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Hello, I have a question regarding the first equation above.

it says du

^{i}=a

^{i}*dr=a

^{i}*a

^{j}*du

_{j}but I wonder how. (sorry I omitted vector notation because I don't know how to put them on)

if du

^{i}=a

^{i}*dr=a

^{i}*a

^{j}*du

_{j}is true, then

dr=a

^{j}*du

_{j}

|dr|*r

^{hat}=|a

^{j}|*du

_{j}*a

^{j}

^{hat}

where lim |dr|,|du

_{j}|->0

which means r

^{hat}=a

^{j}

^{hat}.

are both the directions actually the same or do we just don't care about directions because both will eventually be small increment of each component?