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}}}

Data Source : WIKIPEDIA
Number of Data columns : 4 Number of Data rows : 3
Categories : economy, demography, politics, knowledge

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Data row number No Name 0 Cartesian Cylindrical Spherical

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Structured data parsed from Wikipedia. Television Fox Sports covered their sixteenth race at the Bristol Motor Speedway. Mike Joy, five time Bristol winner Jeff Gordon and 12 time Bristol winner – and all time Bristol race winner – Darrell Waltrip will have the call in the booth for the race. Jamie Little, Chris Neville, Vince Welch and Matt Yocum will handle the pit road duties for the television side.

television, sports, fox, food, media

Armorial des communes de Lot-et-Garonne - S

From WIKIPEDIA

Structured data parsed from Wikipedia. S Saint Antoine de Ficalba D'argent au tau d'azur chargé sur sa branche transversale de trois besants d'or. 13 décembre 1988 Saint Barthélemy d'Agenais Écartelé : au premier de gueules au léopard d'or, armé et lampassé d'azur, au deuxième et au troisième tiercé en bande d'or, de gueules et d'azur, au quatrième de gueules au château donjonné d'or, ouvert du champ, maçonné de sable. Saint Hilaire de Lusignan Tiercé en pairle au 1) d’argent au lion de gueules couronné d’or, au 2) de gueules au serpent ondoyant en pal d’or, au 3) d’azur à la tierce ondée d’argent accompagnée, en pointe, d’une ancre avec sa gumène d’or.

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