Q1:
Consider the three-dimensional scalar field
h(x, y, z) = 1/(X2 + y2 + z2 ) - z.
(c) Find an expression for the scalar field h in spherical coordinates.
(d) Determine the gradient grad h and the value of |grad h| in spherical coordinates.
Q2:
(a) Evaluate the scalar line integral of the vector field F(x,y, z) = -yi + xj + z2k around the closed circular path C of radius 1 centred at the point (2,1,3) defined by the parametric equations
x = cost + 2, y = sin t+1, z = 3, for 0 ≤ t ≤ 2Π.
Is the vector field F conservative?
(b) A velocity field v is expressed in spherical coordinates as
v(r, θ, Φ) = sinΦsin(2θ)er + 2 sin Φ cos(2θ)eθ + 2cosΦcosθeΦ,
By calculating ∇ x v, show that v is everywhere conservative,
Q3:
Consider the vector field
F(x, y, z) = (2xy2 - yz + 1)i+ (2x2y - xz)j - xy k.
(a) Calculate curl F and state whether or not F is conservative.
(b) Follow Procedure 1 on page 94 of Unit 16 to calculate a potential function U such that F = - grad U.
(c) Using the expression for U(x, y, z) that you found in part (b), calculate - grad U and confirm that F = - grad U.
Q4:
A region in the (x, y)-plane is bounded by the curves
Y = x2/Π2 1 +4 and y = cos x for -Π/2 ≤ x ≤ Π/2.
These curves meet only at x = ±Π/2.
(a) Sketch a diagram showing the region. Indicate a thin vertical strip within the region, and mark the values of y at its endpoints.
(b) Use an area integral to find the area of the region.
Q5:
The classical scholar Archimedes (287-212 Bc) calculated the volumes of many solids in a work called The Method. For many centuries this great work was lost, and all that remained were reports and excerpts in the works of other authors. From reading these reports, mathematicians were intrigued by how close Archimedes' methods were to the calculus invented by Isaac Newton and Gottfried Leibniz centuries later. Remarkably, a palimpsest of the lost work was rediscovered in the 1990s. This question concerns reworking, using modern calculus techniques, Proposition 13 of the palimpsest, which states: The volume of an ungula is one sixth of the volume of an enclosing cube'.
Consider a cube of side 2a positioned so that the centre of one face is at the origin, and take x-, y- and z-axes parallel to the edges of the cube, as shown in the diagram below. This cube is the enclosing cube mentioned in Archimedes' result. Inscribe a circle of radius a in the top and bottom faces of the cube, and imagine cutting these out so that a cylinder of height 2a with base of radius a remains. Now imagine cutting this cylinder by the plane z = 2y (shown in blue in the left-hand diagram) that passes through a diameter of the base circle and along an edge at the top of the cube. The object remaining underneath this plane resembles a horse's hoof and hence it is called an ungula; it is shown in the right-hand diagram below.
Use cylindrical coordinates to calculate the volume of the ungula, and hence verify Archimedes' result.