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Unformatted text preview: HW2 1.(a) Let zxF21and zzyyxxF2. Calculate the divergence of and curl of 1Fand 2FWhich one can be written as the gradient of a scalar? Find a scalar that does the job. Which one can be written as a curl of a vector? Find a suitable vector potential. (b) Show that zxyyzxxyzF3can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function. Solution:(a)21zxF. yxyxxxzyxzyxF222132zzyyxxF. 2zyxzyxzyxF. Because 1F, 1Fcan be written as the curl of a vector potential A, i.e., AF1. zyxAAAzyxzyxA. So xyxA2can do the job. Because 2F, 2Fcan be written as the gradient of a scalar, i.e., 2F. In fact, it is easy to see that )2/()(2122222rzyxzzyyxxF, and 2/2ris a scalar. (b))()()(3zxyyzxxyzF. Thus 3Fcan be written both as the curl of a vector, i.e., zyAxAyxAzAxzAyAAFxyzxyz3. So zzyyyxxxzA222222can do the job....
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 Fall '09
 Qiu
 Vector Calculus, Magnetism, Sin, Cos, Vector field

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