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Q1). F(x, y) = (exp(y), xexp(y) – 2ysin(y2). r(t) = (t3

Added on - 12 Oct 2021

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Q1)
F(x, y) = (exp(y), xexp(y) – 2ysin(y2)
r(t) = (t3, cos(πt)) where -1t1
x = t3, y =cos(πt)
r’(t) = 3t2, -πsin(πt)
F(r(t)) = (exp(y), xexp(y) – 2ysin(y2) * (3t2, -πsin(πt))
=C

F(x,y)dr=1
1
¿¿exp(y), xexp(y) – 2ysin(y2) * (3t2, -πsin(πt))dt
=1
1
3t2¿¿
=1
1
3t2exp(cos(πt))πsin(πt)t3exp(cos(πt)+2πsin(πt)cos(πt)sin(cos(πt))2
]dt
= [cos(2πt)+1
2+t3exp¿¿]-11
= 1.03678 – 0.6322
= 0.40458
Q2)
F(x, y, z) = (z-x, -y, z)
r(t) = (t, t, sin(t))
x = t, y = t, z = sin(t)
r’(t) = 1, 1, cos(t)
F(r(t)) = sin(t)-t, -t, sin(t)
=γ

F(x,y,z)dr=0
1
¿¿sin(t)-t, -t, sin(t)]*[1, 1, cos(t)]dt
=0
1
¿¿1(sin(t)-t)+1* -t +cos(t)* sin(t)]dt
=0
1
¿¿(sin(t) – 2t +cos(t)sin(t)]dt
= [sin2(t)2(cos(t)+t2)
2+C]01
= 0.000304586 – 2*1.99985
= -1.9997
2
Q3)
F(x, y)= [cos)x)-y(x2), xy2,ln(y+1)]
r(t) = (cos(t), sin(t))
X = cos(t), y = sin(t)
r’(t) = [-sin(t), cos(t)]
F(r(t)) = cos(cos(t)) – sin(t)(cos(t))2, cos(t)(sin(t))2+ ln(sin(t) + 1)
=γ

F(x,y)dr=0
1
¿¿cos(cos(t)) – sin(t)(cos(t))2, cos(t)(sin(t))2+ ln(sin(t) + 1)]* [-sin(t),
cos(t)]dt
=0
1
¿¿
[(sin(t)+1)ln(sin(t)+1)+sin(cos(t))- sin(4t)/16 – sin(t)+t/4+C]0π
= -0.034905 +π
4
Q4)
a)U(x, y, z) = (xyz – x2, y2+ z2, 2xz)
u=(
x,
y,
z)(xyzx2,y2+z2,2xz)

x(xyzx2)+
y(y2+z2)+
z(2xz)
yz−2x+2y+2x
¿yz+2y
ifr=(x,y,z)r=¿r¿
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