082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

864 Chapter 26 Vector calculus
9 Ascalar field φ and avector fieldFare given by
φ=xyz 2 F=x 2 i+2j+zk
(a) Find ∇φ.
(b) Find ∇ ·F.
(c) Calculate φ(∇ ·F) +F · (∇φ).[Hint:recall the
dot product oftwo vectors.]
(d) State φF.
(e) Calculate ∇ · (φF).
(f) What do you conclude from(c)and (e)?
Solutions
3 j−k
(d) (6x 2 yz +2xy 2 z 2 )i + (2x 3 z +2yx 2 z 2 )j
+(2x 3 y +2y 2 x 2 z)k
4 ∇φ=yzi+xzj+xyk
∇·v=6x+6y 2
(e) same as(d)
∇·(φv) =9x 2 yz+8xy 3 z+x 2 y 2
(f) ∇(φ 1 φ 2 ) = φ 1 ∇φ 2 +φ 2 ∇φ 1
9 (a) yz 2 i +xz 2 j +2xyzk
7 (a) ∇(∇·F) (b) ∇ ×(∇φ)
(b) 2x+1
(c) ∇×(∇×F) (d) ∇·(∇×F)
(e) ∇·(∇φ)
(c) 3x 2 yz 2 +2xz 2 +3xyz 2
8 (a) 2yi + (2x +2yz)j +y 2 (d) x 3 yz 2 i +2xyz 2 j +xyz 3 k
k
(b) 2xzi +x 2 k
(e) same as(c)
(c) 2x 3 yz +y 2 x 2 z 2 (f) ∇·(φF)=φ(∇·F)+F·(∇φ)
26.7 VECTORCALCULUSANDELECTROMAGNETISM
Vector calculus provides a useful mechanism for expressing the fundamental laws of
electromagnetisminaconcisemanner.Theselawscanbesummarizedbymeansoffour
equations,knownasMaxwell’sequations.Muchofelectromagnetismisconcernedwith
solving Maxwell’s equations for different boundary conditions.
Equation1
divD=∇·D=ρ
where D = electric flux density, and ρ = charge density. This equation is a general
formofGauss’stheoremwhichstatesthatthetotalelectricfluxflowingoutofaclosed
surface isproportional tothe electriccharge enclosed by thatsurface.
Equation2
divB=∇·B=0
whereBisthe magnetic fluxdensity.
This equation arises from the observation that all magnetic poles occur in pairs and
therefore magnetic field lines are continuous; that is, there are no isolated magnetic
poles.Incontrast,electricfieldlinesoriginateonpositivechargesandterminateonnegative
charges and so a net positive charge in a region leads to an outflow of electric
flux.