \ theta \ sin \ varphi \ mathbf (\ hat (y)) + \ cos \ theta \ mathbf (\ hat (z))) </Dd> </Dl>

Which unit vector points in the positive direction of the z axis
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