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Del in cylindrical and spherical coordinates
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<h2 id="notes">Notes</h2> <ul><li>This article uses the standard notation <a href="/facts/ISO_80000-2/k1nrIjMW">ISO 80000-2</a>, which supersedes <a href="/facts/ISO_31-11/IjFTjgw4">ISO 31-11</a>, for <a href="/facts/Spherical_coordinate_system/5cVnNof3">spherical coordinates</a> (other sources may reverse the definitions of <i>θ</i> and <i>φ</i>): <ul><li>The polar angle is denoted by θ ∈ [ 0 , π ] {\displaystyle \theta \in [0,\pi ]} : it is the angle between the <i>z</i>-axis and the radial vector connecting the origin to the point in question.</li> <li>The azimuthal angle is denoted by φ ∈ [ 0 , 2 π ] {\displaystyle \varphi \in [0,2\pi ]} : it is the angle between the <i>x</i>-axis and the projection of the radial vector onto the <i>xy</i>-plane.</li></ul></li> <li>The function <a href="/facts/Atan2/0JDnUYfP">atan2</a>(<i>y</i>, <i>x</i>) can be used instead of the mathematical function <a href="/facts/Arctan/bVwU9FPG">arctan</a>(<i>y</i>/<i>x</i>) owing to its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> and <a href="/facts/Image_(mathematics)/KANefMAl">image</a>. The classical arctan function has an image of (−π/2, +π/2), whereas atan2 is defined to have an image of (−π, π].</li></ul> <h2 id="coordinate-conversions">Coordinate conversions</h2> Conversion between Cartesian, cylindrical, and spherical coordinates<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><table><tbody><tr><th colspan="2" rowspan="2"></th><th colspan="3">From</th></tr><tr><th>Cartesian</th><th>Cylindrical</th><th>Spherical</th></tr><tr><th rowspan="3">To</th><th>Cartesian</th><td> x = x y = y z = z {\displaystyle {\begin{aligned}x&=x\\y&=y\\z&=z\\\end{aligned}}} </td><td> x = ρ cos φ y = ρ sin φ z = z {\displaystyle {\begin{aligned}x&=\rho \cos \varphi \\y&=\rho \sin \varphi \\z&=z\end{aligned}}} </td><td> x = r sin θ cos φ y = r sin θ sin φ z = r cos θ {\displaystyle {\begin{aligned}x&=r\sin \theta \cos \varphi \\y&=r\sin \theta \sin \varphi \\z&=r\cos \theta \\\end{aligned}}} </td></tr><tr><th>Cylindrical</th><td> ρ = x 2 + y 2 φ = arctan ( y x ) z = z {\displaystyle {\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\varphi &=\arctan \left({\frac {y}{x}}\right)\\z&=z\end{aligned}}} </td><td> ρ = ρ φ = φ z = z {\displaystyle {\begin{aligned}\rho &=\rho \\\varphi &=\varphi \\z&=z\\\end{aligned}}} </td><td> ρ = r sin θ φ = φ z = r cos θ {\displaystyle {\begin{aligned}\rho &=r\sin \theta \\\varphi &=\varphi \\z&=r\cos \theta \end{aligned}}} </td></tr><tr><th>Spherical</th><td> r = x 2 + y 2 + z 2 θ = arctan ( x 2 + y 2 z ) φ = arctan ( y x ) {\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arctan \left({\frac {\sqrt {x^{2}+y^{2}}}{z}}\right)\\\varphi &=\arctan \left({\frac {y}{x}}\right)\end{aligned}}} </td><td> r = ρ 2 + z 2 θ = arctan ( ρ z ) φ = φ {\displaystyle {\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {\left({\frac {\rho }{z}}\right)}\\\varphi &=\varphi \end{aligned}}} </td><td> r = r θ = θ φ = φ {\displaystyle {\begin{aligned}r&=r\\\theta &=\theta \\\varphi &=\varphi \end{aligned}}} </td></tr></tbody></table> <p>Note that the operation arctan ( A B ) {\displaystyle \arctan \left({\frac {A}{B}}\right)} must be interpreted as the two-argument inverse tangent, <a href="/facts/Atan2/0JDnUYfP">atan2</a>. </p> <h2 id="unit-vector-conversions">Unit vector conversions</h2> Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of <i>destination</i> coordinates<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><table><tbody><tr><th></th><th>Cartesian</th><th>Cylindrical</th><th>Spherical</th></tr><tr><th>Cartesian</th><td> x ^ = x ^ y ^ = y ^ z ^ = z ^ {\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&={\hat {\mathbf {x} }}\\{\hat {\mathbf {y} }}&={\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\\\end{aligned}}} </td><td> 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}}} </td><td> 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}}} </td></tr><tr><th>Cylindrical</th><td> 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}}} </td><td> ρ ^ = ρ ^ φ ^ = φ ^ z ^ = z ^ {\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\hat {\boldsymbol {\rho }}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\\\end{aligned}}} </td><td> 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}}} </td></tr><tr><th>Spherical</th><td> 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}}} </td><td> 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}}} </td><td> r ^ = r ^ θ ^ = θ ^ φ ^ = φ ^ {\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&={\hat {\mathbf {r} }}\\{\hat {\boldsymbol {\theta }}}&={\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\\end{aligned}}} </td></tr></tbody></table> Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of <i>source</i> coordinates<table><tbody><tr><th></th><th>Cartesian</th><th>Cylindrical</th><th>Spherical</th></tr><tr><th>Cartesian</th><td> x ^ = x ^ y ^ = y ^ z ^ = z ^ {\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&={\hat {\mathbf {x} }}\\{\hat {\mathbf {y} }}&={\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\\\end{aligned}}} </td><td> x ^ = x ρ ^ − y φ ^ x 2 + y 2 y ^ = y ρ ^ + x φ ^ x 2 + y 2 z ^ = z ^ {\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&={\frac {x{\hat {\boldsymbol {\rho }}}-y{\hat {\boldsymbol {\varphi }}}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {y} }}&={\frac {y{\hat {\boldsymbol {\rho }}}+x{\hat {\boldsymbol {\varphi }}}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}} </td><td> x ^ = x ( x 2 + y 2 r ^ + z θ ^ ) − y x 2 + y 2 + z 2 φ ^ x 2 + y 2 x 2 + y 2 + z 2 y ^ = y ( x 2 + y 2 r ^ + z θ ^ ) + x x 2 + y 2 + z 2 φ ^ x 2 + y 2 x 2 + y 2 + z 2 z ^ = z r ^ − x 2 + y 2 θ ^ x 2 + y 2 + z 2 {\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&={\frac {x\left({\sqrt {x^{2}+y^{2}}}{\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}\right)-y{\sqrt {x^{2}+y^{2}+z^{2}}}{\hat {\boldsymbol {\varphi }}}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\mathbf {y} }}&={\frac {y\left({\sqrt {x^{2}+y^{2}}}{\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}\right)+x{\sqrt {x^{2}+y^{2}+z^{2}}}{\hat {\boldsymbol {\varphi }}}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\mathbf {z} }}&={\frac {z{\hat {\mathbf {r} }}-{\sqrt {x^{2}+y^{2}}}{\hat {\boldsymbol {\theta }}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\end{aligned}}} </td></tr><tr><th>Cylindrical</th><td> ρ ^ = cos φ x ^ + sin φ y ^ φ ^ = − sin φ x ^ + cos φ y ^ z ^ = z ^ {\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\\{\hat {\boldsymbol {\varphi }}}&=-\sin \varphi {\hat {\mathbf {x} }}+\cos \varphi {\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}} </td><td> ρ ^ = ρ ^ φ ^ = φ ^ z ^ = z ^ {\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\hat {\boldsymbol {\rho }}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\\\end{aligned}}} </td><td> ρ ^ = ρ r ^ + z θ ^ ρ 2 + z 2 φ ^ = φ ^ z ^ = z r ^ − ρ θ ^ ρ 2 + z 2 {\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {\rho {\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&={\frac {z{\hat {\mathbf {r} }}-\rho {\hat {\boldsymbol {\theta }}}}{\sqrt {\rho ^{2}+z^{2}}}}\end{aligned}}} </td></tr><tr><th>Spherical</th><td> r ^ = sin θ ( cos φ x ^ + sin φ y ^ ) + cos θ z ^ θ ^ = cos θ ( cos φ x ^ + sin φ y ^ ) − sin θ z ^ φ ^ = − sin φ x ^ + cos φ y ^ {\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta \left(\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right)+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }}}&=\cos \theta \left(\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right)-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}&=-\sin \varphi {\hat {\mathbf {x} }}+\cos \varphi {\hat {\mathbf {y} }}\end{aligned}}} </td><td> r ^ = sin θ ρ ^ + cos θ z ^ θ ^ = cos θ ρ ^ − sin θ z ^ φ ^ = φ ^ {\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta {\hat {\boldsymbol {\rho }}}+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }}}&=\cos \theta {\hat {\boldsymbol {\rho }}}-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\end{aligned}}} </td><td> r ^ = r ^ θ ^ = θ ^ φ ^ = φ ^ {\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&={\hat {\mathbf {r} }}\\{\hat {\boldsymbol {\theta }}}&={\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\\end{aligned}}} </td></tr></tbody></table> <h2 id="del-formula">Del formula</h2> Table with the <a href="/facts/Del/0xEmQeBo">del</a> operator in cartesian, cylindrical and spherical coordinates<table><tbody><tr><th>Operation</th><th><a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> (<i>x</i>, <i>y</i>, <i>z</i>)</th><th><a href="/facts/Cylindrical_coordinates/1Vc6lhxD">Cylindrical coordinates</a> (<i>ρ</i>, <i>φ</i>, <i>z</i>)</th><th><a href="/facts/Spherical_coordinates/5cVnNof3">Spherical coordinates</a> (<i>r</i>, <i>θ</i>, <i>φ</i>),where <i>θ</i> is the polar angle and <i>φ</i> is the azimuthal angleα</th></tr><tr><th><a href="/facts/Vector_field/ccFNuPPp">Vector field</a> A</th><td> A x x ^ + A y y ^ + A z z ^ {\displaystyle A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}{\hat {\mathbf {z} }}} </td><td> A ρ ρ ^ + A φ φ ^ + A z z ^ {\displaystyle A_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\varphi }{\hat {\boldsymbol {\varphi }}}+A_{z}{\hat {\mathbf {z} }}} </td><td> A r r ^ + A θ θ ^ + A φ φ ^ {\displaystyle A_{r}{\hat {\mathbf {r} }}+A_{\theta }{\hat {\boldsymbol {\theta }}}+A_{\varphi }{\hat {\boldsymbol {\varphi }}}} </td></tr><tr><th><a href="/facts/Gradient/TsR7uukj">Gradient</a> ∇<i>f</i><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></th><td> ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^ {\displaystyle {\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}} </td><td> ∂ f ∂ ρ ρ ^ + 1 ρ ∂ f ∂ φ φ ^ + ∂ f ∂ z z ^ {\displaystyle {\partial f \over \partial \rho }{\hat {\boldsymbol {\rho }}}+{1 \over \rho }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}} </td><td> ∂ f ∂ r r ^ + 1 r ∂ f ∂ θ θ ^ + 1 r sin θ ∂ f ∂ φ φ ^ {\displaystyle {\partial f \over \partial r}{\hat {\mathbf {r} }}+{1 \over r}{\partial f \over \partial \theta }{\hat {\boldsymbol {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}}} </td></tr><tr><th><a href="/facts/Divergence/zpwwkFoU">Divergence</a> ∇ ⋅ A<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></th><td> ∂ A x ∂ x + ∂ A y ∂ y + ∂ A z ∂ z {\displaystyle {\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}} </td><td> 1 ρ ∂ ( ρ A ρ ) ∂ ρ + 1 ρ ∂ A φ ∂ φ + ∂ A z ∂ z {\displaystyle {1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z}} </td><td> 1 r 2 ∂ ( r 2 A r ) ∂ r + 1 r sin θ ∂ ∂ θ ( A θ sin θ ) + 1 r sin θ ∂ A φ ∂ φ {\displaystyle {1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\varphi } \over \partial \varphi }} </td></tr><tr><th><a href="/facts/Curl_(mathematics)/GuS8dc7o">Curl</a> ∇ × A<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></th><td> ( ∂ A z ∂ y − ∂ A y ∂ z ) x ^ + ( ∂ A x ∂ z − ∂ A z ∂ x ) y ^ + ( ∂ A y ∂ x − ∂ A x ∂ y ) z ^ {\displaystyle {\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)&{\hat {\mathbf {x} }}\\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)&{\hat {\mathbf {y} }}\\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)&{\hat {\mathbf {z} }}\end{aligned}}} </td><td> ( 1 ρ ∂ A z ∂ φ − ∂ A φ ∂ z ) ρ ^ + ( ∂ A ρ ∂ z − ∂ A z ∂ ρ ) φ ^ + 1 ρ ( ∂ ( ρ A φ ) ∂ ρ − ∂ A ρ ∂ φ ) z ^ {\displaystyle {\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right)&{\hat {\boldsymbol {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\hat {\boldsymbol {\varphi }}}\\+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\varphi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \varphi }}\right)&{\hat {\mathbf {z} }}\end{aligned}}} </td><td> 1 r sin θ ( ∂ ∂ θ ( A φ sin θ ) − ∂ A θ ∂ φ ) r ^ + 1 r ( 1 sin θ ∂ A r ∂ φ − ∂ ∂ r ( r A φ ) ) θ ^ + 1 r ( ∂ ∂ r ( r A θ ) − ∂ A r ∂ θ ) φ ^ {\displaystyle {\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\varphi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&{\hat {\mathbf {r} }}\\{}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {\partial }{\partial r}}\left(rA_{\varphi }\right)\right)&{\hat {\boldsymbol {\theta }}}\\{}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}} </td></tr><tr><th><a href="/facts/Laplace_operator/kov9u3PT">Laplace operator</a> ∇2<i>f</i> ≡ ∆<i>f</i><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></th><td> ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 {\displaystyle {\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}} </td><td> 1 ρ ∂ ∂ ρ ( ρ ∂ f ∂ ρ ) + 1 ρ 2 ∂ 2 f ∂ φ 2 + ∂ 2 f ∂ z 2 {\displaystyle {1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \varphi ^{2}}+{\partial ^{2}f \over \partial z^{2}}} </td><td> 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin θ ∂ ∂ θ ( sin θ ∂ f ∂ θ ) + 1 r 2 sin 2 θ ∂ 2 f ∂ φ 2 {\displaystyle {1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}} </td></tr><tr><th><a href="/facts/Vector_calculus_identities/QWAaYA7t">Vector gradient</a> ∇Aβ</th><td> ∂ A x ∂ x x ^ ⊗ x ^ + ∂ A x ∂ y x ^ ⊗ y ^ + ∂ A x ∂ z x ^ ⊗ z ^ + ∂ A y ∂ x y ^ ⊗ x ^ + ∂ A y ∂ y y ^ ⊗ y ^ + ∂ A y ∂ z y ^ ⊗ z ^ + ∂ A z ∂ x z ^ ⊗ x ^ + ∂ A z ∂ y z ^ ⊗ y ^ + ∂ A z ∂ z z ^ ⊗ z ^ {\displaystyle {\begin{aligned}{}&{\frac {\partial A_{x}}{\partial x}}{\hat {\mathbf {x} }}\otimes {\hat {\mathbf {x} }}+{\frac {\partial A_{x}}{\partial y}}{\hat {\mathbf {x} }}\otimes {\hat {\mathbf {y} }}+{\frac {\partial A_{x}}{\partial z}}{\hat {\mathbf {x} }}\otimes {\hat {\mathbf {z} }}\\{}+&{\frac {\partial A_{y}}{\partial x}}{\hat {\mathbf {y} }}\otimes {\hat {\mathbf {x} }}+{\frac {\partial A_{y}}{\partial y}}{\hat {\mathbf {y} }}\otimes {\hat {\mathbf {y} }}+{\frac {\partial A_{y}}{\partial z}}{\hat {\mathbf {y} }}\otimes {\hat {\mathbf {z} }}\\{}+&{\frac {\partial A_{z}}{\partial x}}{\hat {\mathbf {z} }}\otimes {\hat {\mathbf {x} }}+{\frac {\partial A_{z}}{\partial y}}{\hat {\mathbf {z} }}\otimes {\hat {\mathbf {y} }}+{\frac {\partial A_{z}}{\partial z}}{\hat {\mathbf {z} }}\otimes {\hat {\mathbf {z} }}\end{aligned}}} </td><td> ∂ A ρ ∂ ρ ρ ^ ⊗ ρ ^ + ( 1 ρ ∂ A ρ ∂ φ − A φ ρ ) ρ ^ ⊗ φ ^ + ∂ A ρ ∂ z ρ ^ ⊗ z ^ + ∂ A φ ∂ ρ φ ^ ⊗ ρ ^ + ( 1 ρ ∂ A φ ∂ φ + A ρ ρ ) φ ^ ⊗ φ ^ + ∂ A φ ∂ z φ ^ ⊗ z ^ + ∂ A z ∂ ρ z ^ ⊗ ρ ^ + 1 ρ ∂ A z ∂ φ z ^ ⊗ φ ^ + ∂ A z ∂ z z ^ ⊗ z ^ {\displaystyle {\begin{aligned}{}&{\frac {\partial A_{\rho }}{\partial \rho }}{\hat {\boldsymbol {\rho }}}\otimes {\hat {\boldsymbol {\rho }}}+\left({\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \varphi }}-{\frac {A_{\varphi }}{\rho }}\right){\hat {\boldsymbol {\rho }}}\otimes {\hat {\boldsymbol {\varphi }}}+{\frac {\partial A_{\rho }}{\partial z}}{\hat {\boldsymbol {\rho }}}\otimes {\hat {\mathbf {z} }}\\{}+&{\frac {\partial A_{\varphi }}{\partial \rho }}{\hat {\boldsymbol {\varphi }}}\otimes {\hat {\boldsymbol {\rho }}}+\left({\frac {1}{\rho }}{\frac {\partial A_{\varphi }}{\partial \varphi }}+{\frac {A_{\rho }}{\rho }}\right){\hat {\boldsymbol {\varphi }}}\otimes {\hat {\boldsymbol {\varphi }}}+{\frac {\partial A_{\varphi }}{\partial z}}{\hat {\boldsymbol {\varphi }}}\otimes {\hat {\mathbf {z} }}\\{}+&{\frac {\partial A_{z}}{\partial \rho }}{\hat {\mathbf {z} }}\otimes {\hat {\boldsymbol {\rho }}}+{\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}{\hat {\mathbf {z} }}\otimes {\hat {\boldsymbol {\varphi }}}+{\frac {\partial A_{z}}{\partial z}}{\hat {\mathbf {z} }}\otimes {\hat {\mathbf {z} }}\end{aligned}}} </td><td> ∂ A r ∂ r r ^ ⊗ r ^ + ( 1 r ∂ A r ∂ θ − A θ r ) r ^ ⊗ θ ^ + ( 1 r sin θ ∂ A r ∂ φ − A φ r ) r ^ ⊗ φ ^ + ∂ A θ ∂ r θ ^ ⊗ r ^ + ( 1 r ∂ A θ ∂ θ + A r r ) θ ^ ⊗ θ ^ + ( 1 r sin θ ∂ A θ ∂ φ − cot θ A φ r ) θ ^ ⊗ φ ^ + ∂ A φ ∂ r φ ^ ⊗ r ^ + 1 r ∂ A φ ∂ θ φ ^ ⊗ θ ^ + ( 1 r sin θ ∂ A φ ∂ φ + cot θ A θ r + A r r ) φ ^ ⊗ φ ^ {\displaystyle {\begin{aligned}{}&{\frac {\partial A_{r}}{\partial r}}{\hat {\mathbf {r} }}\otimes {\hat {\mathbf {r} }}+\left({\frac {1}{r}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {A_{\theta }}{r}}\right){\hat {\mathbf {r} }}\otimes {\hat {\boldsymbol {\theta }}}+\left({\frac {1}{r\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {A_{\varphi }}{r}}\right){\hat {\mathbf {r} }}\otimes {\hat {\boldsymbol {\varphi }}}\\{}+&{\frac {\partial A_{\theta }}{\partial r}}{\hat {\boldsymbol {\theta }}}\otimes {\hat {\mathbf {r} }}+\left({\frac {1}{r}}{\frac {\partial A_{\theta }}{\partial \theta }}+{\frac {A_{r}}{r}}\right){\hat {\boldsymbol {\theta }}}\otimes {\hat {\boldsymbol {\theta }}}+\left({\frac {1}{r\sin \theta }}{\frac {\partial A_{\theta }}{\partial \varphi }}-\cot \theta {\frac {A_{\varphi }}{r}}\right){\hat {\boldsymbol {\theta }}}\otimes {\hat {\boldsymbol {\varphi }}}\\{}+&{\frac {\partial A_{\varphi }}{\partial r}}{\hat {\boldsymbol {\varphi }}}\otimes {\hat {\mathbf {r} }}+{\frac {1}{r}}{\frac {\partial A_{\varphi }}{\partial \theta }}{\hat {\boldsymbol {\varphi }}}\otimes {\hat {\boldsymbol {\theta }}}+\left({\frac {1}{r\sin \theta }}{\frac {\partial A_{\varphi }}{\partial \varphi }}+\cot \theta {\frac {A_{\theta }}{r}}+{\frac {A_{r}}{r}}\right){\hat {\boldsymbol {\varphi }}}\otimes {\hat {\boldsymbol {\varphi }}}\end{aligned}}} </td></tr><tr><th><a href="/facts/Vector_Laplacian/kov9u3PT">Vector Laplacian</a> ∇2A ≡ ∆A<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></th><td> ∇ 2 A x x ^ + ∇ 2 A y y ^ + ∇ 2 A z z ^ {\displaystyle \nabla ^{2}A_{x}{\hat {\mathbf {x} }}+\nabla ^{2}A_{y}{\hat {\mathbf {y} }}+\nabla ^{2}A_{z}{\hat {\mathbf {z} }}} </td><td><p> ( ∇ 2 A ρ − A ρ ρ 2 − 2 ρ 2 ∂ A φ ∂ φ ) ρ ^ + ( ∇ 2 A φ − A φ ρ 2 + 2 ρ 2 ∂ A ρ ∂ φ ) φ ^ + ∇ 2 A z z ^ {\displaystyle {\begin{aligned}{\mathopen {}}\left(\nabla ^{2}A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right){\mathclose {}}&{\hat {\boldsymbol {\rho }}}\\+{\mathopen {}}\left(\nabla ^{2}A_{\varphi }-{\frac {A_{\varphi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\mathclose {}}&{\hat {\boldsymbol {\varphi }}}\\{}+\nabla ^{2}A_{z}&{\hat {\mathbf {z} }}\end{aligned}}} </p></td><td><p> ( ∇ 2 A r − 2 A r r 2 − 2 r 2 sin θ ∂ ( A θ sin θ ) ∂ θ − 2 r 2 sin θ ∂ A φ ∂ φ ) r ^ + ( ∇ 2 A θ − A θ r 2 sin 2 θ + 2 r 2 ∂ A r ∂ θ − 2 cos θ r 2 sin 2 θ ∂ A φ ∂ φ ) θ ^ + ( ∇ 2 A φ − A φ r 2 sin 2 θ + 2 r 2 sin θ ∂ A r ∂ φ + 2 cos θ r 2 sin 2 θ ∂ A θ ∂ φ ) φ ^ {\displaystyle {\begin{aligned}\left(\nabla ^{2}A_{r}-{\frac {2A_{r}}{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial \left(A_{\theta }\sin \theta \right)}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right)&{\hat {\mathbf {r} }}\\+\left(\nabla ^{2}A_{\theta }-{\frac {A_{\theta }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right)&{\hat {\boldsymbol {\theta }}}\\+\left(\nabla ^{2}A_{\varphi }-{\frac {A_{\varphi }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}} </p></td></tr><tr><th><a href="/facts/Directional_derivative/Fjc3JSVn">Directional derivative</a> (A ⋅ ∇)B<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></th><td> A ⋅ ∇ B x x ^ + A ⋅ ∇ B y y ^ + A ⋅ ∇ B z z ^ {\displaystyle \mathbf {A} \cdot \nabla B_{x}{\hat {\mathbf {x} }}+\mathbf {A} \cdot \nabla B_{y}{\hat {\mathbf {y} }}+\mathbf {A} \cdot \nabla B_{z}{\hat {\mathbf {z} }}} </td><td> ( A ρ ∂ B ρ ∂ ρ + A φ ρ ∂ B ρ ∂ φ + A z ∂ B ρ ∂ z − A φ B φ ρ ) ρ ^ + ( A ρ ∂ B φ ∂ ρ + A φ ρ ∂ B φ ∂ φ + A z ∂ B φ ∂ z + A φ B ρ ρ ) φ ^ + ( A ρ ∂ B z ∂ ρ + A φ ρ ∂ B z ∂ φ + A z ∂ B z ∂ z ) z ^ {\displaystyle {\begin{aligned}\left(A_{\rho }{\frac {\partial B_{\rho }}{\partial \rho }}+{\frac {A_{\varphi }}{\rho }}{\frac {\partial B_{\rho }}{\partial \varphi }}+A_{z}{\frac {\partial B_{\rho }}{\partial z}}-{\frac {A_{\varphi }B_{\varphi }}{\rho }}\right)&{\hat {\boldsymbol {\rho }}}\\+\left(A_{\rho }{\frac {\partial B_{\varphi }}{\partial \rho }}+{\frac {A_{\varphi }}{\rho }}{\frac {\partial B_{\varphi }}{\partial \varphi }}+A_{z}{\frac {\partial B_{\varphi }}{\partial z}}+{\frac {A_{\varphi }B_{\rho }}{\rho }}\right)&{\hat {\boldsymbol {\varphi }}}\\+\left(A_{\rho }{\frac {\partial B_{z}}{\partial \rho }}+{\frac {A_{\varphi }}{\rho }}{\frac {\partial B_{z}}{\partial \varphi }}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right)&{\hat {\mathbf {z} }}\end{aligned}}} </td><td><p> ( A r ∂ B r ∂ r + A θ r ∂ B r ∂ θ + A φ r sin θ ∂ B r ∂ φ − A θ B θ + A φ B φ r ) r ^ + ( A r ∂ B θ ∂ r + A θ r ∂ B θ ∂ θ + A φ r sin θ ∂ B θ ∂ φ + A θ B r r − A φ B φ cot θ r ) θ ^ + ( A r ∂ B φ ∂ r + A θ r ∂ B φ ∂ θ + A φ r sin θ ∂ B φ ∂ φ + A φ B r r + A φ B θ cot θ r ) φ ^ {\displaystyle {\begin{aligned}\left(A_{r}{\frac {\partial B_{r}}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{r}}{\partial \theta }}+{\frac {A_{\varphi }}{r\sin \theta }}{\frac {\partial B_{r}}{\partial \varphi }}-{\frac {A_{\theta }B_{\theta }+A_{\varphi }B_{\varphi }}{r}}\right)&{\hat {\mathbf {r} }}\\+\left(A_{r}{\frac {\partial B_{\theta }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\theta }}{\partial \theta }}+{\frac {A_{\varphi }}{r\sin \theta }}{\frac {\partial B_{\theta }}{\partial \varphi }}+{\frac {A_{\theta }B_{r}}{r}}-{\frac {A_{\varphi }B_{\varphi }\cot \theta }{r}}\right)&{\hat {\boldsymbol {\theta }}}\\+\left(A_{r}{\frac {\partial B_{\varphi }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\varphi }}{\partial \theta }}+{\frac {A_{\varphi }}{r\sin \theta }}{\frac {\partial B_{\varphi }}{\partial \varphi }}+{\frac {A_{\varphi }B_{r}}{r}}+{\frac {A_{\varphi }B_{\theta }\cot \theta }{r}}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}} </p></td></tr><tr><th><a href="/facts/Divergence/zpwwkFoU">Tensor divergence</a> ∇ ⋅ Tγ</th><td><p> ( ∂ T x x ∂ x + ∂ T y x ∂ y + ∂ T z x ∂ z ) x ^ + ( ∂ T x y ∂ x + ∂ T y y ∂ y + ∂ T z y ∂ z ) y ^ + ( ∂ T x z ∂ x + ∂ T y z ∂ y + ∂ T z z ∂ z ) z ^ {\displaystyle {\begin{aligned}\left({\frac {\partial T_{xx}}{\partial x}}+{\frac {\partial T_{yx}}{\partial y}}+{\frac {\partial T_{zx}}{\partial z}}\right)&{\hat {\mathbf {x} }}\\+\left({\frac {\partial T_{xy}}{\partial x}}+{\frac {\partial T_{yy}}{\partial y}}+{\frac {\partial T_{zy}}{\partial z}}\right)&{\hat {\mathbf {y} }}\\+\left({\frac {\partial T_{xz}}{\partial x}}+{\frac {\partial T_{yz}}{\partial y}}+{\frac {\partial T_{zz}}{\partial z}}\right)&{\hat {\mathbf {z} }}\end{aligned}}} </p></td><td><p> [ ∂ T ρ ρ ∂ ρ + 1 ρ ∂ T φ ρ ∂ φ + ∂ T z ρ ∂ z + 1 ρ ( T ρ ρ − T φ φ ) ] ρ ^ + [ ∂ T ρ φ ∂ ρ + 1 ρ ∂ T φ φ ∂ φ + ∂ T z φ ∂ z + 1 ρ ( T ρ φ + T φ ρ ) ] φ ^ + [ ∂ T ρ z ∂ ρ + 1 ρ ∂ T φ z ∂ φ + ∂ T z z ∂ z + T ρ z ρ ] z ^ {\displaystyle {\begin{aligned}\left[{\frac {\partial T_{\rho \rho }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi \rho }}{\partial \varphi }}+{\frac {\partial T_{z\rho }}{\partial z}}+{\frac {1}{\rho }}(T_{\rho \rho }-T_{\varphi \varphi })\right]&{\hat {\boldsymbol {\rho }}}\\+\left[{\frac {\partial T_{\rho \varphi }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {\partial T_{z\varphi }}{\partial z}}+{\frac {1}{\rho }}(T_{\rho \varphi }+T_{\varphi \rho })\right]&{\hat {\boldsymbol {\varphi }}}\\+\left[{\frac {\partial T_{\rho z}}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi z}}{\partial \varphi }}+{\frac {\partial T_{zz}}{\partial z}}+{\frac {T_{\rho z}}{\rho }}\right]&{\hat {\mathbf {z} }}\end{aligned}}} </p></td><td><p> [ ∂ T r r ∂ r + 2 T r r r + 1 r ∂ T θ r ∂ θ + cot θ r T θ r + 1 r sin θ ∂ T φ r ∂ φ − 1 r ( T θ θ + T φ φ ) ] r ^ + [ ∂ T r θ ∂ r + 2 T r θ r + 1 r ∂ T θ θ ∂ θ + cot θ r T θ θ + 1 r sin θ ∂ T φ θ ∂ φ + T θ r r − cot θ r T φ φ ] θ ^ + [ ∂ T r φ ∂ r + 2 T r φ r + 1 r ∂ T θ φ ∂ θ + 1 r sin θ ∂ T φ φ ∂ φ + T φ r r + cot θ r ( T θ φ + T φ θ ) ] φ ^ {\displaystyle {\begin{aligned}\left[{\frac {\partial T_{rr}}{\partial r}}+2{\frac {T_{rr}}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta r}}{\partial \theta }}+{\frac {\cot \theta }{r}}T_{\theta r}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi r}}{\partial \varphi }}-{\frac {1}{r}}(T_{\theta \theta }+T_{\varphi \varphi })\right]&{\hat {\mathbf {r} }}\\+\left[{\frac {\partial T_{r\theta }}{\partial r}}+2{\frac {T_{r\theta }}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta \theta }}{\partial \theta }}+{\frac {\cot \theta }{r}}T_{\theta \theta }+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi \theta }}{\partial \varphi }}+{\frac {T_{\theta r}}{r}}-{\frac {\cot \theta }{r}}T_{\varphi \varphi }\right]&{\hat {\boldsymbol {\theta }}}\\+\left[{\frac {\partial T_{r\varphi }}{\partial r}}+2{\frac {T_{r\varphi }}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta \varphi }}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {T_{\varphi r}}{r}}+{\frac {\cot \theta }{r}}(T_{\theta \varphi }+T_{\varphi \theta })\right]&{\hat {\boldsymbol {\varphi }}}\end{aligned}}} </p></td></tr><tr><th>Differential displacement <i>dℓ</i><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></th><td> d x x ^ + d y y ^ + d z z ^ {\displaystyle dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\,{\hat {\mathbf {z} }}} </td><td> d ρ ρ ^ + ρ d φ φ ^ + d z z ^ {\displaystyle d\rho \,{\hat {\boldsymbol {\rho }}}+\rho \,d\varphi \,{\hat {\boldsymbol {\varphi }}}+dz\,{\hat {\mathbf {z} }}} </td><td> d r r ^ + r d θ θ ^ + r sin θ d φ φ ^ {\displaystyle dr\,{\hat {\mathbf {r} }}+r\,d\theta \,{\hat {\boldsymbol {\theta }}}+r\,\sin \theta \,d\varphi \,{\hat {\boldsymbol {\varphi }}}} </td></tr><tr><th>Differential normal area <i>d</i>S</th><td> d y d z x ^ + d x d z y ^ + d x d y z ^ {\displaystyle {\begin{aligned}dy\,dz&\,{\hat {\mathbf {x} }}\\{}+dx\,dz&\,{\hat {\mathbf {y} }}\\{}+dx\,dy&\,{\hat {\mathbf {z} }}\end{aligned}}} </td><td> ρ d φ d z ρ ^ + d ρ d z φ ^ + ρ d ρ d φ z ^ {\displaystyle {\begin{aligned}\rho \,d\varphi \,dz&\,{\hat {\boldsymbol {\rho }}}\\{}+d\rho \,dz&\,{\hat {\boldsymbol {\varphi }}}\\{}+\rho \,d\rho \,d\varphi &\,{\hat {\mathbf {z} }}\end{aligned}}} </td><td> r 2 sin θ d θ d φ r ^ + r sin θ d r d φ θ ^ + r d r d θ φ ^ {\displaystyle {\begin{aligned}r^{2}\sin \theta \,d\theta \,d\varphi &\,{\hat {\mathbf {r} }}\\{}+r\sin \theta \,dr\,d\varphi &\,{\hat {\boldsymbol {\theta }}}\\{}+r\,dr\,d\theta &\,{\hat {\boldsymbol {\varphi }}}\end{aligned}}} </td></tr><tr><th>Differential volume <i>dV</i><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></th><td> d x d y d z {\displaystyle dx\,dy\,dz} </td><td> ρ d ρ d φ d z {\displaystyle \rho \,d\rho \,d\varphi \,dz} </td><td> r 2 sin θ d r d θ d φ {\displaystyle r^{2}\sin \theta \,dr\,d\theta \,d\varphi } </td></tr></tbody></table> ^α This page uses θ {\displaystyle \theta } for the polar angle and φ {\displaystyle \varphi } for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses θ {\displaystyle \theta } for the azimuthal angle and φ {\displaystyle \varphi } for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch θ {\displaystyle \theta } and φ {\displaystyle \varphi } in the formulae shown in the table above. ^β Defined in Cartesian coordinates as ∂ i A ⊗ e i {\displaystyle \partial _{i}\mathbf {A} \otimes \mathbf {e} _{i}} . An alternative definition is e i ⊗ ∂ i A {\displaystyle \mathbf {e} _{i}\otimes \partial _{i}\mathbf {A} } . ^γ Defined in Cartesian coordinates as e i ⋅ ∂ i T {\displaystyle \mathbf {e} _{i}\cdot \partial _{i}\mathbf {T} } . An alternative definition is ∂ i T ⋅ e i {\displaystyle \partial _{i}\mathbf {T} \cdot \mathbf {e} _{i}} . <h3>Calculation rules</h3> <ol><li> div grad f ≡ ∇ ⋅ ∇ f ≡ ∇ 2 f {\displaystyle \operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f\equiv \nabla ^{2}f} </li> <li> curl grad f ≡ ∇ × ∇ f = 0 {\displaystyle \operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0} } </li> <li> div curl A ≡ ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0} </li> <li> curl curl A ≡ ∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A {\displaystyle \operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} } (<a href="/facts/Triple_product/46u8v6SV">Lagrange's formula</a> for del)</li> <li> ∇ 2 ( f g ) = f ∇ 2 g + 2 ∇ f ⋅ ∇ g + g ∇ 2 f {\displaystyle \nabla ^{2}(fg)=f\nabla ^{2}g+2\nabla f\cdot \nabla g+g\nabla ^{2}f} </li> <li> ∇ 2 ( P ⋅ Q ) = Q ⋅ ∇ 2 P − P ⋅ ∇ 2 Q + 2 ∇ ⋅ [ ( P ⋅ ∇ ) Q + P × ∇ × Q ] {\displaystyle \nabla ^{2}\left(\mathbf {P} \cdot \mathbf {Q} \right)=\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} -\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} +2\nabla \cdot \left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} \right]\quad } (From <a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> )</li></ol> <h2 id="cartesian-derivation">Cartesian derivation</h2> <p> div A = lim V → 0 ∬ ∂ V A ⋅ d S ∭ V d V = A x ( x + d x ) d y d z − A x ( x ) d y d z + A y ( y + d y ) d x d z − A y ( y ) d x d z + A z ( z + d z ) d x d y − A z ( z ) d x d y d x d y d z = ∂ A x ∂ x + ∂ A y ∂ y + ∂ A z ∂ z {\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} =\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}&={\frac {A_{x}(x+dx)\,dy\,dz-A_{x}(x)\,dy\,dz+A_{y}(y+dy)\,dx\,dz-A_{y}(y)\,dx\,dz+A_{z}(z+dz)\,dx\,dy-A_{z}(z)\,dx\,dy}{dx\,dy\,dz}}\\&={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}} </p><p> ( curl A ) x = lim S ⊥ x ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A z ( y + d y ) d z − A z ( y ) d z + A y ( z ) d y − A y ( z + d z ) d y d y d z = ∂ A z ∂ y − ∂ A y ∂ z {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{x}=\lim _{S^{\perp \mathbf {\hat {x}} }\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{z}(y+dy)\,dz-A_{z}(y)\,dz+A_{y}(z)\,dy-A_{y}(z+dz)\,dy}{dy\,dz}}\\&={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\end{aligned}}} </p><p>The expressions for ( curl A ) y {\displaystyle (\operatorname {curl} \mathbf {A} )_{y}} and ( curl A ) z {\displaystyle (\operatorname {curl} \mathbf {A} )_{z}} are found in the same way. </p> <h2 id="cylindrical-derivation">Cylindrical derivation</h2> <p> div A = lim V → 0 ∬ ∂ V A ⋅ d S ∭ V d V = A ρ ( ρ + d ρ ) ( ρ + d ρ ) d ϕ d z − A ρ ( ρ ) ρ d ϕ d z + A ϕ ( ϕ + d ϕ ) d ρ d z − A ϕ ( ϕ ) d ρ d z + A z ( z + d z ) d ρ ( ρ + d ρ / 2 ) d ϕ − A z ( z ) d ρ ( ρ + d ρ / 2 ) d ϕ ρ d ϕ d ρ d z = 1 ρ ∂ ( ρ A ρ ) ∂ ρ + 1 ρ ∂ A ϕ ∂ ϕ + ∂ A z ∂ z {\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\&={\frac {A_{\rho }(\rho +d\rho )(\rho +d\rho )\,d\phi \,dz-A_{\rho }(\rho )\rho \,d\phi \,dz+A_{\phi }(\phi +d\phi )\,d\rho \,dz-A_{\phi }(\phi )\,d\rho \,dz+A_{z}(z+dz)\,d\rho \,(\rho +d\rho /2)\,d\phi -A_{z}(z)\,d\rho (\rho +d\rho /2)\,d\phi }{\rho \,d\phi \,d\rho \,dz}}\\&={\frac {1}{\rho }}{\frac {\partial (\rho A_{\rho })}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}} </p><p> ( curl A ) ρ = lim S ⊥ ρ ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A ϕ ( z ) ( ρ + d ρ ) d ϕ − A ϕ ( z + d z ) ( ρ + d ρ ) d ϕ + A z ( ϕ + d ϕ ) d z − A z ( ϕ ) d z ( ρ + d ρ ) d ϕ d z = − ∂ A ϕ ∂ z + 1 ρ ∂ A z ∂ ϕ {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\rho }&=\lim _{S^{\perp {\hat {\boldsymbol {\rho }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d{\boldsymbol {\ell }}}{\iint _{S}dS}}\\[1ex]&={\frac {A_{\phi }(z)\left(\rho +d\rho \right)\,d\phi -A_{\phi }(z+dz)\left(\rho +d\rho \right)\,d\phi +A_{z}(\phi +d\phi )\,dz-A_{z}(\phi )\,dz}{\left(\rho +d\rho \right)\,d\phi \,dz}}\\[1ex]&=-{\frac {\partial A_{\phi }}{\partial z}}+{\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}\end{aligned}}} </p><p> ( curl A ) ϕ = lim S ⊥ ϕ ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A z ( ρ ) d z − A z ( ρ + d ρ ) d z + A ρ ( z + d z ) d ρ − A ρ ( z ) d ρ d ρ d z = − ∂ A z ∂ ρ + ∂ A ρ ∂ z {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }&=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d{\boldsymbol {\ell }}}{\iint _{S}dS}}\\&={\frac {A_{z}(\rho )\,dz-A_{z}(\rho +d\rho )\,dz+A_{\rho }(z+dz)\,d\rho -A_{\rho }(z)\,d\rho }{d\rho \,dz}}\\&=-{\frac {\partial A_{z}}{\partial \rho }}+{\frac {\partial A_{\rho }}{\partial z}}\end{aligned}}} </p><p> ( curl A ) z = lim S ⊥ z ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A ρ ( ϕ ) d ρ − A ρ ( ϕ + d ϕ ) d ρ + A ϕ ( ρ + d ρ ) ( ρ + d ρ ) d ϕ − A ϕ ( ρ ) ρ d ϕ ρ d ρ d ϕ = − 1 ρ ∂ A ρ ∂ ϕ + 1 ρ ∂ ( ρ A ϕ ) ∂ ρ {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{z}&=\lim _{S^{\perp {\hat {\boldsymbol {z}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\[1ex]&={\frac {A_{\rho }(\phi )\,d\rho -A_{\rho }(\phi +d\phi )\,d\rho +A_{\phi }(\rho +d\rho )(\rho +d\rho )\,d\phi -A_{\phi }(\rho )\rho \,d\phi }{\rho \,d\rho \,d\phi }}\\[1ex]&=-{\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}{\frac {\partial (\rho A_{\phi })}{\partial \rho }}\end{aligned}}} </p><p> curl A = ( curl A ) ρ ρ ^ + ( curl A ) ϕ ϕ ^ + ( curl A ) z z ^ = ( 1 ρ ∂ A z ∂ ϕ − ∂ A ϕ ∂ z ) ρ ^ + ( ∂ A ρ ∂ z − ∂ A z ∂ ρ ) ϕ ^ + 1 ρ ( ∂ ( ρ A ϕ ) ∂ ρ − ∂ A ρ ∂ ϕ ) z ^ {\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {A} &=(\operatorname {curl} \mathbf {A} )_{\rho }{\hat {\boldsymbol {\rho }}}+(\operatorname {curl} \mathbf {A} )_{\phi }{\hat {\boldsymbol {\phi }}}+(\operatorname {curl} \mathbf {A} )_{z}{\hat {\boldsymbol {z}}}\\[1ex]&=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right){\hat {\boldsymbol {\rho }}}+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\hat {\boldsymbol {\phi }}}+{\frac {1}{\rho }}\left({\frac {\partial (\rho A_{\phi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\hat {\boldsymbol {z}}}\end{aligned}}} </p> <h2 id="spherical-derivation">Spherical derivation</h2> <p> div A = lim V → 0 ∬ ∂ V A ⋅ d S ∭ V d V = A r ( r + d r ) ( r + d r ) d θ ( r + d r ) sin θ d ϕ − A r ( r ) r d θ r sin θ d ϕ + A θ ( θ + d θ ) sin ( θ + d θ ) r d r d ϕ − A θ ( θ ) sin ( θ ) r d r d ϕ + A ϕ ( ϕ + d ϕ ) r d r d θ − A ϕ ( ϕ ) r d r d θ d r r d θ r sin θ d ϕ = 1 r 2 ∂ ( r 2 A r ) ∂ r + 1 r sin θ ∂ ( A θ sin θ ) ∂ θ + 1 r sin θ ∂ A ϕ ∂ ϕ {\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\&={\frac {A_{r}(r+dr)(r+dr)\,d\theta \,(r+dr)\sin \theta \,d\phi -A_{r}(r)r\,d\theta \,r\sin \theta \,d\phi +A_{\theta }(\theta +d\theta )\sin(\theta +d\theta )r\,dr\,d\phi -A_{\theta }(\theta )\sin(\theta )r\,dr\,d\phi +A_{\phi }(\phi +d\phi )r\,dr\,d\theta -A_{\phi }(\phi )r\,dr\,d\theta }{dr\,r\,d\theta \,r\sin \theta \,d\phi }}\\&={\frac {1}{r^{2}}}{\frac {\partial (r^{2}A_{r})}{\partial r}}+{\frac {1}{r\sin \theta }}{\frac {\partial (A_{\theta }\sin \theta )}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\end{aligned}}} </p><p> ( curl A ) r = lim S ⊥ r ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A θ ( ϕ ) r d θ + A ϕ ( θ + d θ ) r sin ( θ + d θ ) d ϕ − A θ ( ϕ + d ϕ ) r d θ − A ϕ ( θ ) r sin ( θ ) d ϕ r d θ r sin θ d ϕ = 1 r sin θ ∂ ( A ϕ sin θ ) ∂ θ − 1 r sin θ ∂ A θ ∂ ϕ {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{r}=\lim _{S^{\perp {\boldsymbol {\hat {r}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\theta }(\phi )r\,d\theta +A_{\phi }(\theta +d\theta )r\sin(\theta +d\theta )\,d\phi -A_{\theta }(\phi +d\phi )r\,d\theta -A_{\phi }(\theta )r\sin(\theta )\,d\phi }{r\,d\theta \,r\sin \theta \,d\phi }}\\&={\frac {1}{r\sin \theta }}{\frac {\partial (A_{\phi }\sin \theta )}{\partial \theta }}-{\frac {1}{r\sin \theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\end{aligned}}} </p><p> ( curl A ) θ = lim S ⊥ θ ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A ϕ ( r ) r sin θ d ϕ + A r ( ϕ + d ϕ ) d r − A ϕ ( r + d r ) ( r + d r ) sin θ d ϕ − A r ( ϕ ) d r d r r sin θ d ϕ = 1 r sin θ ∂ A r ∂ ϕ − 1 r ∂ ( r A ϕ ) ∂ r {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\theta }=\lim _{S^{\perp {\boldsymbol {\hat {\theta }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\phi }(r)r\sin \theta \,d\phi +A_{r}(\phi +d\phi )\,dr-A_{\phi }(r+dr)(r+dr)\sin \theta \,d\phi -A_{r}(\phi )\,dr}{dr\,r\sin \theta \,d\phi }}\\&={\frac {1}{r\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {1}{r}}{\frac {\partial (rA_{\phi })}{\partial r}}\end{aligned}}} </p><p> ( curl A ) ϕ = lim S ⊥ ϕ ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A r ( θ ) d r + A θ ( r + d r ) ( r + d r ) d θ − A r ( θ + d θ ) d r − A θ ( r ) r d θ r d r d θ = 1 r ∂ ( r A θ ) ∂ r − 1 r ∂ A r ∂ θ {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{r}(\theta )\,dr+A_{\theta }(r+dr)(r+dr)\,d\theta -A_{r}(\theta +d\theta )\,dr-A_{\theta }(r)r\,d\theta }{r\,dr\,d\theta }}\\&={\frac {1}{r}}{\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {1}{r}}{\frac {\partial A_{r}}{\partial \theta }}\end{aligned}}} </p><p> curl A = ( curl A ) r r ^ + ( curl A ) θ θ ^ + ( curl A ) ϕ ϕ ^ = 1 r sin θ ( ∂ ( A ϕ sin θ ) ∂ θ − ∂ A θ ∂ ϕ ) r ^ + 1 r ( 1 sin θ ∂ A r ∂ ϕ − ∂ ( r A ϕ ) ∂ r ) θ ^ + 1 r ( ∂ ( r A θ ) ∂ r − ∂ A r ∂ θ ) ϕ ^ {\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {A} &=(\operatorname {curl} \mathbf {A} )_{r}\,{\hat {\boldsymbol {r}}}+(\operatorname {curl} \mathbf {A} )_{\theta }\,{\hat {\boldsymbol {\theta }}}+(\operatorname {curl} \mathbf {A} )_{\phi }\,{\hat {\boldsymbol {\phi }}}\\[1ex]&={\frac {1}{r\sin \theta }}\left({\frac {\partial (A_{\phi }\sin \theta )}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial \phi }}\right){\hat {\boldsymbol {r}}}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial (rA_{\phi })}{\partial r}}\right){\hat {\boldsymbol {\theta }}}+{\frac {1}{r}}\left({\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {\partial A_{r}}{\partial \theta }}\right){\hat {\boldsymbol {\phi }}}\end{aligned}}} </p> <h2 id="unit-vector-conversion-formula">Unit vector conversion formula</h2> <p>The unit vector of a coordinate parameter <i>u</i> is defined in such a way that a small positive change in <i>u</i> causes the position vector r {\displaystyle \mathbf {r} } to change in u {\displaystyle \mathbf {u} } direction. </p><p>Therefore, ∂ r ∂ u = ∂ s ∂ u u {\displaystyle {\frac {\partial {\mathbf {r} }}{\partial u}}={\frac {\partial {s}}{\partial u}}\mathbf {u} } where s is the arc length parameter. </p><p>For two sets of coordinate systems u i {\displaystyle u_{i}} and v j {\displaystyle v_{j}} , according to <a href="/facts/Differential_of_a_function/337XJVMC">chain rule</a>, d r = ∑ i ∂ r ∂ u i d u i = ∑ i ∂ s ∂ u i u ^ i d u i = ∑ j ∂ s ∂ v j v ^ j d v j = ∑ j ∂ s ∂ v j v ^ j ∑ i ∂ v j ∂ u i d u i = ∑ i ∑ j ∂ s ∂ v j ∂ v j ∂ u i v ^ j d u i . {\displaystyle d\mathbf {r} =\sum _{i}{\frac {\partial \mathbf {r} }{\partial u_{i}}}\,du_{i}=\sum _{i}{\frac {\partial s}{\partial u_{i}}}{\hat {\mathbf {u} }}_{i}du_{i}=\sum _{j}{\frac {\partial s}{\partial v_{j}}}{\hat {\mathbf {v} }}_{j}\,dv_{j}=\sum _{j}{\frac {\partial s}{\partial v_{j}}}{\hat {\mathbf {v} }}_{j}\sum _{i}{\frac {\partial v_{j}}{\partial u_{i}}}\,du_{i}=\sum _{i}\sum _{j}{\frac {\partial s}{\partial v_{j}}}{\frac {\partial v_{j}}{\partial u_{i}}}{\hat {\mathbf {v} }}_{j}\,du_{i}.} </p><p>Now, we isolate the i {\displaystyle i} th component. For i ≠ k {\displaystyle i{\neq }k} , let d u k = 0 {\displaystyle \mathrm {d} u_{k}=0} . Then divide on both sides by d u i {\displaystyle \mathrm {d} u_{i}} to get: ∂ s ∂ u i u ^ i = ∑ j ∂ s ∂ v j ∂ v j ∂ u i v ^ j . {\displaystyle {\frac {\partial s}{\partial u_{i}}}{\hat {\mathbf {u} }}_{i}=\sum _{j}{\frac {\partial s}{\partial v_{j}}}{\frac {\partial v_{j}}{\partial u_{i}}}{\hat {\mathbf {v} }}_{j}.} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Del/0xEmQeBo">Del</a></li> <li><a href="/facts/Orthogonal_coordinates/LLoa2lAJ">Orthogonal coordinates</a></li> <li><a href="/facts/Curvilinear_coordinates/RQD7Cajd">Curvilinear coordinates</a></li> <li><a href="/facts/Vector_fields_in_cylindrical_and_spherical_coordinates/rBfvz6HI">Vector fields in cylindrical and spherical coordinates</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.csulb.edu/~woollett/">Maxima Computer Algebra system scripts</a> to generate some of these operators in cylindrical and spherical coordinates.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2. <a href="978-0-321-85656-2" target="_blank">978-0-321-85656-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2. <a href="978-0-321-85656-2" target="_blank">978-0-321-85656-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2. <a href="978-0-321-85656-2" target="_blank">978-0-321-85656-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2. <a href="978-0-321-85656-2" target="_blank">978-0-321-85656-2</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2. <a href="978-0-321-85656-2" target="_blank">978-0-321-85656-2</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2. <a href="978-0-321-85656-2" target="_blank">978-0-321-85656-2</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Arfken, George; Weber, Hans; Harris, Frank (2012). Mathematical Methods for Physicists (Seventh ed.). Academic Press. p. 192. ISBN 9789381269558. <a href="9789381269558" target="_blank">9789381269558</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011. <a href="http://mathworld.wolfram.com/ConvectiveOperator.html" target="_blank">http://mathworld.wolfram.com/ConvectiveOperator.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2. <a href="978-0-321-85656-2" target="_blank">978-0-321-85656-2</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2. <a href="978-0-321-85656-2" target="_blank">978-0-321-85656-2</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Fernández-Guasti, M. (2012). "Green's Second Identity for Vector Fields". ISRN Mathematical Physics. 2012. Hindawi Limited: 1–7. doi:10.5402/2012/973968. ISSN 2090-4681. <a href="https://doi.org/10.5402%2F2012%2F973968" target="_blank">https://doi.org/10.5402%2F2012%2F973968</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> </ol>