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求解大學數vector
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發問:
In cylindrical polar coordinates, vector fields F and G are defined as F = r^2 sin$ er and G= r z^2 e$ Verify, using cylindrical polar coordinates that div(F x G) = G . (curl F) - F.(curl G)
最佳解答:
Actually the equation required to be verified isa property relating div and curl. Now use the same convention of notation stated in yourquestion. firstwe know FxG =(r^3 * sin $ * z^2) ez With the formula list in wolfram alpha,[reference link: http://mathworld.wolfram.com/CylindricalCoordinates.html (no. 116)Remark: one notation is different in the abovewebsite, theta as $]for any F can be written as F = F_r er + F_z er +F_$ e$,[kindly note that F_r is the scalar component ofF in the direction on r, and so on]Div F= 1/r * d/dr (r*F_r) + 1/r* d/d$ (F_$) + d/dz(F_z) R.H.S.=div (F x G)=0+0+d/dz (r^3 * sin $ * z^2)=2 * r^3 * z * sin $ L.H.S.=G . (curl F) – F . (curl G)=G . { (0 – 0) er + [ d/dz (r^2 * sin$)- 0]e$ + 1/r*[ 0 - d/d$ (r^2 * sin$)] ez }-F. { [0 – d/dz (r*z^2)] er + (0- 0) e$ + 1/r*[d/dr (r^2 * z^2) - 0] ez }[formula of curl is also in the formulalist (no. 117)]=[r* z^2 e$] . [-r*cos$ ez] – [r^2 *sin$ er]. [-2*r*z er + 2*z^2 ez]=2*r^3 *z*sin$ done.
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