91影视

Green鈥檚 first and second identities are given.

$鈭�\underset{\text{inside}蟽}{\text{volume}}\left(蠒{\mathrm{鈭�}}^2蠄+\mathrm{鈭�}蠒脳\mathrm{鈭�}蠄\right)d蟿=\text{鈭�}\underset{\text{surface}蟽}{\text{dosed}}\left(蠒\mathrm{鈭�}蠄\right)脳\mathbf{nd}d蟽$

${鈭�}_{\begin{array}{c}\text{volume}\\ \text{inside}蟽\end{array}}鈥�\left(蠒{\mathrm{鈭�}}^2蠄鈭�蠄{\mathrm{鈭�}}^2蠒\right)d蟿={\text{鈭�}}_{\begin{array}{c}\text{closed}\\ \text{surface}蟽\end{array}}(蠒\mathrm{鈭�}蠄鈭�蠄\mathrm{鈭�}蠒)脳\text{nd}d$

Divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem in vector calculus, is a theorem that connects the flux of a vector field through a closed surface to the field's divergence in the volume enclosed.

Part (a):

Apply the Gauss theorem.

${\text{鈭�}}_{\mathrm{鈭�}z}\left(蠒\mathrm{鈭�}蠄\right)脳\mathrm{n}d蟽={鈭�}_z鈥�\mathrm{鈭�}脳\left(蠒\mathrm{鈭�}蠄\right)d蟿$

Find the value of $\mathrm{鈭�}脳\left(蠒\mathrm{鈭�}蠄\right)$.

$\mathrm{鈭�}脳\left(蠒\mathrm{鈭�}蠄\right)=蠒(鈭�脳鈭�蠄)+\mathrm{鈭�}蠄脳\mathrm{鈭�}蠒$

$=蠒{\mathrm{鈭�}}^2蠄+\mathrm{鈭�}蠄脳\mathrm{鈭�}蠒$

Substitute equation $\left(2\right)$in $\left(1\right)$.

${\text{鈭�}}_{\mathrm{鈭�}z}\left(蠒\mathrm{鈭�}蠄\right)脳\mathrm{n}d蟽={鈭�}_z鈥�\left(蠒{\mathrm{鈭�}}^2蠄+\mathrm{鈭�}蠄脳\mathrm{鈭�}蠒\right)d蟿$

Hence, the result has been proved by using Gauss Theorem.

Part (b):

It is known that

${\text{鈭�}}_{\mathrm{鈭�}z}\left[\right(蠒\mathrm{鈭�}蠄)鈭�(蠒\mathrm{鈭�}蠄\left)\right]脳\mathrm{nd}鈦�蟽={\text{鈭�}}_{\mathrm{鈭�}z}\left(蠒\mathrm{鈭�}蠄\right)脳\mathrm{n}d蟽鈭�{\text{鈭�}}_{\mathrm{鈭�}z}\left(蠄\mathrm{鈭�}蠒\right)脳\mathrm{n}d蟽$

Use the result from part (a) in equation (3).

${\text{鈭�}}_{\mathrm{鈭�}z}\left[\left(蠒\mathrm{鈭�}蠄\right)鈭�\left(蠒\mathrm{鈭�}蠄\right]脳\mathrm{n}d蟽={鈭�}_z鈥�\left(蠒{\mathrm{鈭�}}^2蠄+\mathrm{鈭�}蠄脳\mathrm{鈭�}蠒\right)d蟿鈭�{鈭�}_z鈥�\left(蠄{\mathrm{鈭�}}^2蠒+\mathrm{鈭�}蠒脳\mathrm{鈭�}蠄\right)d蟿\right.$

${\text{鈭�}}_{\mathrm{鈭�}z}\left[\left(蠒\mathrm{鈭�}蠄\right)鈭�\left(蠒\mathrm{鈭�}蠄\right]脳\mathrm{nd}蟽={鈭�}_z鈥�\left(蠒{\mathrm{鈭�}}^2蠄鈭�蠄{\mathrm{鈭�}}^2蠒\right)d蟿\right.$

Hence, the result has been proved by using Gauss Theorem.