(adsbygoogle = window.adsbygoogle || []).push({}); 1. By considering, seperately, each component of the vector A, show that [tex] \iint A(u.n) ds = \iiint {(u.\nabla)A + A(\nabla.u)} dV [/tex] (A,u and n are vectors)

2. Relevant equations

3. Attempt at solution

L.H.S.

Let A = [tex] a\vec{i} + b\vec{j} + c\vec{k}[/tex]

[tex]

\iint (a\vec{i} + b\vec{j} + c\vec{k})[(u_1\vec{i} + u_2 \vec{j} + u_3 \vec{k}).(n_1 \vec{i} + n_2 \vec {j} + n_3 \vec{k})] ds

= \iint (a\vec{i} + b\vec{j} + c\vec{k})(u_1 n_1 + u_2 n_2 + u_3 n_3) ds

= \iint au_1n_1\vec{i} + au_2n_2\vec{i} + au_3n_3\vec{i} + bu_1n_1\vec{j} [/tex][tex]

+ bu_2n_2\vec{j} + bu_3n_3\vec{j} + cu_1n_1\vec{k} + cu_2n_2\vec{k} + cu_3n_3\vec{k} ds[/tex]

R.H.S.[tex]

\iiint(u_1a_x\vec{i}+u_1b_x\vec{j}+u_1c_x\vec{k}+u_2a_y\vec{i}+u_2b_y\vec{j}+u_2c_y\vec{k}+u_3a_z\vec{i}+u_3b_z\vec{j}+u_3c_z\vec{k}+u_1a_x\vec{i}+u_1b_y\vec{i}+u_1c_z\vec{i}+u_2a_x\vec{j}+u_2b_y\vec{j}+u_2c_z\vec{j}+u_3a_x\vec{k}+u_3b_y\vec{k}+u_3c_z\vec{k})dv[/tex]

Is this right? Where to now? Thanks

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# Homework Help: Integral Identity

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