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MA231 Vector Analysis
Example Sheet 1
2010, term 1 Stefan Adams
Students should hand in solutions to questions B1, B2, B3 and B4 by 3pm Monday of week 4 to the maths pigeonloft. Maths students hand in solutions to their supervisors and maths/physics students hand solutions into the slots marked Vector Analysis Maths+Physics.
A1 Level sets of scalar fields Sketch level sets f −^1 (c), for c = 0 and for some values c > 0 and c < 0 , of the following functions: (a) f (x, y) = y^2 + x, (b) f (x, y) = xy.
A2 Visualizing planar vector fields
(a) Sketch the vector field v(x, y) = (− 1 , 2 y). Compare with your sketch of the level sets of f = y^2 − x to confirm it looks like the gradient vector field of f. (b) Sketch the vector fields (i) v(x, y) = (−x, y) and (ii) v(x, y) =
( (^) −x √ x^2 +y^2 , √ y x^2 +y^2
A3 Gradients of scalar fields
(a) Find the gradient vector field ∇f for each of the following scalar fields:
(i) f (x, y) = 2xy + y^2 , (ii) f (x, y) = xy cos(πy).
(b) What is the directional derivative of the function f (x, y) = 2xy + y^2 at the point (2, 3) in the direction (− 1 , 5)?
A4 Line integrals
(a) Find the arclength of the curve parameterized by (t^2 , t^3 ) for t ∈ [0, 1]. (b) Let v be the vector field v(x, y) = (x + y^2 , y − 1). Let C be the curve consisting of the line along the x-axis in the plane joining the points (− 2 , 0) and (2, 0) together with the upper semicircle of radius 2, centered at the origin. Find a parameterization for each part of C. Then evaluate the tangential line integral
C v^ ·^ T dsˆ , where C is traversed in the anticlockwise direction.
A5 Gradient vector fields For the following vector fields v, find a scalar field f so that v = ∇f.
(a) v(x, y) = (2xy + 3x^2 , x^2 ) (b) v(x, y, z) = (2xyz + z, x^2 z + 1, x^2 y + x).
(c) Show that the vector field v(x, y) = (3y, x + y) is not of gradient type.
A6 Finding unit normals to surfaces
(a) Find a unit normal to the surface z = xy + 1 at the point (2, 2 , 5). (b) Find a unit normal to the surface parameterized by x(s, t) = (st, s^2 + t^2 , t^2 s).
A7 Surface integrals
(a) The surface S is parameterized by (s, t, s^2 + t) over s ∈ [0, 1], t ∈ [− 1 , 1]. Calculate the integral
S x^ dS. (b) Compute the surface area of the part of the paraboloid z = x^2 + y^2 that lies between the planes z = 0 and z = L.
B1 Visualization of functions
(a) Sketch level sets f −^1 (c), for c = 0 and some c > 0 and c < 0 , and the graphs of the following functions:
(i) f (x, y) = x−y+2, (ii) g(x, y) = x^2 − 4 y^2 , (ii) h(x, y, z) =
x^2 + y^2 + 3−z.
(b) Sketch or describe the surfaces in R^3 of the following equations:
(i) x^2 + y^2 − 2 x = 0, (ii) z = x^2.
(c) Using polar coordinates, describe the level sets of the function f : R^2 → Rdefined by
f (x, y) =
2 xy (x^2 +y^2 ) if^ (x, y)^6 = (0,^ 0) 0 if (x, y) = (0, 0).
(d) Sketch the following vector fields in the plane
(i) u(x, y) = (1, − x 2
), (ii) v(x, y) = (y, − sin x)
B2 Gradients and Directional Derivatives
(a) Find the gradient vector field ∇f for each of the following scalar fields f : R^3 → R; recall ‖x‖ =
x^21 + x^22 + x^23 :
(i) f (x) = log ‖x‖ for x 6 = 0, (ii) f (x) =
‖x‖
for x 6 = 0,
(iii) f (x, y, z) = x^3 3(y^2 + 1)
sin(3xz).
(b) What is the directional derivative of the function (i) f (x, y, z) = x^2 yz + 4xz^2 at the point (1, − 2 , −1) in direction (2, − 1 , −2)? (ii) g(x, y, z) = ex^ + yz at the point (1, 1 , 1) in direction (1, − 1 , 1). (c) In what direction from (0, 1) does f (x, y) = x^2 − y^2 increase the fastest? (Justify your answer!)
B3 Gradient Vector Fields
(a) Find a potential v : R^3 → R for the following vector fields f : R^3 → R^3 :
(i) f (x, y, z) = (0, y, 0), (ii) f (x 1 , x 2 , x 3 ) = 2‖x‖^4
x 1 x 2 x 3
(^) , x = (x 1 , x 2 , x 3 ) ∈ R^3
(b) Show that f : R^3 → R^3 defined by f (x, y, z) = (z, 0 , 0) is no gradient vector field.
B4 Line integrals
(a) Evaluate the line integral (^) ∫
C
(x + y^2 ),
where C is the parabola y = x^2 in the plane z = 0 connecting the points (0, 0 , 0) and (2, 4 , 0). (b) Calculate the tangent line integral of the vector field
v(x, y, z) =
(x − 1)(z − 3), xyz, x + z
along the straight line from (1, 1 , 1) to (1, 3 , 9). (c) Consider the half circle C =
y^2 + z^2 = 1, z ≥ 0 , x = 0
⊆ R^3 and the vector field f (x, y, z) = (0, y, 0). Use the fundamental theorem of calculus for gradient vector fields to calculate the tangent line integral of f along C from (0, − 1 , 0) to (0, 1 , 0).
