Electromagnetic Theory volume 1

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But observe that, by the trans- formation (164), we may also write it 2TH = 2VNV. H, . . . . (171) similarly to (169), in which the operand is a scalar. This is mnemonically useful, but (170) is more practically useful.
(d). Take F(T) = VTH. Here, with normal i, the first of (166) gives /= V 2 VkH - V 3 VjH = V (kV 2 - JV 3 )H.
But here we have kV 2 - j V 3 = ViV, so that /=V(ViV;H; and therefore, generally, putting N for i, 2VTH = 2V(VNV)H. . . . (172) (e). Let the quantity in the circuital su
...mmation be a vector of length P (a scalar function of position) drawn perpendi- cularly to the plane of T and N. That is, F(T) = (VTN)P. We then find, taking N = i, and using the first of (166), /= V 2 Vki . P - V 3 Vji . P = (j V 2 + kV 3 )P = VP - i (iV) P. In general, therefore, 2 VTN . P = 2 (VP - N . NVP) = 2 V a P = 2 V (VNV)N. P . (173) The element of the surface-summation is V 8 P, meaning the slope of P on the surface itself, disregarding any variation it ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS.

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