Del in cylindrical and spherical coordinates - Unit vector conversions

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Unit vector conversions Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates destination x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ z ^ = z ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=cos varphi {hat {boldsymbol {rho }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin varphi {hat {boldsymbol {rho }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ z ^ = z ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=cos varphi {hat {boldsymbol {rho }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin varphi {hat {boldsymbol {rho }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ z ^ = z ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=cos varphi {hat {boldsymbol {rho }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin varphi {hat {boldsymbol {rho }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ z ^ = z ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=cos varphi {hat {boldsymbol {rho }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin varphi {hat {boldsymbol {rho }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ z ^ = z ^ x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ z ^ = z ^ x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ z ^ = z ^ x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ z ^ = z ^ x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ x ^ x ^ x ^ x ^ x x ^ = cos ⁡ φ ρ ^ − sin ⁡ φ φ ^ = cos ⁡ φ ρ ^ ρ ^ ρ ^ ρ ^ − sin ⁡ φ φ ^ φ ^ φ ^ φ ^ y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ y ^ y ^ y ^ y ^ y y ^ = sin ⁡ φ ρ ^ + cos ⁡ φ φ ^ = sin ⁡ φ ρ ^ ρ ^ ρ ^ ρ ^ + cos ⁡ φ φ ^ φ ^ φ ^ φ ^ z ^ = z ^ z ^ z ^ z ^ z ^ z z ^ = z ^ = z ^ z ^ z ^ z z ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=cos varphi {hat {boldsymbol {rho }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin varphi {hat {boldsymbol {rho }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=sin theta cos varphi {hat {mathbf {r} }}+cos theta cos varphi {hat {boldsymbol {theta }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin theta sin varphi {hat {mathbf {r} }}+cos theta sin varphi {hat {boldsymbol {theta }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=sin theta cos varphi {hat {mathbf {r} }}+cos theta cos varphi {hat {boldsymbol {theta }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin theta sin varphi {hat {mathbf {r} }}+cos theta sin varphi {hat {boldsymbol {theta }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=sin theta cos varphi {hat {mathbf {r} }}+cos theta cos varphi {hat {boldsymbol {theta }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin theta sin varphi {hat {mathbf {r} }}+cos theta sin varphi {hat {boldsymbol {theta }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=sin theta cos varphi {hat {mathbf {r} }}+cos theta cos varphi {hat {boldsymbol {theta }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin theta sin varphi {hat {mathbf {r} }}+cos theta sin varphi {hat {boldsymbol {theta }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ x ^ x ^ x ^ x ^ x x ^ = sin ⁡ θ cos ⁡ φ r ^ + cos ⁡ θ cos ⁡ φ θ ^ − sin ⁡ φ φ ^ = sin ⁡ θ cos ⁡ φ r ^ r ^ r ^ r r ^ + cos ⁡ θ cos ⁡ φ θ ^ θ ^ θ ^ θ ^ − sin ⁡ φ φ ^ φ ^ φ ^ φ ^ y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ y ^ y ^ y ^ y ^ y y ^ = sin ⁡ θ sin ⁡ φ r ^ + cos ⁡ θ sin ⁡ φ θ ^ + cos ⁡ φ φ ^ = sin ⁡ θ sin ⁡ φ r ^ r ^ r ^ r r ^ + cos ⁡ θ sin ⁡ φ θ ^ θ ^ θ ^ θ ^ + cos ⁡ φ φ ^ φ ^ φ ^ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ z ^ z ^ z ^ z ^ z z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ = cos ⁡ θ r ^ r ^ r ^ r r ^ − sin ⁡ θ θ ^ θ ^ θ ^ θ ^ {displaystyle {begin{aligned}{hat {mathbf {x} }}&=sin theta cos varphi {hat {mathbf {r} }}+cos theta cos varphi {hat {boldsymbol {theta }}} sin varphi {hat {boldsymbol {varphi }}}{hat {mathbf {y} }}&=sin theta sin varphi {hat {mathbf {r} }}+cos theta sin varphi {hat {boldsymbol {theta }}}+cos varphi {hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} ρ ^ = x x ^ + y y ^ x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 z ^ = z ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} ρ ^ = x x ^ + y y ^ x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 z ^ = z ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} ρ ^ = x x ^ + y y ^ x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 z ^ = z ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} ρ ^ = x x ^ + y y ^ x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 z ^ = z ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} ρ ^ = x x ^ + y y ^ x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 z ^ = z ^ ρ ^ = x x ^ + y y ^ x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 z ^ = z ^ ρ ^ = x x ^ + y y ^ x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 z ^ = z ^ ρ ^ = x x ^ + y y ^ x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 z ^ = z ^ ρ ^ = x x ^ + y y ^ x 2 + y 2 ρ ^ ρ ^ ρ ^ ρ ^ ρ ^ = x x ^ + y y ^ x 2 + y 2 = x x ^ + y y ^ x 2 + y 2 x x ^ + y y ^ x 2 + y 2 x x ^ + y y ^ x x ^ x ^ x ^ x x ^ + y y ^ y ^ y ^ y y ^ x 2 + y 2 x 2 x 2 2 + y 2 y 2 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 φ ^ φ ^ φ ^ φ ^ φ ^ = − y x ^ + x y ^ x 2 + y 2 = − y x ^ + x y ^ x 2 + y 2 − y x ^ + x y ^ x 2 + y 2 − y x ^ + x y ^ − y x ^ x ^ x ^ x x ^ + x y ^ y ^ y ^ y y ^ x 2 + y 2 x 2 x 2 2 + y 2 y 2 2 z ^ = z ^ z ^ z ^ z ^ z ^ z z ^ = z ^ = z ^ z ^ z ^ z z ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}{hat {mathbf {z} }}&={hat {mathbf {z} }}end{aligned}}} ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ φ ^ = φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&=sin theta {hat {mathbf {r} }}+cos theta {hat {boldsymbol {theta }}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ φ ^ = φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&=sin theta {hat {mathbf {r} }}+cos theta {hat {boldsymbol {theta }}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ φ ^ = φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&=sin theta {hat {mathbf {r} }}+cos theta {hat {boldsymbol {theta }}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ φ ^ = φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&=sin theta {hat {mathbf {r} }}+cos theta {hat {boldsymbol {theta }}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ φ ^ = φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ φ ^ = φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ φ ^ = φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ φ ^ = φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ ρ ^ ρ ^ ρ ^ ρ ^ ρ ^ = sin ⁡ θ r ^ + cos ⁡ θ θ ^ = sin ⁡ θ r ^ r ^ r ^ r r ^ + cos ⁡ θ θ ^ θ ^ θ ^ θ ^ φ ^ = φ ^ φ ^ φ ^ φ ^ φ ^ φ ^ = φ ^ = φ ^ φ ^ φ ^ φ ^ z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ z ^ z ^ z ^ z ^ z z ^ = cos ⁡ θ r ^ − sin ⁡ θ θ ^ = cos ⁡ θ r ^ r ^ r ^ r r ^ − sin ⁡ θ θ ^ θ ^ θ ^ θ ^ {displaystyle {begin{aligned}{hat {boldsymbol {rho }}}&=sin theta {hat {mathbf {r} }}+cos theta {hat {boldsymbol {theta }}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }} sin theta {hat {boldsymbol {theta }}}end{aligned}}} r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}+z{hat {mathbf {z} }}}{sqrt {x^{2}+y^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {left(x{hat {mathbf {x} }}+y{hat {mathbf {y} }}right)z left(x^{2}+y^{2}right){hat {mathbf {z} }}}{{sqrt {x^{2}+y^{2}+z^{2}}}{sqrt {x^{2}+y^{2}}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}end{aligned}}} r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}+z{hat {mathbf {z} }}}{sqrt {x^{2}+y^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {left(x{hat {mathbf {x} }}+y{hat {mathbf {y} }}right)z left(x^{2}+y^{2}right){hat {mathbf {z} }}}{{sqrt {x^{2}+y^{2}+z^{2}}}{sqrt {x^{2}+y^{2}}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}end{aligned}}} r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}+z{hat {mathbf {z} }}}{sqrt {x^{2}+y^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {left(x{hat {mathbf {x} }}+y{hat {mathbf {y} }}right)z left(x^{2}+y^{2}right){hat {mathbf {z} }}}{{sqrt {x^{2}+y^{2}+z^{2}}}{sqrt {x^{2}+y^{2}}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}end{aligned}}} r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}+z{hat {mathbf {z} }}}{sqrt {x^{2}+y^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {left(x{hat {mathbf {x} }}+y{hat {mathbf {y} }}right)z left(x^{2}+y^{2}right){hat {mathbf {z} }}}{{sqrt {x^{2}+y^{2}+z^{2}}}{sqrt {x^{2}+y^{2}}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}end{aligned}}} r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 r ^ r ^ r ^ r ^ r r ^ = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 = x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 x x ^ + y y ^ + z z ^ x 2 + y 2 + z 2 x x ^ + y y ^ + z z ^ x x ^ x ^ x ^ x x ^ + y y ^ y ^ y ^ y y ^ + z z ^ z ^ z ^ z z ^ x 2 + y 2 + z 2 x 2 x 2 2 + y 2 y 2 2 + z 2 z 2 2 θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 θ ^ θ ^ θ ^ θ ^ θ ^ = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 = ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ x 2 + y 2 + z 2 x 2 + y 2 ( x x ^ + y y ^ ) z − ( x 2 + y 2 ) z ^ ( x x ^ + y y ^ ) ( x x ^ + y y ^ x x ^ x ^ x ^ x x ^ + y y ^ y ^ y ^ y y ^ ) z − ( x 2 + y 2 ) ( x 2 + y 2 x 2 x 2 2 + y 2 y 2 2 ) z ^ z ^ z ^ z z ^ x 2 + y 2 + z 2 x 2 + y 2 x 2 + y 2 + z 2 x 2 + y 2 + z 2 x 2 x 2 2 + y 2 y 2 2 + z 2 z 2 2 x 2 + y 2 x 2 + y 2 x 2 x 2 2 + y 2 y 2 2 φ ^ = − y x ^ + x y ^ x 2 + y 2 φ ^ φ ^ φ ^ φ ^ φ ^ = − y x ^ + x y ^ x 2 + y 2 = − y x ^ + x y ^ x 2 + y 2 − y x ^ + x y ^ x 2 + y 2 − y x ^ + x y ^ − y x ^ x ^ x ^ x x ^ + x y ^ y ^ y ^ y y ^ x 2 + y 2 x 2 x 2 2 + y 2 y 2 2 {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}+z{hat {mathbf {z} }}}{sqrt {x^{2}+y^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {left(x{hat {mathbf {x} }}+y{hat {mathbf {y} }}right)z left(x^{2}+y^{2}right){hat {mathbf {z} }}}{{sqrt {x^{2}+y^{2}+z^{2}}}{sqrt {x^{2}+y^{2}}}}}{hat {boldsymbol {varphi }}}&={frac { y{hat {mathbf {x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}end{aligned}}} r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 φ ^ = φ ^ {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {rho {hat {boldsymbol {rho }}}+z{hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {z{hat {boldsymbol {rho }}} rho {hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}end{aligned}}} r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 φ ^ = φ ^ {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {rho {hat {boldsymbol {rho }}}+z{hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {z{hat {boldsymbol {rho }}} rho {hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}end{aligned}}} r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 φ ^ = φ ^ {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {rho {hat {boldsymbol {rho }}}+z{hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {z{hat {boldsymbol {rho }}} rho {hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}end{aligned}}} r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 φ ^ = φ ^ {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {rho {hat {boldsymbol {rho }}}+z{hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {z{hat {boldsymbol {rho }}} rho {hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}end{aligned}}} r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 φ ^ = φ ^ r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 φ ^ = φ ^ r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 φ ^ = φ ^ r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 φ ^ = φ ^ r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 r ^ r ^ r ^ r ^ r r ^ = ρ ρ ^ + z z ^ ρ 2 + z 2 = ρ ρ ^ + z z ^ ρ 2 + z 2 ρ ρ ^ + z z ^ ρ 2 + z 2 ρ ρ ^ + z z ^ ρ ρ ^ ρ ^ ρ ^ ρ ^ + z z ^ z ^ z ^ z z ^ ρ 2 + z 2 ρ 2 ρ 2 2 + z 2 z 2 2 θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 θ ^ θ ^ θ ^ θ ^ θ ^ = z ρ ^ − ρ z ^ ρ 2 + z 2 = z ρ ^ − ρ z ^ ρ 2 + z 2 z ρ ^ − ρ z ^ ρ 2 + z 2 z ρ ^ − ρ z ^ z ρ ^ ρ ^ ρ ^ ρ ^ − ρ z ^ z ^ z ^ z z ^ ρ 2 + z 2 ρ 2 ρ 2 2 + z 2 z 2 2 φ ^ = φ ^ φ ^ φ ^ φ ^ φ ^ φ ^ = φ ^ = φ ^ φ ^ φ ^ φ ^ {displaystyle {begin{aligned}{hat {mathbf {r} }}&={frac {rho {hat {boldsymbol {rho }}}+z{hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {theta }}}&={frac {z{hat {boldsymbol {rho }}} rho {hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}end{aligned}}}

Dataset

Data row number	No Name 0	Cartesian	Cylindrical	Spherical

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Data Columns

Name	Description	Data Type
No Name 0		text
Cartesian		text
Cylindrical		text
Spherical		text

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