The derivative of the basis vector e_r with respect to theta in polar coordinates is computed as (1/r)e_theta. This arises from the relationship between normalized basis vectors and their non-normalized counterparts, where the non-normalized basis vector E_theta is expressed as r * e_theta. The confusion regarding the factor of 1/r stems from the distinction between normalized and non-normalized basis vectors, specifically in the context of polar coordinates.
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Based on some of his other threads, he is not using normalised basis vectors but ##\vec e_i = \partial_i \vec x## (which I usually would denote ##\vec E_i## or similar to underline that they are not necessarily unit vectors - however, there are some places in the literature where ##\vec e_i## is used and instead ##\hat i## or similar is used for normalised basis vectors). With my preferred notation, where ##\vec e_i## are normalised and ##\vec E_i = \partial_i \vec x##, you would have ##\vec E_r = \vec e_r## but ##\vec E_\theta = r \vec e_\theta## so indeed you would have ##\partial_\theta \vec E_r = \vec E_\theta/r##.vela said:You shouldn't get a factor of 1/r. Why do you think there is one?