Del en coordenadas cilíndricas y esféricas - Del in cylindrical and spherical coordinates

${\ Displaystyle {\ begin {alineado} \ operatorname {div} \ mathbf {A} = \ lim _ {V \ to 0} {\ frac {\ iint _ {\ parcial V} \ mathbf {A} \ cdot d \ mathbf {S}} {\ iiint _ {V} dV}} & = {\ frac {A_ {x} (x + dx) \, dy \, dz-A_ {x} (x) \, dy \, dz + A_ {y} (y + dy) \, dx \, dz-A_ {y} (y) \, dx \, dz + A_ {z} (z + dz) \, dx \, dy-A_ {z } (z) \, dx \, dy} {dx \, dy \, dz}} \\ & = {\ frac {\ parcial A_ {x}} {\ parcial x}} + {\ frac {\ parcial A_ {y}} {\ y parcial}} + {\ frac {\ parcial A_ {z}} {\ parcial z}} \ end {alineado}}}$

${\ Displaystyle {\ begin {alineado} (\ operatorname {curl} \ mathbf {A}) _ {x} = \ lim _ {S ^ {\ perp \ mathbf {\ hat {x}}} \ to 0} { \ frac {\ int _ {\ parcial S} \ mathbf {A} \ cdot d \ mathbf {\ ell}} {\ iint _ {S} dS}} & = {\ frac {A_ {z} (y + dy ) \, dz-A_ {z} (y) \, dz + A_ {y} (z) \, dy-A_ {y} (z + dz) \, dy} {dy \, dz}} \\ & = {\ frac {\ parcial A_ {z}} {\ parcial y}} - {\ frac {\ parcial A_ {y}} {\ parcial z}} \ end {alineado}}}$

${\ Displaystyle {\ begin {alineado} \ operatorname {div} \ mathbf {A} & = \ lim _ {V \ to 0} {\ frac {\ iint _ {\ partial V} \ mathbf {A} \ cdot d \ mathbf {S}} {\ iiint _ {V} dV}} \\ & = {\ frac {A _ {\ rho} (\ rho + d \ rho) (\ rho + d \ rho) \, d \ phi \, dz-A _ {\ rho} (\ rho) \ rho \, d \ phi \, dz + A _ {\ phi} (\ phi + d \ phi) \, d \ rho \, dz-A _ {\ phi } (\ phi) \, d \ rho \, dz + A_ {z} (z + dz) \, d \ rho \, (\ rho + d \ rho / 2) \, d \ phi -A_ {z} (z) \, d \ rho (\ rho + d \ rho / 2) \, d \ phi} {\ rho \, d \ phi \, d \ rho \, dz}} \\ & = {\ frac { 1} {\ rho}} {\ frac {\ parcial (\ rho A _ {\ rho})} {\ parcial \ rho}} + {\ frac {1} {\ rho}} {\ frac {\ parcial A_ { \ phi}} {\ parcial \ phi}} + {\ frac {\ parcial A_ {z}} {\ parcial z}} \ end {alineado}}}$

${\ Displaystyle {\ begin {alineado} (\ operatorname {curl} \ mathbf {A}) _ {\ rho} & = \ lim _ {S ^ {\ perp {\ boldsymbol {\ hat {\ rho}}}} \ to 0} {\ frac {\ int _ {\ parcial S} \ mathbf {A} \ cdot d \ mathbf {\ ell}} {\ iint _ {S} dS}} \\ & = {\ frac {A_ {\ phi} (z) (\ rho + d \ rho) \, d \ phi -A _ {\ phi} (z + dz) (\ rho + d \ rho) \, d \ phi + A_ {z} ( \ phi + d \ phi) \, dz-A_ {z} (\ phi) \, dz} {(\ rho + d \ rho) \, d \ phi \, dz}} \\ & = - {\ frac {\ A _ {parcial} {\ phi}} {\ parcial z}} + {\ frac {1} {\ rho}} {\ frac {\ parcial A_ {z}} {\ parcial \ phi}} \ end {alineado} }}$

${\ Displaystyle {\ begin {alineado} (\ operatorname {curl} \ mathbf {A}) _ {\ phi} & = \ lim _ {S ^ {\ perp {\ boldsymbol {\ hat {\ phi}}}} \ to 0} {\ frac {\ int _ {\ parcial S} \ mathbf {A} \ cdot d \ mathbf {\ ell}} {\ iint _ {S} dS}} \\ & = {\ frac {A_ {z} (\ rho) \, dz-A_ {z} (\ rho + d \ rho) \, dz + A _ {\ rho} (z + dz) \, d \ rho -A _ {\ rho} (z ) \, d \ rho} {d \ rho \, dz}} \\ & = - {\ frac {\ parcial A_ {z}} {\ parcial \ rho}} + {\ frac {\ parcial A _ {\ rho }} {\ parcial z}} \ end {alineado}}}$

${\ Displaystyle {\ begin {alineado} (\ operatorname {curl} \ mathbf {A}) _ {z} & = \ lim _ {S ^ {\ perp {\ boldsymbol {\ hat {z}}}} \ to 0} {\ frac {\ int _ {\ parcial S} \ mathbf {A} \ cdot d \ mathbf {\ ell}} {\ iint _ {S} dS}} \\ & = {\ frac {A _ {\ rho} (\ phi) \, d \ rho -A _ {\ rho} (\ phi + d \ phi) \, d \ rho + A _ {\ phi} (\ rho + d \ rho) (\ rho + d \ rho) \, d \ phi -A _ {\ phi} (\ rho) \ rho \, d \ phi} {\ rho \, d \ rho \, d \ phi}} \\ & = - {\ frac {1 } {\ rho}} {\ frac {\ parcial A _ {\ rho}} {\ parcial \ phi}} + {\ frac {1} {\ rho}} {\ frac {\ parcial (\ rho A _ {\ phi })} {\ parcial \ rho}} \ end {alineado}}}$

${\ Displaystyle {\ begin {alineado} \ operatorname {curl} \ mathbf {A} & = (\ operatorname {curl} \ mathbf {A}) _ {\ rho} {\ hat {\ boldsymbol {\ rho}}} + (\ operatorname {curl} \ mathbf {A}) _ {\ phi} {\ hat {\ boldsymbol {\ phi}}} + (\ operatorname {curl} \ mathbf {A}) _ {z} {\ hat {\ boldsymbol {z}}} \\ & = \ left ({\ frac {1} {\ rho}} {\ frac {\ parcial A_ {z}} {\ parcial \ phi}} - {\ frac {\ A _ {\ phi}} {\ z} parcial \ right) {\ hat {\ boldsymbol {\ rho}}} + \ left ({\ frac {\ parcial A _ {\ rho}} {\ parcial z}} - {\ frac {\ parcial A_ {z}} {\ parcial \ rho}} \ derecha) {\ hat {\ boldsymbol {\ phi}}} + {\ frac {1} {\ rho}} \ left ({ \ frac {\ parcial (\ rho A _ {\ phi})} {\ parcial \ rho}} - {\ frac {\ parcial A _ {\ rho}} {\ parcial \ phi}} \ derecha) {\ hat {\ símbolo en negrita {z}}} \ end {alineado}}}$

${\ Displaystyle {\ begin {alineado} \ operatorname {div} \ mathbf {A} & = \ lim _ {V \ to 0} {\ frac {\ iint _ {\ partial V} \ mathbf {A} \ cdot d \ mathbf {S}} {\ iiint _ {V} dV}} \\ & = {\ frac {A_ {r} (r + dr) (r + dr) \, d \ theta \, (r + dr) \ sin \ theta \, d \ phi -A_ {r} (r) r \, d \ theta \, r \ sin \ theta \, d \ phi + A _ {\ theta} (\ theta + d \ theta) \ sin (\ theta + d \ theta) r \, dr \, d \ phi -A _ {\ theta} (\ theta) \ sin (\ theta) r \, dr \, d \ phi + A _ {\ phi} ( \ phi + d \ phi) r \, dr \, d \ theta -A _ {\ phi} (\ phi) r \, dr \, d \ theta} {dr \, r \, d \ theta \, r \ sin \ theta \, d \ phi}} \\ & = {\ frac {1} {r ^ {2}}} {\ frac {\ partial (r ^ {2} A_ {r})} {\ partid r }} + {\ frac {1} {r \ sin \ theta}} {\ frac {\ parcial (A _ {\ theta} \ sin \ theta)} {\ parcial \ theta}} + {\ frac {1} { r \ sin \ theta}} {\ frac {\ parcial A _ {\ phi}} {\ parcial \ phi}} \ end {alineado}}}$

${\ displaystyle {\ begin {alineado} (\ operatorname {curl} \ mathbf {A}) _ {r} = \ lim _ {S ^ {\ perp {\ boldsymbol {\ hat {r}}}} \ to 0 } {\ frac {\ int _ {\ parcial S} \ mathbf {A} \ cdot d \ mathbf {\ ell}} {\ iint _ {S} dS}} & = {\ frac {A _ {\ theta} ( \ phi) r \, d \ theta + A _ {\ phi} (\ theta + d \ theta) r \ sin (\ theta + d \ theta) \, d \ phi -A _ {\ theta} (\ phi + d \ phi) r \, d \ theta -A _ {\ phi} (\ theta) r \ sin (\ theta) \, d \ phi} {r \, d \ theta \, r \ sin \ theta \, d \ phi}} \\ & = {\ frac {1} {r \ sin \ theta}} {\ frac {\ parcial (A _ {\ phi} \ sin \ theta)} {\ parcial \ theta}} - {\ frac {1} {r \ sin \ theta}} {\ frac {\ parcial A _ {\ theta}} {\ parcial \ phi}} \ end {alineado}}}$

${\ displaystyle {\ begin {alineado} (\ operatorname {curl} \ mathbf {A}) _ {\ theta} = \ lim _ {S ^ {\ perp {\ boldsymbol {\ hat {\ theta}}}} \ a 0} {\ frac {\ int _ {\ parcial S} \ mathbf {A} \ cdot d \ mathbf {\ ell}} {\ iint _ {S} dS}} & = {\ frac {A _ {\ phi } (r) r \ sin \ theta \, d \ phi + A_ {r} (\ phi + d \ phi) \, dr-A _ {\ phi} (r + dr) (r + dr) \ sin \ theta \, d \ phi -A_ {r} (\ phi) \, dr} {dr \, r \ sin \ theta \, d \ phi}} \\ & = {\ frac {1} {r \ sin \ theta }} {\ frac {\ parcial A_ {r}} {\ parcial \ phi}} - {\ frac {1} {r}} {\ frac {\ parcial (rA _ {\ phi})} {\ parcial r} } \ end {alineado}}}$

${\ Displaystyle {\ begin {alineado} (\ operatorname {curl} \ mathbf {A}) _ {\ phi} = \ lim _ {S ^ {\ perp {\ boldsymbol {\ hat {\ phi}}}} \ a 0} {\ frac {\ int _ {\ parcial S} \ mathbf {A} \ cdot d \ mathbf {\ ell}} {\ iint _ {S} dS}} & = {\ frac {A_ {r} (\ theta) \, dr + A _ {\ theta} (r + dr) (r + dr) \, d \ theta -A_ {r} (\ theta + d \ theta) \, dr-A _ {\ theta} (r) r \, d \ theta} {r \, dr \, d \ theta}} \\ & = {\ frac {1} {r}} {\ frac {\ parcial (rA _ {\ theta})} {\ parcial r}} - {\ frac {1} {r}} {\ frac {\ parcial A_ {r}} {\ parcial \ theta}} \ end {alineado}}}$

${\ Displaystyle \ operatorname {curl} \ mathbf {A} = (\ operatorname {curl} \ mathbf {A}) _ {r} \, {\ hat {\ boldsymbol {r}}} + (\ operatorname {curl} \ mathbf {A}) _ {\ theta} \, {\ hat {\ boldsymbol {\ theta}}} + (\ operatorname {curl} \ mathbf {A}) _ {\ phi} \, {\ hat {\ símbolo en negrita {\ phi}}} = {\ frac {1} {r \ sin \ theta}} \ left ({\ frac {\ partial (A _ {\ phi} \ sin \ theta)} {\ partial \ theta}} - {\ frac {\ parcial A _ {\ theta}} {\ parcial \ phi}} \ right) {\ hat {\ boldsymbol {r}}} + {\ frac {1} {r}} \ left ({\ frac {1} {\ sin \ theta}} {\ frac {\ parcial A_ {r}} {\ parcial \ phi}} - {\ frac {\ parcial (rA _ {\ phi})} {\ parcial r}} \ right) {\ hat {\ boldsymbol {\ theta}}} + {\ frac {1} {r}} \ left ({\ frac {\ parcial (rA _ {\ theta})} {\ parcial r}} - {\ frac {\ parcial A_ {r}} {\ parcial \ theta}} \ right) {\ hat {\ boldsymbol {\ phi}}}}$