Vector Laplacian/Proofs

The following is the proof that

:$abla\; ^2\; left(\; \{mathbf\{u\; ight)\; =\; abla\; left(\; \{\; abla\; cdot\{mathbf\{u\}\; ight)\; -\; abla\; imes\; left(\; \{\; abla\; imes\; \{mathbf\{u\}\; ight)\; =\; leftlangle\; \{\; abla\; ^2\; u\_x\; ,\; abla\; ^2\; u\_y\; ,\; abla\; ^2\; u\_z\; \}\; ight\; angle.$

This is a proof in Cartesian coordinates. This identity is only valid for Cartesian coordinates; completing the calculations in other coordinate systems will result in other solutions.

Let $\{mathbf\{u\; =\; u,\{mathbf\{hat\; e\_\{mathbf\{x\; +\; v,\{mathbf\{hat\; e\_\{mathbf\{y\; +\; w,\{mathbf\{hat\; e\_\{mathbf\{z\; =\; leftlangle\; \{u,v,w\}\; ight\; angle$
where $u\; =\; u\; left(\; \{x,y,z\}\; ight)$, $v\; =\; v\; left(\; \{x,y,z\}\; ight)$ and $w=\; w\; left(\; \{x,y,z\}\; ight)$

:

abla ^2 left( {mathbf{u ight) &= abla left( { abla cdot leftlangle {u,v,w} ight angle } ight) - abla imes left( { abla imes leftlangle {u,v,w} ight angle } ight) \

&= abla left( {fracpartial upartial z} ight) - abla imes left| {egin{array}{*{20}c} mathbf{hat e_{mathbf{x } & mathbf{hat e_{mathbf{y } & mathbf{hat e_{mathbf{z } \ {frac{partial }partial x} & {frac{partial }partial y} & {frac{partial }partial z} \ u & v & w \

end{array} } ight| \

&= leftlangle {fracpartial ^2 upartial y} ight angle \ &= left( {fracpartial ^2 upartial ypartial z} ight){mathbf{hat e_{mathbf{z - left| {egin{array}{*{20}c} mathbf{hat e_{mathbf{x } & mathbf{hat e_{mathbf{y } & mathbf{hat e_{mathbf{z } \ {frac{partial }partial x} & {frac{partial }partial y} & {frac{partial }partial z} \ {fracpartial wpartial y} \

end{array} } ight| \ &= left( {egin{array}{*{20}c} {fracpartial ^2 upartial xpartial z} \ {fracpartial ^2 vpartial ypartial z} \ {fracpartial ^2 wpartial ypartial z} \

end{array} } ight)left{ mathbf{hat e_{mathbf{x ,{mathbf{hat e_{mathbf{y ,{mathbf{hat e_{mathbf{z } ight} + left( {egin{array}{*{20}c} { - fracpartial ^2 vpartial xpartial z} \ {fracpartial ^2 vpartial z^2 } \ { - fracpartial ^2 upartial ypartial z} \

end{array} } ight),left{ mathbf{hat e_{mathbf{x ,,{mathbf{hat e_{mathbf{y ,,{mathbf{hat e_{mathbf{z } ight} \ &= left( {egin{array}{*{20}c} {fracpartial ^2 upartial z^2 } \ {fracpartial ^2 vpartial y^2 } \ {fracpartial ^2 wpartial z^2 } \

end{array} } ight)left{ mathbf{hat e_{mathbf{x ,{mathbf{hat e_{mathbf{y ,{mathbf{hat e_{mathbf{z } ight} \

&= abla ^2 u,{mathbf{hat e_{mathbf{x + abla ^2 v,{mathbf{hat e_{mathbf{y + abla ^2 w,{mathbf{hat e_{mathbf{z \ end{align}