User:Egm6936.s09/Test Word macros

$$\begin{align} \overleftarrow{\text{u}} \\ \text{u}\times \text{u}\overleftarrow{\nabla }\times \text{n}=\left( {{u}_{r}}{{\text{e}}_{r}}+{{u}_{\theta }}{{\text{e}}_{\theta }} \right)\times \left[ \frac{\partial {{u}_{r}}}{\partial r}{{\text{e}}_{r}}{{\text{e}}_{r}}+\frac{1}{r}\left( \frac{\partial {{u}_{r}}}{\partial \theta }-{{u}_{\theta }} \right){{\text{e}}_{r}}{{\text{e}}_{\theta }}+\frac{\partial {{u}_{\theta }}}{\partial r}{{\text{e}}_{\theta }}{{\text{e}}_{r}}+\frac{1}{r}\left( \frac{\partial {{u}_{\theta }}}{\partial \theta }+{{u}_{r}} \right){{\text{e}}_{\theta }}{{\text{e}}_{\theta }} \right]\times {{\text{e}}_{r}}=-\frac{1}{r}\left( u_{r}^{2}+u_{\theta }^{2}+{{u}_{r}}{{u}_{\theta ,\theta }}-{{u}_{\theta }}{{u}_{r,\theta }} \right){{\text{e}}_{z}}{{\text{e}}_{z}} \\ \end{align}$$

$$\begin{align} & {{\text{n}}_{t}}d{{s}_{t}}=\det \left( {{\text{F}}_{t}} \right)\text{F}_{t}^{-T}\bullet \text{n}ds \\ & {{\text{F}}_{t}}=\frac{\partial {{\text{x}}_{t}}}{\partial \text{x}}=\frac{\partial \left( \text{x}+\text{u}t \right)}{\partial \text{x}}=\text{I}+t\frac{\partial \text{u}}{\partial \text{x}}=\text{I}+t\text{u}\overleftarrow{\nabla } \\ & \det \left( {{\text{F}}_{t}} \right)=1+t{{I}_{1}}+{{t}^{2}}{{I}_{2}}+{{t}^{3}}{{I}_{3}}\,\,\,\,\,\,\text{u}\overleftarrow{\nabla }=\frac{\partial \text{u}}{\partial \text{x}} \\ & {{I}_{1}}=tr\left( \text{u}\overleftarrow{\nabla } \right),\,\,{{I}_{2}}=\frac{1}{2}\left[ t{{r}^{2}}\left( \text{u}\overleftarrow{\nabla } \right)-tr\left( \text{u}\overleftarrow{\nabla }\bullet \text{u}\overleftarrow{\nabla } \right) \right],\,\,{{I}_{3}}=\det \left( \text{u}\overleftarrow{\nabla } \right) \\ & \text{F}_{t}^{-1}={{\left( \text{I}+t\text{u}\overleftarrow{\nabla } \right)}^{-1}}\approx \text{I}-t\text{u}\overleftarrow{\nabla }+{{t}^{2}}{{\left( \text{u}\overleftarrow{\nabla } \right)}^{2}}-{{t}^{3}}{{\left( \text{u}\overleftarrow{\nabla } \right)}^{3}}+\cdots \\ & \text{F}_{t}^{-T}={{\left( \text{I}+t\text{u}\overleftarrow{\nabla } \right)}^{-T}}\approx \text{I}-t\nabla \text{u}+{{t}^{2}}{{\left( \nabla \text{u} \right)}^{2}}-{{t}^{3}}{{\left( \nabla \text{u} \right)}^{3}}+\cdots \,\,\,\,\,\nabla \text{u}={{\left( \frac{\partial \text{u}}{\partial \text{x}} \right)}^{T}}\,\,\,\left| \text{u}\overleftarrow{\nabla } \right|<<1 \\ & \text{u}\bullet {{\text{n}}_{t}}d{{s}_{t}}=\det \left( {{\text{F}}_{t}} \right)\text{u}\bullet \text{F}_{t}^{-T}\bullet \text{n}ds\approx \left( 1+t{{I}_{1}}+{{t}^{2}}{{I}_{2}}+{{t}^{3}}{{I}_{3}} \right)\text{u}\bullet \left( \text{I}-t\nabla \text{u}+{{t}^{2}}{{\left( \nabla \text{u} \right)}^{2}}-{{t}^{3}}{{\left( \nabla \text{u} \right)}^{3}}+\cdots \right)\bullet \text{n}ds \\ & =\text{u}\bullet \left( \text{I}+t{{I}_{1}}\text{I}-t\nabla \text{u}+{{t}^{2}}{{\left( \nabla \text{u} \right)}^{2}}-{{t}^{2}}{{I}_{1}}\nabla \text{u}+{{t}^{2}}{{I}_{2}}\text{I}+\cdots \right)\bullet \text{n}ds=\text{u}\bullet \left\{ \text{I}+t\left( {{I}_{1}}\text{I}-\nabla \text{u} \right)+{{t}^{2}}\left[ {{\left( \nabla \text{u} \right)}^{2}}-{{I}_{1}}\nabla \text{u}+{{I}_{2}}\text{I} \right]+\cdots  \right\}\bullet \text{n}ds \\ & A_{e}^{1}=-P\int\limits_{0}^{1}{dt}\int\limits_{\det \left( {{\text{F}}_{t}} \right)\text{u}\bullet \text{F}_{t}^{-T}\bullet \text{n}ds}=-P\int\limits_{\text{u}\bullet \int\limits_{0}^{1}{\left\{ \text{I}+t\left( {{I}_{1}}\text{I}-\nabla \text{u} \right)+{{t}^{2}}\left[ {{\left( \nabla \text{u} \right)}^{2}}-{{I}_{1}}\nabla \text{u}+{{I}_{2}}\text{I} \right]+\cdots \right\}dt}\bullet \text{n}ds} \\ & =-P\int\limits_{\text{u}\bullet \left\{ \text{I}+\frac{1}{2}\left( {{I}_{1}}\text{I}-\nabla \text{u} \right)+\frac{1}{3}\left[ {{\left( \nabla \text{u} \right)}^{2}}-{{I}_{1}}\nabla \text{u}+{{I}_{2}}\text{I} \right]+\cdots \right\}\bullet \text{n}ds}\approx -P\int\limits_{\text{u}\bullet \left[ \text{I}+\frac{1}{2}\left( {{I}_{1}}\text{I}-\nabla \text{u} \right) \right]\bullet \text{n}ds} \\ & {{\left( {{\text{n}}_{i}} \right)}_{t}}{{\left( ds \right)}_{t}}=\frac{1}{2}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}\frac{\partial \left( {{x}_{j}}+t{{u}_{j}} \right)}{\partial {{x}_{m}}}\frac{\partial \left( {{x}_{k}}+t{{u}_{k}} \right)}{\partial {{x}_{n}}}{{n}_{p}}ds=\frac{1}{2}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}\frac{\partial \left( {{x}_{j}}+t{{u}_{j}} \right)}{\partial {{x}_{m}}}\frac{\partial \left( {{x}_{k}}+t{{u}_{k}} \right)}{\partial {{x}_{n}}}{{n}_{p}}ds \\ & =\frac{1}{2}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}\left( {{\delta }_{jm}}+t\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}} \right)\left( {{\delta }_{kn}}+t\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}} \right){{n}_{p}}ds=\frac{1}{2}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}\left( {{\delta }_{jm}}{{\delta }_{kn}}+t{{\delta }_{jm}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}}+t{{\delta }_{kn}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}+{{t}^{2}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}} \right){{n}_{p}}ds \\ & =\frac{1}{2}\left( {{\varepsilon }_{jki}}{{\varepsilon }_{jkp}}+t{{\varepsilon }_{jki}}{{\varepsilon }_{jnp}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}}+t{{\varepsilon }_{jki}}{{\varepsilon }_{mkp}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}+{{t}^{2}}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}} \right){{n}_{p}}ds \\ & =\frac{1}{2}\left[ \left( {{\delta }_{kk}}{{\delta }_{ip}}-{{\delta }_{ki}}{{\delta }_{kp}} \right)+t\left( {{\delta }_{kn}}{{\delta }_{ip}}-{{\delta }_{kp}}{{\delta }_{in}} \right)\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}}+t\left( {{\delta }_{jm}}{{\delta }_{ip}}-{{\delta }_{im}}{{\delta }_{jp}} \right)\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}+{{t}^{2}}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}} \right]{{n}_{p}}ds \\ & =\frac{1}{2}\left[ \left( 3{{\delta }_{ip}}-{{\delta }_{pi}} \right)+t\left( \frac{\partial {{u}_{k}}}{\partial {{x}_{k}}}{{\delta }_{ip}}-\frac{\partial {{u}_{p}}}{\partial {{x}_{i}}} \right)+t\left( \frac{\partial {{u}_{j}}}{\partial {{x}_{j}}}{{\delta }_{ip}}-\frac{\partial {{u}_{p}}}{\partial {{x}_{i}}} \right)+{{t}^{2}}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}} \right]{{n}_{p}}ds=\left( {{\delta }_{ip}}+t\frac{\partial {{u}_{k}}}{\partial {{x}_{k}}}{{\delta }_{ip}}-t\frac{\partial {{u}_{p}}}{\partial {{x}_{i}}}+\frac{1}{2}{{t}^{2}}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}} \right){{n}_{p}}ds \\ & {{u}_{i}}{{\left( {{\text{n}}_{i}} \right)}_{t}}{{\left( ds \right)}_{t}}={{u}_{i}}\left( {{\delta }_{ip}}+t\frac{\partial {{u}_{k}}}{\partial {{x}_{k}}}{{\delta }_{ip}}-\frac{\partial {{u}_{p}}}{\partial {{x}_{i}}}+\frac{1}{2}{{t}^{2}}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}} \right){{n}_{p}}ds=\left( {{u}_{i}}{{n}_{i}}+t{{u}_{i}}{{n}_{i}}\frac{\partial {{u}_{k}}}{\partial {{x}_{k}}}-t{{u}_{i}}\frac{\partial {{u}_{p}}}{\partial {{x}_{i}}}{{n}_{p}}+\frac{1}{2}{{t}^{2}}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}{{u}_{i}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}}{{n}_{p}} \right)ds \\ & A_{e}^{1}=-P\int\limits_{0}^{1}{dt}\int\limits_=-P\int\limits_{0}^{1}{dt}\int\limits_{\left( {{u}_{i}}{{n}_{i}}+t{{u}_{i}}{{n}_{i}}\frac{\partial {{u}_{k}}}{\partial {{x}_{k}}}-t{{u}_{i}}\frac{\partial {{u}_{p}}}{\partial {{x}_{i}}}{{n}_{p}}+\frac{1}{2}{{t}^{2}}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}{{u}_{i}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}}{{n}_{p}} \right)ds} \\ & =-P\int\limits_{\left( {{u}_{i}}{{n}_{i}}+\frac{1}{2}{{u}_{i}}{{n}_{i}}\frac{\partial {{u}_{k}}}{\partial {{x}_{k}}}-\frac{1}{2}{{u}_{i}}\frac{\partial {{u}_{p}}}{\partial {{x}_{i}}}{{n}_{p}}+\frac{1}{6}{{\varepsilon }_{jki}}{{\varepsilon }_{mnp}}{{u}_{i}}\frac{\partial {{u}_{j}}}{\partial {{x}_{m}}}\frac{\partial {{u}_{k}}}{\partial {{x}_{n}}}{{n}_{p}} \right)ds}=-P\int\limits_{\left\{ \text{u}\bullet \text{n}ds+\frac{1}{2}\left[ \left( \nabla \bullet \text{u} \right)\text{u}\bullet \text{n}-\text{u}\bullet \left( \nabla \text{u} \right)\bullet \text{n} \right]+\frac{1}{6}\left( \text{u}\times \text{u}\overleftarrow{\nabla }\text{:u}\overleftarrow{\nabla }\times \text{n} \right) \right\}ds} \\ \end{align}$$