I'm confused by the following passage in our book (translated).(adsbygoogle = window.adsbygoogle || []).push({});

An alternative too choosing the normed tangent vectors ##\vec e_i = \frac{1}{h_1}\frac{\partial \vec r}{\partial u_i}## with scale factors ##h_i = \left| \frac{\partial \vec r }{\partial u_i} \right| ## is to choose the normal vector ##\nabla u_i##.

If the system is ortogonal the vectors ##\nabla u_i## and ##\frac{\partial \vec r}{\partial u_i}## point in the same direction so we can write ##\vec e_i = h_i \nabla u_i##.

##\{ u_i \}_{i=1}^3## is supposed to be curvilinear coordinates with a transformation describing the position ##\vec r = \vec r (u_1,u_2,u_3)##.

What does it mean to take ##\nabla u_i##? am I supposed to express ##u_i## in cartesian coordinates and then take the gradient? And why multiply with the scale factors instead of dividing?

Another question I'm wondering about is if it's always true that the jacobian ##J## is equal too ##h_1h_2h_3##.

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# Understanding dual basis & scale factors

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