Exercise 4.7\ Consider the vector field

F(x,y)=(2-y,x)

and path

\\\\alpha (t)=(t-sint,1-cost)

,\

tin[0,2\\\\pi ]

. Compute the line integral

\\\\int F*d\\\\alpha

.\ Exercise 4.8\ The vector field

F(x,y)=(3x^(2)y,x^(3))

is conservative on

R^(2)

. Find

\\\\phi

such that

F=grad\\\\phi

.\ Exercise 4.9\ Consider the vector field\

f(x,y)=([-y(x^(2)+y^(2))^(-1)],[x(x^(2)+y^(2))^(-1)])

\ defined on

S=(R^(2))/(/)(0,0)

. Let

\\\\alpha (t)

denote the path which traverses clockwise the circle\ of radius

r>0

centred at the origin. Evaluate the line integral

\\\\int f*d\\\\alpha

.\ Exercise 4.10\ Evaluate

\\\\int (x^(2)-2y)d\\\\alpha

where

d\\\\alpha

is the path defined as

\\\\alpha (t)=(4t^(4),t^(4))

for

tin[-1,0]

\ Exercise 4.11\ Determine if the vector-field

G(x,y)=(2y^(2),x+2)

is conservative on

R^(2)

.\ Exercise 4.12\ Evaluate

\\\\int gradf*d\\\\alpha

where

f(x,y)=ye^(x^(2)-1)+4xy

and the path is\

\\\\alpha (t)=(1-t,2t^(2)-2t)

for

0

.

Consider the vector field F(x,y)=(2y,x) and path (t)=(tsint,1cost), t[0,2]. Compute the line integral Fd. Exercise 4.8 The vector field F(x,y)=(3x2y,x3) is conservative on R2. Find such that F=. Exercise 4.9 Consider the vector field f(x,y)=(y(x2+y2)1x(x2+y2)1) defined on S=R2\(0,0). Let (t) denote the path which traverses clockwise the circle of radius r>0 centred at the origin. Evaluate the line integral fd. Exercise 4.10 Evaluate (x22y)d where d is the path defined as (t)=(4t4,t4) for t[1,0] Exercise 4.11 Determine if the vector-field G(x,y)=(2y2,x+2) is conservative on R2. Exercise 4.12 Evaluate fd where f(x,y)=yex21+4xy and the path is (t)=(1t,2t22t) for 0t2