<P> The spherical unit vectors depend on both φ (\ displaystyle \ varphi) and θ (\ displaystyle \ theta), and hence there are 5 possible non-zero derivatives . For a more complete description, see Jacobian matrix and determinant . The non-zero derivatives are: </P> <Dl> <Dd> ∂ r ^ ∂ φ = − sin ⁡ θ sin ⁡ φ x ^ + sin ⁡ θ cos ⁡ φ y ^ = sin ⁡ θ φ ^ (\ displaystyle (\ frac (\ partial \ mathbf (\ hat (r))) (\ partial \ varphi)) = - \ sin \ theta \ sin \ varphi \ mathbf (\ hat (x)) + \ sin \ theta \ cos \ varphi \ mathbf (\ hat (y)) = \ sin \ theta (\ boldsymbol (\ hat (\ varphi)))) </Dd> </Dl> <Dd> ∂ r ^ ∂ φ = − sin ⁡ θ sin ⁡ φ x ^ + sin ⁡ θ cos ⁡ φ y ^ = sin ⁡ θ φ ^ (\ displaystyle (\ frac (\ partial \ mathbf (\ hat (r))) (\ partial \ varphi)) = - \ sin \ theta \ sin \ varphi \ mathbf (\ hat (x)) + \ sin \ theta \ cos \ varphi \ mathbf (\ hat (y)) = \ sin \ theta (\ boldsymbol (\ hat (\ varphi)))) </Dd> <Dl> <Dd> ∂ r ^ ∂ θ = cos ⁡ θ cos ⁡ φ x ^ + cos ⁡ θ sin ⁡ φ y ^ − sin ⁡ θ z ^ = θ ^ (\ displaystyle (\ frac (\ partial \ mathbf (\ hat (r))) (\ partial \ theta)) = \ cos \ theta \ cos \ varphi \ mathbf (\ hat (x)) + \ cos \ theta \ sin \ varphi \ mathbf (\ hat (y)) - \ sin \ theta \ mathbf (\ hat (z)) = (\ boldsymbol (\ hat (\ theta)))) </Dd> </Dl>

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