Be in cylindrical coordinates![{\ displaystyle (\ rho, \ varphi, z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b915f15da61cb0537189d6795ab55d9377537056)
![{\ displaystyle {\ hat {e}} _ {\ rho} = {\ begin {pmatrix} \ cos \ varphi \\\ sin \ varphi \\ 0 \ end {pmatrix}}, \ quad {\ hat {e} } _ {\ varphi} = {\ begin {pmatrix} - \ sin \ varphi \\\ cos \ varphi \\ 0 \ end {pmatrix}}, \ quad {\ hat {e}} _ {z} = {\ begin {pmatrix} 0 \\ 0 \\ 1 \ end {pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/051c60fdbea2551de53f4b311768b17a1aa4d6ee)
taken as orthonormal basis vectors. Their derivatives are:
Here, as in the following, an index after a comma means a derivative according to the specified coordinate, for example
the application of the Laplace operator
to a vector field results in:
i.e. the formula specified in the text.
![{\ displaystyle {\ hat {e}} _ {\ rho, \ varphi} = {\ hat {e}} _ {\ varphi} \ quad {\ text {and}} \ quad {\ hat {e}} _ {\ varphi, \ varphi} = - {\ hat {e}} _ {\ rho}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63fd5bcfd70c6d89beef93b494cd992f2ae119b2)
![{\ displaystyle {\ hat {e}} _ {\ rho, \ varphi}: = {\ frac {\ partial} {\ partial \ varphi}} {\ hat {e}} _ {\ rho}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dddda8abfc0bc4c6ba9d9a12d0366b5bd03ea703)
![{\ displaystyle \ Delta = {\ frac {\ partial ^ {2}} {\ partial \ rho ^ {2}}} + {\ frac {1} {\ rho}} {\ frac {\ partial} {\ partial \ rho}} + {\ frac {1} {\ rho ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} + {\ frac {\ partial ^ {2}} {\ partial z ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3cb70168041c7b20aa243e5721542c393772498)
![{\ displaystyle {\ begin {aligned} & \ left ({\ frac {\ partial ^ {2}} {\ partial \ rho ^ {2}}} + {\ frac {1} {\ rho}} {\ frac {\ partial} {\ partial \ rho}} + {\ frac {1} {\ rho ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} + {\ frac {\ partial ^ {2}} {\ partial z ^ {2}}} \ right) (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e}} _ {z}) \\ = & {\ frac {\ partial ^ {2}} {\ partial \ rho ^ {2 }}} (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e} } _ {z}) + {\ frac {1} {\ rho}} {\ frac {\ partial} {\ partial \ rho}} (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e}} _ {z}) \\ & + {\ frac {1} {\ rho ^ { 2}}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e}} _ {z}) + {\ frac {\ partial ^ {2}} {\ partial z ^ {2}} } (v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z} {\ hat {e}} _ {z}) \\ = & v _ {\ rho, \ rho \ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ rho \ rho} {\ hat {e}} _ {\ varphi} + v_ {z, \ rho \ rho} {\ hat {e}} _ {z} + {\ frac {1} {\ rho}} (v _ {\ rho, \ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ rho} {\ hat {e}} _ {\ varphi } + v_ {z, \ rho} {\ hat {e}} _ {z}) \\ & + {\ frac {1} {\ rho ^ {2}}} {\ frac {\ partial} {\ partial \ varphi}} (v _ {\ rho, \ varphi} {\ hat {e}} _ {\ rho} + v _ {\ rho} {\ hat {e}} _ {\ varphi} + v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ varphi} -v _ {\ varphi} {\ hat {e}} _ {\ rho} + v_ {z, \ varphi} {\ hat {e}} _ {z }) \\ & + v _ {\ rho, zz} {\ hat {e}} _ {\ rho} + v _ {\ varphi, zz} {\ hat {e}} _ {\ varphi} + v_ {z, zz} {\ hat {e}} _ {z} \\ = & v _ {\ rho, \ rho \ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ rho \ rho} { \ hat {e}} _ {\ varphi} + v_ {z, \ rho \ rho} {\ hat {e}} _ {z} + {\ frac {1} {\ rho}} (v _ {\ rho, \ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ rho} {\ hat {e}} _ {\ varphi} + v_ {z, \ rho} {\ hat {e} } _ {z}) \\ & + {\ frac {1} {\ rho ^ {2}}} (v _ {\ rho, \ varphi \ varphi} {\ hat {e}} _ {\ rho} + 2v_ {\ rho, \ varphi} {\ hat {e}} _ {\ varphi} -v _ {\ rho} {\ hat {e}} _ {\ rho} + v _ {\ varphi, \ varphi \ varphi} {\ hat {e}} _ {\ varphi} -2v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ rho} -v _ {\ varphi} {\ hat {e}} _ {\ varphi} + v_ {z, \ varphi \ varphi} {\ hat {e}} _ {z}) \\ & + v _ {\ rho , zz} {\ hat {e}} _ {\ rho} + v _ {\ varphi, zz} {\ hat {e}} _ {\ varphi} + v_ {z, zz} {\ hat {e}} _ {z} \\ = & + \ left (\ Delta v _ {\ rho} - {\ frac {1} {\ rho ^ {2}}} v _ {\ rho} - {\ frac {2} {\ rho ^ {2}}} v _ {\ varphi, \ varphi} \ right) {\ hat {e}} _ {\ rho} + \ left (\ Delta v _ {\ varphi} - {\ frac {1} {\ rho ^ {2}}} v _ {\ varphi} + {\ frac {2} {\ rho ^ {2}}} v _ {\ rho, \ varphi} \ right) {\ hat {e}} _ {\ varphi} + \ Delta v_ {z} {\ hat {e}} _ {z}, \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/849fc62a2cdfcb15fe4547088e009971fe983881)
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The basis vectors
![{\ displaystyle {\ hat {e}} _ {r} = {\ begin {pmatrix} \ sin \ theta \ cos \ varphi \\\ sin \ theta \ sin \ varphi \\\ cos \ theta \ end {pmatrix} }, \ qquad {\ hat {e}} _ {\ theta} = {\ begin {pmatrix} \ cos \ theta \ cos \ varphi \\\ cos \ theta \ sin \ varphi \\ - \ sin \ theta \ end {pmatrix}}, \ qquad {\ hat {e}} _ {\ varphi} = {\ begin {pmatrix} - \ sin \ varphi \\\ cos \ varphi \\ 0 \ end {pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/357f3467c184c23b27d7eb7bc85e1ad57e9352ce)
be used. These vectors gave the derivatives
Application of the Laplace operator
to a vector field:
thus the same result as given in the text.
![{\ displaystyle {\ begin {aligned} {\ hat {e}} _ {r, \ theta} = & {\ begin {pmatrix} \ cos \ theta \ cos \ varphi \\\ cos \ theta \ sin \ varphi \ \ - \ sin \ theta \ end {pmatrix}} = {\ hat {e}} _ {\ theta} \ ,, \ quad {\ hat {e}} _ {r, \ varphi} = {\ begin {pmatrix } - \ sin \ theta \ sin \ varphi \\\ sin \ theta \ cos \ varphi \\ 0 \ end {pmatrix}} = \ sin \ theta {\ hat {e}} _ {\ varphi} \\ {\ hat {e}} _ {\ theta, \ theta} = & {\ begin {pmatrix} - \ sin \ theta \ cos \ varphi \\ - \ sin \ theta \ sin \ varphi \\ - \ cos \ theta \ end {pmatrix}} = - {\ hat {e}} _ {r} \ ,, \ quad {\ hat {e}} _ {\ theta, \ varphi} = {\ begin {pmatrix} - \ cos \ theta \ sin \ varphi \\\ cos \ theta \ cos \ varphi \\ 0 \ end {pmatrix}} = \ cos \ theta {\ hat {e}} _ {\ varphi} \\ {\ hat {e}} _ { \ varphi, \ varphi} = & {\ begin {pmatrix} - \ cos \ varphi \\ - \ sin \ varphi \\ 0 \ end {pmatrix}} = {\ hat {e}} _ {z} \ times { \ hat {e}} _ {\ varphi} = - \ sin \ theta {\ hat {e}} _ {r} - \ cos \ theta {\ hat {e}} _ {\ theta} \ end {aligned} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cbff6a386e55aacd5e5770e64f27d01de44595)
![{\ displaystyle \ Delta = {\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} + {\ frac {2} {r}} {\ frac {\ partial} {\ partial r} } + {\ frac {1} {r ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} + {\ frac {1} {r ^ {2 } \ tan \ theta}} {\ frac {\ partial} {\ partial \ theta}} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4fee04ac3e668ad0f765499480c34c111be7a3a)
![{\ displaystyle {\ begin {aligned} & \ left ({\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} + {\ frac {2} {r}} {\ frac {\ partial} {\ partial r}} + {\ frac {1} {r ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} + {\ frac { 1} {r ^ {2} \ tan \ theta}} {\ frac {\ partial} {\ partial \ theta}} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta} } {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} \ right) \ cdot (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ varphi}) \\ = & {\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat { e}} _ {\ varphi}) + {\ frac {2} {r}} {\ frac {\ partial} {\ partial r}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ varphi}) + {\ frac {1} {r ^ {2}} } {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e }} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ varphi}) \\ & + {\ frac {1} {r ^ {2} \ tan \ theta}} {\ frac {\ partial} {\ partial \ theta}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ v arphi}) + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ varphi ^ {2}}} (v_ {r} {\ hat {e}} _ {r} + v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi} {\ hat {e}} _ {\ varphi} ) \\ = & v_ {r, rr} {\ hat {e}} _ {r} + v _ {\ theta, rr} {\ hat {e}} _ {\ theta} + v _ {\ varphi, rr} { \ hat {e}} _ {\ varphi} + {\ frac {2} {r}} v_ {r, r} {\ hat {e}} _ {r} + {\ frac {2} {r}} v _ {\ theta, r} {\ hat {e}} _ {\ theta} + {\ frac {2} {r}} v _ {\ varphi, r} {\ hat {e}} _ {\ varphi} \ \ & + {\ frac {1} {r ^ {2}}} {\ frac {\ partial} {\ partial \ theta}} (v_ {r, \ theta} {\ hat {e}} _ {r} + v_ {r} {\ hat {e}} _ {\ theta} + v _ {\ theta, \ theta} {\ hat {e}} _ {\ theta} -v _ {\ theta} {\ hat {e} } _ {r} + v _ {\ varphi, \ theta} {\ hat {e}} _ {\ varphi}) \\ & + {\ frac {1} {r ^ {2} \ tan \ theta}} ( v_ {r, \ theta} {\ hat {e}} _ {r} + v_ {r} {\ hat {e}} _ {\ theta} + v _ {\ theta, \ theta} {\ hat {e} } _ {\ theta} -v _ {\ theta} {\ hat {e}} _ {r} + v _ {\ varphi, \ theta} {\ hat {e}} _ {\ varphi}) \\ & + { \ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial} {\ partial \ varphi}} (v_ {r, \ varphi} {\ hat {e}} _ {r} + \ sin \ theta v_ {r} {\ hat {e}} _ {\ varphi} + v _ {\ theta, \ varphi} {\ hat {e }} _ {\ theta} + \ cos \ theta v _ {\ theta} {\ hat {e}} _ {\ varphi} + v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ varphi} - \ sin \ theta v _ {\ varphi} {\ hat {e}} _ {r} - \ cos \ theta v _ {\ varphi} {\ hat {e}} _ {\ theta}) \\ = & v_ {r , rr} {\ hat {e}} _ {r} + v _ {\ theta, rr} {\ hat {e}} _ {\ theta} + v _ {\ varphi, rr} {\ hat {e}} _ {\ varphi} + {\ frac {2} {r}} v_ {r, r} {\ hat {e}} _ {r} + {\ frac {2} {r}} v _ {\ theta, r} {\ hat {e}} _ {\ theta} + {\ frac {2} {r}} v _ {\ varphi, r} {\ hat {e}} _ {\ varphi} \\ & + {\ frac { 1} {r ^ {2}}} (v_ {r, \ theta \ theta} {\ hat {e}} _ {r} + v_ {r, \ theta} {\ hat {e}} _ {\ theta } + v_ {r, \ theta} {\ hat {e}} _ {\ theta} -v_ {r} {\ hat {e}} _ {r} + v _ {\ theta, \ theta \ theta} {\ hat {e}} _ {\ theta} -v _ {\ theta, \ theta} {\ hat {e}} _ {r} -v _ {\ theta, \ theta} {\ hat {e}} _ {r} -v _ {\ theta} {\ hat {e}} _ {\ theta} + v _ {\ varphi, \ theta \ theta} {\ hat {e}} _ {\ varphi}) \\ & + {\ frac { 1} {r ^ {2} \ tan \ theta}} (v_ {r, \ theta} {\ hat {e}} _ {r} + v_ {r} {\ hat {e}} _ {\ theta} + v _ {\ theta, \ theta} {\ hat {e}} _ {\ theta} -v _ {\ theta} {\ hat {e}} _ {r} + v _ {\ varphi, \ theta} {\ hat {e}} _ {\ varphi}) \\ & + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} (v_ {r, \ varphi \ varphi} {\ hat {e}} _ {r} + \ sin \ theta v_ {r, \ varphi} {\ hat {e}} _ {\ varphi} + \ sin \ theta v_ {r, \ varphi} {\ hat {e}} _ {\ varphi} - \ sin ^ {2} \ theta v_ {r} {\ hat {e}} _ {r} - \ sin \ theta \ cos \ theta v_ {r } {\ hat {e}} _ {\ theta} \\ & + v _ {\ theta, \ varphi \ varphi} {\ hat {e}} _ {\ theta} + \ cos \ theta v _ {\ theta, \ varphi} {\ hat {e}} _ {\ varphi} + \ cos \ theta v _ {\ theta, \ varphi} {\ hat {e}} _ {\ varphi} - \ sin \ theta \ cos \ theta v_ { \ theta} {\ hat {e}} _ {r} - \ cos ^ {2} \ theta v _ {\ theta} {\ hat {e}} _ {\ theta} \\ & + v _ {\ varphi, \ varphi \ varphi} {\ hat {e}} _ {\ varphi} - \ sin \ theta v _ {\ varphi, \ varphi} {\ hat {e}} _ {r} - \ cos \ theta v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ theta} - \ sin \ theta v _ {\ varphi, \ varphi} {\ hat {e}} _ {r} - \ sin ^ {2} \ theta v_ { \ varphi} {\ hat {e}} _ {\ varphi} \\ & - \ cos \ theta v _ {\ varphi, \ varphi} {\ hat {e}} _ {\ theta} - \ cos ^ {2} \ theta v _ {\ varphi} {\ hat {e}} _ {\ varphi}) \\ = & {\ Bigl (} v_ {r, rr} + {\ frac {2} {r}} v_ {r, r} + {\ frac {1} {r ^ {2}}} v_ {r, \ theta \ theta} + {\ frac {1} {r ^ {2} \ tan \ theta}} v_ {r, \ theta} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} v_ {r, \ varphi \ varphi} \\ & \ qquad - {\ frac {1} {r ^ {2}}} v_ {r} - {\ frac {1} {r ^ {2}}} v _ {\ theta, \ theta } - {\ frac {1} {r ^ {2}}} v _ {\ theta, \ theta} - {\ frac {1} {r ^ {2} \ tan \ theta}} v _ {\ theta} - { \ frac {1} {r ^ {2}}} v_ {r} - {\ frac {\ cos \ theta} {r ^ {2} \ sin \ theta}} v _ {\ theta} - {\ frac {1 } {r ^ {2} \ sin \ theta}} v _ {\ varphi, \ varphi} - {\ frac {1} {r ^ {2} \ sin \ theta}} v _ {\ varphi, \ varphi} {\ Bigr)} {\ hat {e}} _ {r} \\ & + {\ Bigl (} v _ {\ theta, rr} + {\ frac {2} {r}} v _ {\ theta, r} + { \ frac {1} {r ^ {2}}} v _ {\ theta, \ theta \ theta} + {\ frac {1} {r ^ {2} \ tan \ theta}} v _ {\ theta, \ theta} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta, \ varphi \ varphi} \\ & \ qquad + {\ frac {2} {r ^ {2 }}} v_ {r, \ theta} - {\ frac {1} {r ^ {2}}} v _ {\ theta} + {\ frac {1} {r ^ {2} \ tan \ theta}} v_ {r} - {\ frac {\ cos \ theta} {r ^ {2} \ sin \ theta}} v_ {r} - {\ frac {\ cos ^ {2} \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta} - {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ varphi, \ varphi} { \ Bigr)} {\ hat {e}} _ {\ theta} \\ & + {\ Bigl (} v _ {\ varphi, rr} + {\ frac {2} {r}} v _ {\ varphi, r} + {\ frac {1} {r ^ {2}}} v _ {\ varphi, \ theta \ theta } + {\ frac {1} {r ^ {2} \ tan \ theta}} v _ {\ varphi, \ theta} + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta} } v _ {\ varphi, \ varphi \ varphi} \\ & \ qquad + {\ frac {2} {r ^ {2} \ sin \ theta}} v_ {r, \ varphi} + {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta, \ varphi} - {\ frac {\ sin ^ {2} \ theta + \ cos ^ {2} \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ varphi} {\ Bigr)} {\ hat {e}} _ {\ varphi} \\ = & \ left (\ Delta v_ {r } - {\ frac {2} {r ^ {2}}} v_ {r} - {\ frac {2} {r ^ {2}}} v _ {\ theta, \ theta} - {\ frac {2} {r ^ {2} \ tan \ theta}} v _ {\ theta} - {\ frac {2} {r ^ {2} \ sin \ theta}} v _ {\ varphi, \ varphi} \ right) {\ hat {e}} _ {r} \\ & + \ left (\ Delta v _ {\ theta} + {\ frac {2} {r ^ {2}}} v_ {r, \ theta} - {\ frac {1 } {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta} - {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v_ {\ varphi, \ varphi} \ right) {\ hat {e}} _ {\ theta} \\ & + \ left (\ Delta v _ {\ varphi} + {\ frac {2 \ cos \ theta} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ theta, \ varphi} - {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} v _ {\ varphi} + {\ frac {2} {r ^ {2} \ sin \ theta}} v_ {r, \ varphi} \ right) {\ hat {e}} _ {\ varphi} \ end {aligned}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd51896bf76920babb5e02607ac6efc0cb61403)
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