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

858 Chapter 26 Vector calculus
E
dS
Figure26.1
Theflux ofEthrough δS isa
measure ofthe number oflines
offorce passingthrough δS.
where ρ is the charge density and ε 0
is a constant called the permittivity of free
space. Note that in a charge-free region, ρ = 0, and so ∇ ·E = 0. This means there
isno net flux ofE.
EXERCISES26.4
1 Avector fieldvisgiven by
Find
v =3x 2 yi+2y 3 zj+xz 3 k
(a)v x , v y , v z
(b) ∂v x
∂x , ∂v y
∂y , ∂v z
∂z
(c)∇·v
2 Avector fieldFis defined by
F=(x+y 2 )i+(y 2 −z)j+(y+z 2 )k
(a) Find ∇ ·F.
(b) Calculate ∇ ·F atthe point (3,2, −1).
3 IfA = 3yzi +2xyj +xyzkfind ∇ ·A.
4 Find the divergence ofeachofthe following vector
fields:
(a) v =x 2 i+y 2 j+z 2 k
(b) v = e xy i +2zsin(xy)j +x 3 zk
(c) v=xyi−2yzj+k
(d) v =x 2 y 2 i−y 2 j−xyzk
5 IfE=xi+z 2 j−yzkfind
(a) E·i
(b) E·j
(c) E·k
(d) ∇·E
6 Avector field is given by
F = (3x 2 −z)i + (2x +y)j
+(x +3yz)k
Find ∇ ·F atthe point (1,2,3).
7 Given the scalar field φ =x 2 +y 2 −2z 2 ,find ∇φ and
showthat ∇ ·(∇φ) =0.
8 Forany vectorfield F is ∇ · (−F)the same as
−(∇ ·F)?
9 ThevectorfieldFis given by
F =xz 2 i+2xy 2 zj+xz 3 k
(a) Find ∇ ·F.
(b) State4F.
(c) Find ∇ · (4F).
(d) Is4(∇ ·F)thesameas ∇ · (4F)?
10 Forany vectorfield F andany scalar constantkis
∇ · (kF)thesameask∇ ·F?
11 Give an example ofavectorfield
v = v x i+v y j+v z ksuchthat ∂v x
∂x ≠ 0, ∂v y
∂y ≠ 0,
∂v z
∂z ≠0,but∇·v=0.