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@ -254,15 +254,16 @@ $`=dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z}`$< |
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[EN] y its norm (or length) is thescalar line element :<br> |
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<br>$`||\overrightarrow{dl}||=\sqrt{dl_x^2+dl_y^2+dl_z^2}=\sqrt{dx^2+dy^2+dz^2}`$<br> |
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<br>$`||\overrightarrow{dl}||=\sqrt{\overrightarrow{dl}\cdot\overrightarrow{dl}}`$ |
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$`=\sqrt{(dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z})\cdot |
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(dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z})}`$ |
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$`=\sqrt{dx^2\;(\overrightarrow{e_x}\cdot\overrightarrow{e_x}) |
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+dy^2\;(\overrightarrow{e_y}\cdot\overrightarrow{e_y}) |
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+dz^2\;(\overrightarrow{e_z}\cdot\overrightarrow{e_z})`$ |
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$`+2\,dx\,dy\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_y})`$ |
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$`+2\,dx\,dz\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_z})`$ |
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$`+2\,dy\,dz\,x(\overrightarrow{e_y}\cdot\overrightarrow{e_z})}`$ |
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$`=\sqrt{(dl_x\;\overrightarrow{e_x}+dl_y\;\overrightarrow{e_y}+dl_z\;\overrightarrow{e_z})\cdot |
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(dl_x\;\overrightarrow{e_x}+dl_y\;\overrightarrow{e_y}+dl_z\;\overrightarrow{e_z})}`$ |
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$`=\sqrt{dl_x^2\;(\overrightarrow{e_x}\cdot\overrightarrow{e_x}) |
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+dl_y^2\;(\overrightarrow{e_y}\cdot\overrightarrow{e_y}) |
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+dl_z^2\;(\overrightarrow{e_z}\cdot\overrightarrow{e_z})`$ |
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$`+2\,dl_x\,dl_y\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_y})`$ |
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$`+2\,dl_x\,dl_z\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_z})`$ |
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$`+2\,dl_y\,dl_z\,x(\overrightarrow{e_y}\cdot\overrightarrow{e_z})}`$ |
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$`=\sqrt{dl_x^2+dl_y^2+dl_z^2}`$ |
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$`=\sqrt{dx^2+dy^2+dz^2}`$ |
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* **N3 ($`\rightarrow`$ N4)**<br> |
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