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(MATH102)[2006](f)final~4660^_84970.pdf
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HKUST MATH 102
Second Midterm Examination

Multivariable and Vector Calculus
15 Dec 2006
Answer ALL 8 questions
Time allowed C 180 minutes
Problem 1
(a)
Find an equation of the plane through (.1, 4, .3) and perpendicular to the line x = t +2,y =2t . 3,z = .t.

(b)
Find a rectangular equation for the surface whose spherical equation is . = 2 sin . sin . De-scribe the surface.

(c)
Show that the two lines r = a + vt and r = b + ut, where t is a parameter and a, b, u and v are constant vectors, will intersect if (a . b) (u v) = 0.


Problem2
(a) Describe the graph of the equation r1 (t)= .2 i + t j +(t2 . 1) k.
Find also the vector equation of the tangent line to the curve r1 (t) such that it is parallel to the line r2 (t)= i + (2 + 2t) j + (3 + 4t) k.
22
yz
(b) Sketch the surfaces x + y =4and + =1inthe .rst octant. Find the parametric
42 22
equations of the curve C of intersection of the two surfaces above. Find the parametric equation of the projection curve C onto the xz-plane. Describe the projection curve.
Problem 3
ln(x + y + 1)
(a) Sketch the domain of the function f (x, y)= .
x2 . 1
(b) Determine the largest set on which the function
f (x, y) = . . x2 y x2 + y2 if (x, y) .= (0, 0)
.
. 0 if (x, y) = (0, 0).

is continuous.
2
(c)
Describe the level surfaces of the function f (x, y, z)=(x . 2)2 + y .

(a)
Let f (x, y)= 3x +2y.

(i)
Find the slope of the surface z = f (x, y) in the x-direction at the point (4, 2).

(ii)
Find the slope of the surface z = f (x, y) in the y-direction at the point (4, 2).




Problem 4

2
2 3
(b) Let g(x, y)=(x+ y3 ) . Find gx (x, y), at all points (x, y) in the xy-plane (include the point (0, 0)).
.3
(c) Find f (s 2 . t, s + t2 ) in terms of partial derivatives of f . Assume that f has continuous
.t2 .s
partial derivatives of all orders.
Problem 5
(a)
If f (x, y, z)=(r A) (r B), where r = x i + y j + z k and A and B are constant vectors, show that .f (x, y, z)= P (r A)+ Q (r B). Find P and Q in terms of A, B and r.

(b)
Let r = x i + y j + z k and let r = r. If A and B are constant vectors, show that:


..
1 C
(i) A . = . Find C in terms of A and r.
3
rr
. ...
1 DA B
(ii) B . A . = . . Find D in terms of A, B and r.
53
r rr
Problem 6
A three dimensional surface whose equation is y = f (x) is tangent to the surface z 2 +2xz + y =0 at all points common to the two surfaces. (i) Find f (x). (ii) Find all common points.
Problem 7
(a)
Let f be a nonconstant scalar .eld, di.erentiable everywhere in the plane, and let c be a constant. Assume the Cartesian equation f (x, y)= c describes a curve C having a tangent at each of its points. Prove that f has the following properties at each point of C:

(i)
The gradient vector .f is normal to C.

(ii)
The directional derivative of f is zero along C.


(iii) The di