MAST20029 ENGINEERING MATHEMATICS
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
DEPARTMENT OF MATHEMATICS AND STATISTICS
MAST20029 ENGINEERING MATHEMATICS SAMPLE
MID SEMESTER TEST
1. Consider the double integral
(a) Sketch the region of integration.
(b) By changing the order of integration, evaluate the double integral.
2. Determine the surface area of the part of the surface z = xy that lies inside the cylinder given by x2 + y2 = 1 .
3. Let F(x, y, z) = (sin y cos x)i + (sin x cos y)j + e2zk.
(a) Show that F is a conservative vector field.
(b) Find a scalar function ϕ such that F = ∇ϕ .
(c) Let C be the boundary of the square with vertices (0, 0), (1, 0), (1, 1), (0, 1), in the x-y plane. Determine the work done by F in moving a particle in an anticlockwise direction around C .
4. Consider the surface S of the region V formed by the portion of the cylinder x2 + y2 ≤ 4 which lies between the planes z = 1 and z = 4. let S be oriented with an outward unit normal.
Use Gauss’ (Divergence) Theorem and cylindrical coordinates to express the flux
of the vector field
across the surface as a triple integral.
5. Consider the following system of differential equations
with general solution
Sketch the phase portrait near the critical point at the origin. To sketch the phase portrait, determine:
• any special cases of the orbits,
• how the orbits behave as t → −∞ ,
• how the orbits behave as t → ∞ ,
• the slope of the orbits when x = 0,
• the slope of the orbits when y = 0.
In your sketch, show all the straight line orbits and at least four general orbits.
2022-01-10