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I can not figure out how we obtained the ratio:

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ِA follow up question:

Where do we use the assumption that $u$ and $v$ are linearly independent ? Thanks a lot

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closed as unclear what you're asking by Harry49, Ernie060, Feng Shao, Theoretical Economist, воитель Sep 11 at 21:10

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  • $\begingroup$ I have edited the question. It should be clear and simple now. Thanks $\endgroup$ – Medo Sep 12 at 2:58
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I think this will clarify your doubts.

(I just skipping the previous part)

$$\frac{\partial u}{\partial x}~dx ~+~\frac{\partial u}{\partial y}~dy ~+~\frac{\partial u}{\partial z}~dz ~=~0 \tag1$$ $$\frac{\partial v}{\partial x}~dx ~+~\frac{\partial v}{\partial y}~dy ~+~\frac{\partial v}{\partial z}~dz ~=~0 \tag2$$ By cross-multiplication, $$\frac{dx}{\frac{\partial u}{\partial y}~\frac{\partial v}{\partial z}~-~\frac{\partial u}{\partial z}~\frac{\partial v}{\partial y}}=\frac{dy}{\frac{\partial u}{\partial z}~\frac{\partial v}{\partial x}~-~\frac{\partial u}{\partial x}~\frac{\partial v}{\partial z}}=\frac{dz}{\frac{\partial u}{\partial x}~\frac{\partial v}{\partial y}~-~\frac{\partial u}{\partial y}~\frac{\partial v}{\partial x}}\tag3$$ which can be written in the following ratio form,$$dx:dy:dz=\left(\frac{\partial u}{\partial y}~\frac{\partial v}{\partial z}~-~\frac{\partial u}{\partial z}~\frac{\partial v}{\partial y}\right):\left(\frac{\partial u}{\partial z}~\frac{\partial v}{\partial x}~-~\frac{\partial u}{\partial x}~\frac{\partial v}{\partial z}\right):\left(\frac{\partial u}{\partial x}~\frac{\partial v}{\partial y}~-~\frac{\partial u}{\partial y}~\frac{\partial v}{\partial x}\right)\tag4$$ Again $$\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}\tag5$$can also written in the following ration form $$dx:dy:dz=p:Q:R\tag6$$ Combining $(4)$ and $(6)$, we have $$dx:dy:dz=\left(\frac{\partial u}{\partial y}~\frac{\partial v}{\partial z}~-~\frac{\partial u}{\partial z}~\frac{\partial v}{\partial y}\right):\left(\frac{\partial u}{\partial z}~\frac{\partial v}{\partial x}~-~\frac{\partial u}{\partial x}~\frac{\partial v}{\partial z}\right):\left(\frac{\partial u}{\partial x}~\frac{\partial v}{\partial y}~-~\frac{\partial u}{\partial y}~\frac{\partial v}{\partial x}\right)=P:Q:R$$

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  • $\begingroup$ Thanks a lot for your answer. I still do not understand how we can obtain (3). When you say "cross multiplication" what do mean ? Is it the cross multiplication of two vectors ? So we have two zero vectors (the vectors in (1) and (2)) whose cross multiplication must vanish, so each component of the cross product is zero? $\endgroup$ – Medo Sep 11 at 15:02
  • $\begingroup$ For cross-multiplication, see the link math-only-math.com/cross-multiplication-method.html $\endgroup$ – nmasanta Sep 11 at 16:52
  • $\begingroup$ I see. It is just solving for $dx$, $dy$ and $dz$ after all. Thanks a lot. $\endgroup$ – Medo Sep 12 at 2:44
  • $\begingroup$ You are welcome @Medo $\endgroup$ – nmasanta Sep 12 at 2:45
  • $\begingroup$ Where do we use the assumption that $u$ and $v$ are linearly independent ? I am guessing it implies that the denominator of $dz$ does not vanish..but how? $\endgroup$ – Medo Sep 12 at 3:15

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