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

26.3 The gradient of a scalar field 855
Atthepoint (2,1,0)thisvectorpointsinthedirectionofgreatestrateofdecreasein
potential. This is also the direction of the electrostatic force experienced by the unit
charge atthatpoint.
EXERCISES26.3
1 If φ =x 2 −y 2 −3xyz,(a)find ∇φ,(b)evaluate ∇φ at
the point (0,0,0).
2 If φ =x 2 yz 3 ,find(a) ∇φ,(b) ∇φat (1,2,1),(c) |∇φ|
at (1,2,1).
3 Ifv = ∇φ,find φwhenv = (2x−4y 2 )i−8xyj.
4 Given φ =xyzfind(a) ∇φ,(b) −∇φ,(c) ∇φ
evaluated at (3,0, −1).
5 Find ∇V when
(a)V =x 2 +y 2 +z
( )
2
(b)V =zsin −1 y
x
(c)V = e x+y+z .
6 An electrostatic potential is given byV =xe −√y .
Find ∇V and deduce the direction in which the
decrease ofpotential isgreatest atthe point with
coordinates (1,1,1).
7 Inthe theoryoffluid mechanics the scalar field φ is
known asthevelocity potential ofafluidflow. The
fluid velocityvector, v,atapoint canbe foundfrom
the equationv = ∇φ.Foraparticulartypeofflow
φ =Ux,whereU is aconstant. Show thatthe
corresponding fluidmotion is entirely parallelto thex
axis, andatany point the fluidspeed isU.Finda
velocitypotential forasimilar flow whichisentirely
parallelto theyaxis.
8 Is −∇φ the same as ∇(−φ)?Explain youranswer.
9 Given φ = 3x 2 y +xz,
(a) determine ∇φ
(b) determine ∇(7φ)
(c) determine7∇φ
(d) Is ∇(7φ)thesameas7∇φ?
10 Foranyscalarfield φ andanyconstantk,is ∇(kφ)the
sameask∇φ?
Solutions
1 (a) (2x −3yz)i − (2y +3xz)j −3xyk
(b) 0
2 (a) 2xyz 3 i +x 2 z 3 j +3x 2 yz 2 k
(b) 4i+j+6k
√
(c) 53
3 φ=x 2 −4xy 2 +c
4 (a) yzi +xzj +xyk
(b) −yzi −xzj −xyk
(c) −3j
5 (a) 2xi +2yj +2zk
−zy
(b)
x 2√ 1−y 2 /x 2i
( )
z
+
x √ y
1−y 2 /x 2j +sin−1 k
x
(c) e x+y+z (i +j+k)
6 e −√y i − xe−√ y
2 √ j,−0.368i +0.184j
y
7 φ=Uy
8 yes
9 (a) (6xy +z)i +3x 2 j +xk
(b) (42xy +7z)i +21x 2 j +7xk
(c) (42xy +7z)i +21x 2 j +7xk
(d) yes
10 yes