Skip to content

Additional exercises 4

Exercise

Sketch the vector field F(x,y)=(y,x).

Exercise

Let C denote the curve x2+y2=4. Find a clockwise parameterization of the curve C.

Exercise

Let C denote the line segment from the point a=(1,2,3) to the point b=(5,4,3). Find a parameterization of C, starting from a and finishing at b.

Exercise

Let C denote the curve x2+y2=4. Find an anticlockwise parameterization of the curve C.

Exercise

Let C denote the portion of y=x2+2 from (1,3) to (2,6). Find a parameterization of the curve C.

Exercise

Consider the vector field F(x,y)=(y,x3+y) and the path α(t)=(t2,t3) for t(0,1). Evaluate Fdα.

Exercise

Consider the vector field F(x,y)=(2y,x) and path α(t)=(tsint,1cost), t[0,2π]. Compute the line integral Fdα.

Exercise

The vector field F(x,y)=(3x2y,x3) is conservative on R2. Find φ such that F=φ.

Exercise

Consider the vector field

f(x,y)=(y(x2+y2)1x(x2+y2)1)

defined on S=R2(0,0). Let α(t) denote the path which traverses clockwise the circle of radius r>0 centred at the origin. Evaluate the line integral fdα.

Exercise

Evaluate (x22y) dα where dα is the path defined as α(t)=(4t4,t4) for t[1,0].

Exercise

Determine if the vector-field G(x,y)=(2y2,x+2) is conservative on R2.

Exercise

Evaluate fdα where f(x,y)=yex21+4xy and the path is α(t)=(1t,2t22t) for 0t2.