15.6

1.

(a)
u-y=x*1/2
v-y-axis

(b)
(0,0)(0,7)(10,12)(10,5)

(c)
the line segment(2,3)and(10,8)

(d)
The triangle with vertices(0,1)(2,1)(2,2)

3.

(a)
u-x-axis
v-y-axis

(b)
[0,1][-1,1]

(c)
The line segment y=sqrt(x) from (0,0) to (1,1)

(d)
The image surrouded by y-axis, y=1, y=sqrt(x)

13.
fxu=3
fxv=4
fyu=1
fyv=-2
jacobian=-10

15.
fxr=sint=0
fxt=rcost=-1
fyr=1
fyt=sint=0
jacobian=1

17.
fxr=cos(t)=sqrt(3)/2
fxt=-rsin(t)=-2
fyr=sin(t)=1/2
fyt=rcos(t)=2sqrt(3)
jacobian=4

19.
(4u+2v, u+3v)

23.

(a)
GR=(0,0)(9,3)(14,-7)(5,-10)
area=|<9,3>×<5,-10>|=105

(b)
GR=(7,0)(13,-12)(22,-9)(16,3)
area=|<6,-12>×<9,3>|=126

16.1

1.
F(1,2)=<1,1>
F(-1,-1)=<1,-1>
#Sketch is on the PDF version

3.
F(0,1,1)=<0,1,0>
F(2,1,0)=<2,0,2>
#Sketch is on the PDF version

5.
#Sketch is on the PDF version

7.
#Sketch is on the PDF version

9.
#Sketch is on the PDF version

11.
#Sketch is on the PDF version

17.
F=<1,1,1> corresponds to (C)

23.
i		j		k
d/dx		d/dy		d/dz
xy		yz		y^2-x^3
curl(F)=i(y)-j(3x^2)+k(-x)=<y,3x^2,-x>
div(F)=xyd/dx+yzd/dy+(y^2-x^3)d/dz=y+z

25.
i		j		k
d/dx		d/dy		d/dz
x-2zx^2		z-xy		z^2x^2
curl(F)=i(-1)-j(2xz^2-2x^2)+k(-y)=<-1,2x^2-2xz^2,-y>
div(F)=(x-2zx^2)d/dx+(z-xy)d/dy+(z^2x^2)d/dz=(1-4xz)+(-x)+(2zx^2)

27.
i		j		k
d/dx		d/dy		d/dz
z-y^2		x+z^3		y+x^2
curl(F)=i(1-3z^2)+j(1-2x)+k(1-2y)=<1-3z^2,1-2x,1+2y>
div(F)=(z-y^2)d/dx+(x+z^3)d/dy+(y+x^2)d/dz=0