16.2

3.
(a).
F(r(t))=(t^-2,t^2)
x'(t)=1,y'(t)=-t^-2
dr=(1,-t^-2)dt
(b).
F(r(t))·r'(t)dt=(t^-2,t^2)(1,-t^-2)dt=(t^-2-1)dt
Int(F*dr)=Int(t^-2-1,t=1..2)
=4/3

9.
x(t)=t,	y(t)=t^1/3
x'(t)=1,	y'(t)=1/3(t)^(-2/3)
ds=sqrt(9t^(4/3)+1)/3t^(2/3)dt
f(t)=sqrt(1+9t(t^1/3))
Int(f ds)=Int(sqrt(1+9t(t^1/3))*sqrt(9t^(4/3)+1)/3t^(2/3),t=0..1)
=14/5

11.
x(t)=2t,	y(t)=3t,	z(t)=4t
x'(t)=2	y'(t)=3,	z'(t)=4
ds=sqrt(29)
Int(f ds)=Int(16t^2*sqrt(29),t=0..2)
=128*sqrt(29)/3

13.
from (0,0,1) to (0,2,0)
c=(0,0,1)+t(0,2,-1)=(0,2t,1-t)
x'(t)=0,	y'(t)=2,	z'(t)=-1
ds=sqrt5
Int(f ds)=Int(0,t=0..1)=0
from(0,2,0) to (1,1,1)
c=(0,2,0)+t(1,-1,1)=(t,2-t,t)
x'(t)=1,	y'(t)=-1,	z'(t)=1
ds=sqrt3
Int(f ds)=Int(t*e^(t^2)*sqrt3,t=0..1)
=1.488
0+1.488=1.488
The answer is 1.488

17.
x'(t)=4,	y'(t)=-3,	z'(t)=12
ds=13dt
Int(f ds)=Int(1*13,t=2..5)
=39
This integral represent the distance between (8,-6,24) and (20,-15,60)

27.
x(t)=t,	y(t)=t^2
x'(t)=1,	y'(t)=2t
Int(f ds)=Int(t^2,t=0..2)-Int(t*2t,t=0..2)
=-8/3

29.
c=(0,0,0)+t(1,4,4)=(t,4t,4t)
x'(t)=1,	y'(t)=4,	z'(t)=4
Int(f ds)=Int(-3t,t=0.1)+Int(0,t=0..1)+Int(4t*4,t=0..1)
=6.5

31.
c=(1,0)+t(-1,1)=(1-t,t)
x'(t)=-1,	y'(t)=1
Int(f ds)=Int((t)/((1-t)^2+t^2),t=0..1)+Int((1-t)/((1-t)^2+t^2),t=0..1)
=Pi/2

35.
(0,0,0)->(0,0,1)
f(t)=(0,0,t)
f'(t)=(0,0,1)
Int(F1 ds)=0+0+e^0dt=1
(0,0,1)->(0,1,1)
f(t)=(0,t,1)
f'(t)=(0,1,0)
Int(F2 ds)=0+Int(e^(-t),t=0..1)=1-e^(-1)
(0,1,1)->(-1,1,1)
f(t)=(-t,1,1)
f'(t)=(-1,0,0)
Int(F3 ds)=Int(-e^1,t=0..1)=-e
Int(F ds)=Int(F1 ds)+Int(F2 ds)+Int(F3 ds)=2-e^(-1)-e

16.3

1.
Int(F dr)=1*1*sin(1*Pi)-0=0

3.
fx=3,	fy=6y
F=del(f)
f(t)=3t+12t^-2
Int(F ds)=(12+3/4)-(3+12)=-9/4

5.
fx=ye^z,	fy=xe^z,	fz=xye^z
F=del(f)
f(t)=t^2*t^3*exp(t-1)
Int(F ds)=4*8*e-1=32e-1

9.
i		j		k
d/dx		d/dy		d/dz
y^2		(2xy+e^z)	ye^z
k=i(e^z-e^z)-j(0-0)+k(2y-2y)=0
F is conservative.
fx=y^2
f=xy^2+f(y,z)
fy=(2xy+e^z)
f=xy^2+ye^z+f(z)
fz=ye^z
f=xy^2+ye^z
potential function: f=xy^2+ye^z

13.
i		j		k
d/dx		d/dy		d/dz
z*sec^2(x)	z		y+tan(x)
k=i(1-1)-j(sec^2(x)-tan(x))+k(0-0)=0
F is conservative.
fx=z*sec^2(x)
f=z*sec^2(x)+f(y,z)
fy=z
f=z*sec^2(x)+yz+f(z)
fz=y+tan(x)
f=z*sec^2(x)+yz
potential function: f=z*sec^2(x)+yz

15.
i		j		k
d/dx		d/dy		d/dz
2xy+5		x^2-4z		-4y
k=i(-4+4)-j(0-0)+k(2x-2x)=0
F is conservative.
fx=2xy+5
f=x^2y+5x+f(y,z)
fy=x^2-4z
f=x^2y+5x-4yz+f(z)
fz=-4y
f=x^2y+5x-4yz
potential function: f=x^2y+5x-4yz

17.
i		j		k
d/dx		d/dy		d/dz
2xyz		x^2z		x^2y
k=i(x^2-x^2)-j(2xy-2xy)+k(2xz-2xz)=0
F is conservative.
potential function: f=x^2yz
f(t)=t^4*sin((Pi*t)/4)*e^(t^2-2t)
Int(F ds)=16*1*1-0=16

19.
f(r1)=t^3
Int(F ds)=1-0=1
f(r2)=t^4
Int(F ds)1-0=1
The two paths give the same answer.