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

26.6 Combining the operators grad, div and curl 861
5 Find the curlofeach ofthe following vector fields:
(a) E =x 2 yi +7xyzj +3x 2 k
(b) v =y 2 xi+4xzj+y 2 xk
(c) F = sinxi +cosxj +3xyzk
6 Avector fieldFisgiven by
F=x 3 yi+2y 2 j+(x+z 2 )k
(a) Find ∇ ×F.
(b) State 3F.
(c) Find ∇ × (3F).
(d) Is3(∇ ×F)thesameas ∇ × (3F)?
7 F is avector field andkis aconstant. Is ∇ × (kF)the
sameask(∇ ×F)?
Solutions
1 −3yk
2 (3xz +2y 2 )i −3yzj
4 −sinx,cosxk
5 (a) −7xyi −6xj + (7yz −x 2 )k
(b) x(2y −4)i −y 2 j + (4z −2xy)k
(c) 3xzi −3yzj −sinxk
6 (a) −j−x 3 k
(b) 3x 3 yi +6y 2 j +3(x +z 2 )k
(c) −3j −3x 3 k
(d) yes
7 yes
26.6 COMBININGTHEOPERATORSGRAD,DIVANDCURL
We have now metthreevector operators; these aresummarized inTable 26.1.
Itisimportanttobeabletocombinethethreeoperatorsgrad,divandcurlinsensible
ways.Forinstance,becausethegradientofascalarisavectorwecanconsiderevaluating
its divergence, thatis( ) ∂φ
∇·(∇φ)=∇·
=
∂x , ∂φ
∂y , ∂φ
∂z
( ∂
∂x , ∂ ∂y ∂z)
, ∂ ·
= ∂2 φ
∂x + ∂2 φ
2 ∂y + ∂2 φ
2 ∂z 2
( ∂φ
∂x , ∂φ
∂y , ∂φ
∂z
This lastexpression isvery important and isoften abbreviated tosimply
∇ 2 φ
Table26.1
Thethree vectoroperators.
Operator Acts on Resultisa Definition
grad scalar field vector field ∇φ = grad φ = ∂φ
∂x i + ∂φ
∂y j + ∂φ
∂z k
div vector field scalar field ∇ ·v = divv = ∂v x
∂x + ∂v y
∂y + ∂v z
∂z
∣ i jk ∣∣∣∣∣∣∣
∂ ∂ ∂
curl vector field vector field ∇ ×v = curlv =
∂x ∂y ∂z
∣ v x v y v z
)