Appendix B:Formulas from Vector Calculus

The differential operators in orthogonal, curvilinear coordinates, ðx1; x2; x3Þ:The differntial arc length is ds2 ¼ ðh1dx1Þ2 þ ðh2dx2Þ2 þ ðh3dx3Þ2:The vector V is given by V ¼ u1V1 þ u2V2 þ u3V3:

The gradient of the scalar f is

grad f ¼ rf ¼ ðu1=h1Þqf=qx1 þ ðu2=h2Þqf=qx2 þ ðu3=h3Þqf =qx3:The divergence of the vector V is

divV ¼ r �V ¼ P�1½qðPV1=h1Þ=qx1 þ qðPV2=h2Þ=qx2 þ qðPV3=h3Þ=qx3�; P � h1h2h3:

The curl of V is

curl V ¼ r� V ¼ ðu1=h2h3Þ½qðh3V3Þ=qx2�qðh2V2Þ=qx3�þþðu2=h3h1Þ½qðh1V1Þ=qx3�qðh3V3Þ=qx1�þ ðu3=h1h2Þ½qðh2V2Þ=qx1�qðh1V1Þ=qx2�

The differential operators in cylindrical coordinates, (r, f, z):

grad f ¼ rf ¼ urqf=qrþ ufr�1qf=qfþuzqf=qz

div V ¼ r �V ¼ r�1qðrVrÞ=qrþr�1qVf=qfþ qVz=qz

curl V ¼ r� V ¼ urðr�1qVz=qf�qVf=qzÞþufðqVr=qz�qVz=qrÞ

þuzr�1½qðrVfÞ=qr�qVr=qf�

Electron Cyclotron Heating of Plasmas. Gareth GuestCopyright � 2009 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-40916-7

j243

r �rf ¼ r2f ¼ r�1qðrqf=qrÞ=qrþr�2q2f=qf2 þ q2f=qz2

r �r � V ¼ 0

r�rf ¼ 0

r� ðr � VÞ ¼ rðr �VÞ�r2V

rðfgÞ ¼ grf þ frg

rðU �VÞ ¼ ðU � rÞVþU� ðr � VÞþ ðV � rÞUþV � ðr�UÞ

r � ðfVÞ ¼ rf �Vþ fr �V

r� ðfVÞ ¼ rf � Vþ fr� V

r � ðU� VÞ ¼ ðr �UÞ �V�ðr � VÞ �U

r� ðU� VÞ ¼ ðr �VÞU�ðr �UÞVþðV � rÞU�ðU � rÞV

244j Appendix B: Formulas from Vector Calculus