B.20) Gradient theorem adv=fa B21) aDv=f dv (V×a)dv Vx×AdS a dl Stokes’ s theoren (V×A)·ds A. dI (B.27) 鱼·(V×a)dS=∮dla Greens first identity for scalar fields ds Greens second identity for scalar fields(Green's theorem) Green s first identity for vector fields LI(VxA).(VxB)-A[V x(V xB)dV= V·[Ax(V×B)]dv A×(V×B) (B.31) Green's second identity for vector fields [A×(×B)一B×(V×A)·dS (B32) ②2001 by CRC Press LLCV ∇ · a¯ dV = S nˆ · a¯ d S (B.19) S ∇s · A d S =  mˆ · A dl (B.20) Gradient theorem V ∇adV = S adS (B.21) V ∇A dV = S nAˆ d S (B.22) V ∇sadS =  mˆ a dl (B.23) Curl theorem V (∇ × A) dV = − S A × dS (B.24) V (∇ × a¯) dV = S nˆ × a¯ d S (B.25) S ∇s × A d S =  mˆ × A dl (B.26) Stokes’s theorem S (∇ × A) · dS =  A · dl (B.27) S nˆ · (∇ × a¯) d S =  dl · a¯ (B.28) Green’s first identity for scalar fields V (∇a · ∇b + a∇2 b) dV = S a ∂b ∂n d S (B.29) Green’s second identity for scalar fields (Green’s theorem) V (a∇2 b − b∇2 a) dV = S  a ∂b ∂n − b ∂a ∂n  d S (B.30) Green’s first identity for vector fields V {(∇ × A) · (∇ × B) − A · [∇ × (∇ × B)]} dV = V ∇ · [A × (∇ × B)] dV = S [A × (∇ × B)] · dS (B.31) Green’s second identity for vector fields V {B · [∇ × (∇ × A)] − A · [∇ × (∇ × B)]} dV = S [A × (∇ × B) − B × (∇ × A)] · dS (B.32)