after we differentiate under the integ ne terms V×H D+J by the Fourier integral theorem. This version of Ampere's law involves only the frequency domain fields. By similar reasoning we have V×E=-joB, P (4.9) 0, (4.10) J Equations(4.7)-(4.10) govern the temporal spectra of the electromagnetic fields. We may manipulate them to obtain wave equations, and apply the boundary conditions from the following section. After finding the frequency-domain fields we may find the temporal fields by Fourier inversion. The frequency-domain equations involve one fewer derivative (the time derivative has been replaced by multiplication by jo), hence may be easier solve. However. the inverse transform may be difficult to compute 4.3 Boundary conditions on the frequency-domain fields Several boundary conditions on the source and mediating fields were derived in$ 2.8.2 or example, we found that the tangential electric field must obey n12×E1(r,1)-n12×E2(r,t)=-Jms(r,t) The technique of the previous section gives us 12×[E1(r,o)-E2(r,o)=-Jm(r,o) as the condition satisfied by the frequency-domain electric field. The remaining boundary conditie e treated similarly. Let us summarize the results, including the effects of fictitious magnetic sources H2)=J, n12×(E1-E2) n2·OD1-D2)=5 12·(1-J2)=-V4J,-jop, Here f12 points into region 1 from region 2. 2001 by CRC Press LLCafter we differentiate under the integral signs and combine terms. So ∇ × H˜ = jωD˜ + J˜ (4.7) by the Fourier integral theorem. This version of Ampere’s law involves only the frequency￾domain fields. By similar reasoning we have ∇ × E˜ = − jωB˜ , (4.8) ∇ · D˜ = ρ,˜ (4.9) ∇ · B˜(r,ω) = 0, (4.10) and ∇ · J˜ + jωρ˜ = 0. Equations (4.7)–(4.10) govern the temporal spectra of the electromagnetic fields. We may manipulate them to obtain wave equations, and apply the boundary conditions from the following section. After finding the frequency-domain fields we may find the temporal fields by Fourier inversion. The frequency-domain equations involve one fewer derivative (the time derivative has been replaced by multiplication by jω), hence may be easier to solve. However, the inverse transform may be difficult to compute. 4.3 Boundary conditions on the frequency-domain fields Several boundary conditions on the source and mediating fields were derived in § 2.8.2. For example, we found that the tangential electric field must obey nˆ 12 × E1(r, t) − nˆ 12 × E2(r, t) = −Jms(r, t). The technique of the previous section gives us nˆ 12 × [E˜ 1(r,ω) − E˜ 2(r,ω)] = −J˜ms(r,ω) as the condition satisfied by the frequency-domain electric field. The remaining boundary conditions are treated similarly. Let us summarize the results, including the effects of fictitious magnetic sources: nˆ 12 × (H˜ 1 − H˜ 2) = J˜s, nˆ 12 × (E˜ 1 − E˜ 2) = −J˜ms, nˆ 12 · (D˜ 1 − D˜ 2) = ρ˜s, nˆ 12 · (B˜ 1 − B˜ 2) = ρ˜ms, and nˆ 12 · (J˜1 − J˜2) = −∇s · J˜s − jωρ˜s, nˆ 12 · (J˜m1 − J˜m2) = −∇s · J˜ms − jωρ˜ms. Here nˆ 12 points into region 1 from region 2