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5. Vector Calculus 1
Inregrating, v = i J12 cos2t dt + j J-8sin2t dt + k J161 dt \l::J11 dt = 6sin2t i + 4cos2t j + 8t2 k + c)
Putting v = 0 when t = 0, we fmd 0 = Oi + 4j + Ok + c) and Cl = -4j.
Then v = 6 sin2t i + (4cos2t -4) j + 8t Z k
so that dr = 6sin2t i + (4cos2t 4)j + 8t Z k dt
Integrating, r= i J6sin2tdt + j J(4cos2t-4)dt + k J8t 2 dt
= -3cos2t i + (2sin2t-4t)j +~t3 k + c23
Putting r = 0 when t =0, 0 =-3i + OJ + Ok + C2 and C2= 3i.
Then r = (3 -3cos2t) i + (2 sin2t - 4t) j + ~t3 k . 3
d 2 A3. Evaluate JAX-z-dt.
dt
d dA-(Ax-)=.Ax--+-x-= Ax-
dt 2 dt 2dt dt dt dt
d2A dA dAJdIntegrating, Ax--dt= -(Ax-)dt=Ax-+c.J dt 2 dt dt dt
d 2 A dA dA d 2 A
Advanced Engineering Mathematics [ENG4200j Page 23 of 31
~ 5. Vector Calculus 1 , 2. (a) IfF is a conservative field, prove that curl F = VxF =0 (i.e. F is irrotational). :..(b) Conversely, if V x F = 0 (i.e. F is irrotational), prove that F is conservative. ,(a) If F is a conservative field, then by F = Vel> . C:t~ f ~¥: CV¥):2 0
Thus curlF=VxF=O ~~ (1ft) ,- t -ILi j k I . a a a
(b) If VxF = 0, then lax ay az = 0 and thus
P; F2 1<;/ ~ aF
, '
3 _ aF aF; _ aF3 aF2 =ap; . 2
~
ay - az ' ---, ax ayaz ax
We must prove that F = Vel> follows as a consequence of this.
The work done in moving a particle from (xt'Yl'zJ to (x,y,z) in the force field F
is
.b F;(x,y,z)dx+ F2(x, y,z)dy + F;(x, y,z)dz
where C is a path joining (xl' YI'Zt) and (x, y, z). Let us choose as a particular path the
straight line segments from (xI'YI'Zt) to (x'YI'Zt) to (x,Y,Zt) to (x,y,z) and call
eI>(x, y, z) the work done along this particular path. Then
eI>(x,y,z) =r F;(x'Yl,Zt)dx+ r F2(x,y,z;)dy+ rF3 (x,y,z)dz ! ~! <:1!
It follows that Ct.I] Y/J t:() -7 (f..) il, i-"1)-1 ('I-> '/, Z-l)
C{ (f.)~J~) ~ a<I> =F3 (x,y,z) az
~o-a<I> _ r aF3
.-a C FI (~/ YJ) 21)<iXay -F2 (x,y,Zt)+ ay (x,y,z)dzI o~1 ~ - r aF2 Jfl -- :JF-t- F2 (x, y, Zt) + I az (x, y, z)dz "
~ ---;)l:
Advanced Engineering Mathematics [ENG4200] Page 28 of 31
dF3-(x,y,z)dz
dx .} ~ dE; ddz (x,y,z) z
F' F k d<P . d<l>. d<P k V'n..Th F F. • en = 11 + 2J+ 3 = dx l + dyJ+ dz = '*'.
Thus a necessary and sufficient condition that a field F be conservative is that curIF=V'xF=O.
Advanced Engineering Mathematics [ENG4200} Page 29 of 31
I
F:::\!(A
PCf~~ ~~. ( vecTt;r f(t.U. t 5. Vector Calculus
Tutorial I Assignment
Harmonic Functions
1.
2.
Prove the function u =sin x cosh y is harmonic.
iPu (J2UV 2 u= ax2 + dy2 =-sinxcoshy+sinxcoshy=O
Hencet the field F=+Vu=rcosxcoshyi-tsinxsinhyjmust be conservativet since it arises
from a scalar potential as a gradient. This field is also solenoidal since the divergence V . F :;;;; O.
Thus, fields with a harmonic scalar potential are both conservative and soleoidal.
We see that u satisfies Laplace's equation:
(J2U (J2U -+-=6x-6x=Oax2 dy2
CQYlJ~tk f)~.
\/.f= VIC\lU)~O ~ (j?-vt --. (j
iF C(h-x v>'vp ~1--
>'~ "V')( 5~-~Aydj. r
Adlr.ill1ICed Engineering Mathematics [ENG4200] Page 31 of 31