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WeinGarten-Equation-r-u-r-u-E-r-u-r-v-F-r-v-r-v-G-N-N-1-N-N-u-N-u-N-N-N-u-0-N-N-v-N-v-N-N-N-v-0-N-N-0-N-N-N

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Question Number 225768 by Lara2440 last updated on 09/Nov/25
WeinGarten Equation r_u ∙r_u =E , r_u ∙r_v =F , r_v ∙r_v =G N^� ∙N^� =1 ((∂(N^� ∙N^� ))/∂u)=N_u ^� ∙N^� +N^� ∙N_u ^� =0 ((∂(N^� ∙N^� ))/∂v)=N_v ^� ∙N^� +N^� ∙N_v ^� =0 N_μ ^� ∙N^� =0 ⇋ N_μ ^� ⊥N^� N_u ^� ∙r_u ^� =(Ar_u ^� +Br_v ^� )∙r_u ^� → =Ar_u ^� ∙r_u ^� +Br_u ^� ∙r_v ^� =AE+BF N_u ^� ∙r_v ^� =(Ar_u ^� +Br_v ^� )∙r_v ^� → =Ar_u ^� ∙r_v ^� +Br_v ^� ∙r_v ^� =AF+BG N_v ^� ∙r_u ^� =(Cr_u ^� +Dr_v ^� )∙r_u ^� →=Cr_u ^� ∙r_u ^� +Dr_v ^� ∙r_u ^� =CE+DF N_u ^� ∙r_v ^� =(Cr_u ^� +Dr_v ^� )∙r_v ^� → =Cr_u ^� ∙r_v ^� +Dr_v ^� ∙r_v ^� =CF+DG N_u ^� ∙r_u ^� =−L N_u ^� ∙r_v ^� =−M N_v ^� ∙r_u ^� =−M N_v ^� ∙r_v ^� =−N −L=AE+BF −M=AF+BG −M=CE+DF −N=CF+DG (((−L)),((−M)) )= ((E,F),(F,G) ) ((A),(B) ) (((−M)),((−N)) )= ((E,F),(F,G) ) ((C),(D) ) n_u ^� =((M′F′−G′L′)/(E′G′−F′^2 )) r_u ^� +((L′F′−E′M′)/(E′G′−F′^2 )) r_v ^� n_v ^� =((N′F′−G′M′)/(E′G′−F′^2 )) r_u ^� +((M′F′−E′N′)/(E′G′−F′^2 )) r_v ^�
$$\mathrm{WeinGarten}\:\mathrm{Equation} \\ $$$$\:\boldsymbol{\mathrm{r}}_{{u}} \centerdot\boldsymbol{\mathrm{r}}_{{u}} ={E}\:,\:\boldsymbol{\mathrm{r}}_{{u}} \centerdot\boldsymbol{\mathrm{r}}_{{v}} ={F}\:,\:\boldsymbol{\mathrm{r}}_{{v}} \centerdot\boldsymbol{\mathrm{r}}_{{v}} ={G} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}\centerdot\hat {\boldsymbol{\mathrm{N}}}=\mathrm{1} \\ $$$$\frac{\partial\left(\hat {\boldsymbol{\mathrm{N}}}\centerdot\hat {\boldsymbol{\mathrm{N}}}\right)}{\partial{u}}=\hat {\boldsymbol{\mathrm{N}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{N}}}+\hat {\boldsymbol{\mathrm{N}}}\centerdot\hat {\boldsymbol{\mathrm{N}}}_{{u}} =\mathrm{0} \\ $$$$\frac{\partial\left(\hat {\boldsymbol{\mathrm{N}}}\centerdot\hat {\boldsymbol{\mathrm{N}}}\right)}{\partial{v}}=\hat {\boldsymbol{\mathrm{N}}}_{{v}} \centerdot\hat {\boldsymbol{\mathrm{N}}}+\hat {\boldsymbol{\mathrm{N}}}\centerdot\hat {\boldsymbol{\mathrm{N}}}_{{v}} =\mathrm{0} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}_{\mu} \centerdot\hat {\boldsymbol{\mathrm{N}}}=\mathrm{0}\:\leftrightharpoons\:\hat {\boldsymbol{\mathrm{N}}}_{\mu} \bot\hat {\boldsymbol{\mathrm{N}}} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{u}} =\left({A}\hat {\boldsymbol{\mathrm{r}}}_{{u}} +{B}\hat {\boldsymbol{\mathrm{r}}}_{{v}} \right)\centerdot\hat {\boldsymbol{\mathrm{r}}}_{{u}} \:\rightarrow\:={A}\hat {\boldsymbol{\mathrm{r}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{u}} +{B}\hat {\boldsymbol{\mathrm{r}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} ={AE}+{BF} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} =\left({A}\hat {\boldsymbol{\mathrm{r}}}_{{u}} +{B}\hat {\boldsymbol{\mathrm{r}}}_{{v}} \right)\centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} \:\rightarrow\:={A}\hat {\boldsymbol{\mathrm{r}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} +{B}\hat {\boldsymbol{\mathrm{r}}}_{{v}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} ={AF}+{BG} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}_{{v}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{u}} =\left({C}\hat {\boldsymbol{\mathrm{r}}}_{{u}} +{D}\hat {\boldsymbol{\mathrm{r}}}_{{v}} \right)\centerdot\hat {\boldsymbol{\mathrm{r}}}_{{u}} \:\rightarrow={C}\hat {\boldsymbol{\mathrm{r}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{u}} +{D}\hat {\boldsymbol{\mathrm{r}}}_{{v}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{u}} ={CE}+{DF} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} =\left({C}\hat {\boldsymbol{\mathrm{r}}}_{{u}} +{D}\hat {\boldsymbol{\mathrm{r}}}_{{v}} \right)\centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} \rightarrow\:={C}\hat {\boldsymbol{\mathrm{r}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} +{D}\hat {\boldsymbol{\mathrm{r}}}_{{v}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} ={CF}+{DG} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{u}} =−{L} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}_{{u}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} =−{M} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}_{{v}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{u}} =−{M} \\ $$$$\hat {\boldsymbol{\mathrm{N}}}_{{v}} \centerdot\hat {\boldsymbol{\mathrm{r}}}_{{v}} =−{N} \\ $$$$−{L}={AE}+{BF}\:\:\:\:\:\:\:\: \\ $$$$−{M}={AF}+{BG} \\ $$$$−{M}={CE}+{DF} \\ $$$$−{N}={CF}+{DG} \\ $$$$\begin{pmatrix}{−{L}}\\{−{M}}\end{pmatrix}=\begin{pmatrix}{{E}}&{{F}}\\{{F}}&{{G}}\end{pmatrix}\begin{pmatrix}{{A}}\\{{B}}\end{pmatrix} \\ $$$$\begin{pmatrix}{−{M}}\\{−{N}}\end{pmatrix}=\begin{pmatrix}{{E}}&{{F}}\\{{F}}&{{G}}\end{pmatrix}\begin{pmatrix}{{C}}\\{{D}}\end{pmatrix} \\ $$$$\hat {\boldsymbol{\mathrm{n}}}_{{u}} =\frac{{M}'{F}'−{G}'{L}'}{{E}'{G}'−{F}'^{\mathrm{2}} }\:\hat {\boldsymbol{\mathrm{r}}}_{{u}} +\frac{{L}'{F}'−{E}'{M}'}{{E}'{G}'−{F}'^{\mathrm{2}} }\:\hat {\boldsymbol{\mathrm{r}}}_{{v}} \\ $$$$\hat {\boldsymbol{\mathrm{n}}}_{{v}} =\frac{{N}'{F}'−{G}'{M}'}{{E}'{G}'−{F}'^{\mathrm{2}} }\:\hat {\boldsymbol{\mathrm{r}}}_{{u}} +\frac{{M}'{F}'−{E}'{N}'}{{E}'{G}'−{F}'^{\mathrm{2}} }\:\hat {\boldsymbol{\mathrm{r}}}_{{v}} \\ $$

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