MATH2019 PROBLEM 3-DIV, GRAD, CURL AND LINE INTEGRALS Solution

EXAMPLES3
DIV, GRAD, CURL AND LINE INTEGRALS
1996 1. A moving particle has position vector
r(t) = cos(ωt)i + sin(ωt)j + tk
where ω is a positive constant and t is time.
i) Find the acceleration of the particle and show that it has constant magnitude. ii) Describe the path of the particle.
iii) Evaluate Z F · dr where C is the portion of the path of the particle between t = 0
C
and t = 2π/ω and
F = yz i + xz j + xy k.
[Hint: Show that F = ∇(xyz).]
2014, S1 2. Given the vector field G = yz2i + xz2j + 2xyzk calculate:
i) div G .
ii) curl G .
2014, S1
2014, S2
2015, S1
3. Let r(t) = x(t)i + y(t)j + z(t)k be a path in space embedded within the surface φ(x,y,z) = 1. Assuming that all relevant derivatives exist use the chain rule to show that grad φ is perpendicular to the velocity vector v(t) for all t.
4. Given the vector field F = sinx i + cosx j + xyz k calculate:
i) div F. ii) curl F.
5. Given the vector field F = xz i + y2 j + yz k calculate:
i) div F = ∇ · F,
ii) curl F = ∇ × F and
iii) div(curl F) = ∇ · (∇ × F).
6. i) Suppose that r1(t) = x1(t)i + y1(t)j + z1(t)k
and r2(t) = x2(t)i + y2(t)j + z2(t)k
are two curves in R3. Prove that
.
ii) Suppose that a particle P moves along a curve C in R3 in such a manner that its velocity vector is always perpendicular to its position vector. Using part i) prove that the path C lies on the surface of a sphere whose centre is the origin.
i) Calculate curlF.
ii) Sketch the curve C in R3 for which curlF=0.
F(x,y,z) = sinxsiny k.
i) Calculate ∇ × F.
ii) Calculate ∇ × (∇ × F). iii) Hence, or otherwise, evaluate ∇ × (∇ × (∇ × (∇ × F))).
2014, S1 9. By evaluating an appropriate line integral calculate the work done on a particle traveling in R3 through the vector field F = −yi+xyzj+x2k along the straight line from (1,2,3) to (2,2,5).
2014, S2 10. Let C denote the path taken by a particle travelling in a straight line from point P(−2,3,0) to point Q(−2,0,3).
i) Write down a vector function r(t) that describes the path C and give the value of t at the start and the end of the path.
ii) If F = y2 i + xyz j − z2 k evaluate the line integral Z F · dr.
C
2015, S1 11. Let C denote the path taken by a particle travelling anticlockwise around the unit circle, starting and ending at the point (1,0) [i.e., the particle travels completely around the circle].
i) Write down a vector function r(t) that describes the path C and give the value of t at the start and the end of the path.
ii) If F = −3y i + 3x j evaluate the line integral I F · dr.
C
2015, S2 12. Given a vector field
F = 8e−xi + coshz j − y2k
i) Compute ∇ · F (i.e., div F) and ∇ × F (i.e., curl F).
ii) Calculate the line integral Z F · dr where C is the straight line path from A(0,1,0)
C to B(ln(2) 1, 2).
F(x,y,z) = 3yi − 3xj.
Let C denote the path taken by the particle travelling anticlockwise around the unit circle, starting at (1,0) and ending at (0,1).
i) Write down a vector function r(θ) that describes the path C and give the values of θ at the start and the end of the path.
ii) Calculate the work done on the particle as it moves along the path C by evaluating the line integral Z F · dr.
C
φ(x,y,z) = xez−1 + cosy
and let F = ∇φ.
i) Calculate F. ii) What is ∇ × F?
iii) Hence, or otherwise, calculate the line integral Z F · dr along the straight line path
C
C from (1,0,1) to (5,π,1).
F = yz2i + xz2j + (2xyz + 3)k.
ii) Calculate div F.
iii) Show that F is conservative by evaluating curl F.
iv) The path C in R3 starts at the point (3,4,7) and subsequently travels anticlockwise four complete revolutions around the circle x2 + y2 = 25 within the plane z = 7, returning to the starting point (3,4,7). Using part ii) or otherwise, evaluate the work integral Z F · dr.
C