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    Show that \frac{d}{dt}(\vec{a} \times \vec{b}) = \frac{d\vec{a}}{dt} \times \vec{b} + \vec{a} \times \frac{d\vec{b}}{dt}

    I can do it if I expand everything out, but given that we've just covered the permutation operator, I'm guessing that's not what we're supposed to do.

    I've got \frac{d}{dt}(\vec{a} \times \vec{b}) = \frac{d}{dt}(a_i b_j \epsilon_{ijk} \vec{e}_k) but I'm not sure where to go from there? If I treat \epsilon_{ijk} and \vec{e}_k as constants and apply the chain rule then I end up with a minus error in the answer so I'm guessing I can't treat them as constants. Could someone give a hint on where to go next?
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    (Original post by Plagioclase)
    Show that \frac{d}{dt}(\vec{a} \times \vec{b}) = \frac{d\vec{a}}{dt} \times \vec{b} + \vec{a} \times \frac{d\vec{b}}{dt}

    I can do it if I expand everything out, but given that we've just covered the permutation operator, I'm guessing that's not what we're supposed to do.

    I've got \frac{d}{dt}(\vec{a} \times \vec{b}) = \frac{d}{dt}(a_i b_j \epsilon_{ijk} \vec{e}_k) but I'm not sure where to go from there? If I treat \epsilon_{ijk} and \vec{e}_k as constants and apply the chain rule then I end up with a minus error in the answer so I'm guessing I can't treat them as constants. Could someone give a hint on where to go next?
    Just use the definition of derivative, both u and v are functions \mathbb{R} \to \mathbb{R}^3 so the usual applies.

    \displaystyle

\begin{align*}\dfrac{u(t+h) \times v(t+h) - u(t) \times v(t)}{h} &= \dfrac{u(t+h)\times v(t+h) - u(t+h)\times v(t) + u(t+h)\times v(t) - u(t)v(t)}{h} \\&= u(t+h) \times \dfrac{v(t+h) - v(t)}{h} + \frac{u(t+h) - u(t)}{h} \times v(t) \end{align*}

    then take h \to 0.

    Just realised I've changed the letters to something more familiar to me, I'm sure you get the idea though.
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    (Original post by Plagioclase)
    Show that \frac{d}{dt}(\vec{a} \times \vec{b}) = \frac{d\vec{a}}{dt} \times \vec{b} + \vec{a} \times \frac{d\vec{b}}{dt}

    I can do it if I expand everything out, but given that we've just covered the permutation operator, I'm guessing that's not what we're supposed to do.

    I've got \frac{d}{dt}(\vec{a} \times \vec{b}) = \frac{d}{dt}(a_i b_j \epsilon_{ijk} \vec{e}_k) but I'm not sure where to go from there? If I treat \epsilon_{ijk} and \vec{e}_k as constants and apply the chain rule then I end up with a minus error in the answer so I'm guessing I can't treat them as constants. Could someone give a hint on where to go next?
    In index notation, you usually work on a single component at a time and then this seems to work out trivially. Haven't done this for a long time though:

    \frac{d}{dt}(\vec{a} \times \vec{b})_k = \frac{d}{dt}(a_i b_j \epsilon_{ijk}) = \epsilon_{ijk} \frac{d}{dt}(a_i b_j) =  \epsilon_{ijk} (\dot{a}_i b_j + a_i \dot{b}_j) = \cdots

    I can't see where you'd pick up a -ve sign to be honest. Can you put up your working?
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    Because there seem to be differering approaches suggested, I'm going to throw my weight behind atsruser here; this should be a simple application of product rule to a_ib_j. I don't think there's any need to go back to formal definitions of derivatives etc.
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    (Original post by DFranklin)
    Because there seem to be differering approaches suggested, I'm going to throw my weight behind atsruser here; this should be a simple application of product rule to a_ib_j. I don't think there's any need to go back to formal definitions of derivatives etc.
    Having googled a bit, I can find no specific examples of this derivation using indices. However it's so trivial that I'll be rather depressed if there's an error with it.
 
 
 
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