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

860 Chapter 26 Vector calculus
Example26.7 Ifv =x 2 yzi −2xyj +yzk find ∇ ×v.
i j k
Solution ∇ ×v=
∂ ∂ ∂
∂x ∂y ∂z
∣
x 2 yz −2xy yz
∣
[ ∂(yz)
= − ∂(−2xy) ] [ ] [ ]
∂(yz)
i − − ∂(x2 yz) ∂(−2xy)
j + − ∂(x2 yz)
k
∂y ∂z ∂x ∂z ∂x ∂y
=zi+x 2 yj− (2y+x 2 z)k
Notethatthecurloperationisonlyperformedonavectorfieldandtheresultisanother
vector field.
Adetaileddiscussionofthephysicalinterpretationofthecurlofavectorfieldisbeyondthescopeofthisbook.However,ifthevectorfieldvunderconsiderationrepresents
a fluid flow then it may be shown that curl v is a vector which measures the extent to
whichindividualparticlesofthefluidarespinningorrotating.Forthisreason,avector
field whosecurliszero forallvalues ofx,y andzissaidtobe irrotational.
Example26.8 Show thatthe vectorfield
F=ye xy i+xe xy j+0k
isirrotational.
i j k
Solution ∇ ×F =
∂ ∂ ∂
∂x ∂y ∂z
∣
ye xy xe xy 0
∣
( ∂
=
∂y 0 − ∂ ) ( ∂
∂z xexy i −
∂x 0 − ∂ ) ( ∂
∂z yexy j +
∂x xexy − ∂ )
∂y yexy k
=0i +0j + ((xye xy +e xy ) − (yxe xy +e xy ))k
=0 forallx,yandz
The field istherefore irrotational.
EXERCISES26.5
1 Find the curlofthe vectorfield
v =xi −3xyj +4zk.
2 Ifv =3xi−2y 2 zj+3xyzkfind ∇ ×v.
3 SupposeF =P(x,y)i +Q(x,y)jisa
two-dimensionalvector field. Show thatFis
irrotationalif ∂P
∂y = ∂Q
∂x .
4 Findthe divergence andcurlofthe vector field
E = cosxi +sinxj.