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This is the Exam of Mathematics which includes Solvability, Function, Initial Value Problem, Determine, Regular or Irregular, Equal Difficulty, Incorrect Answers etc. Key important points are: Scalar Projection, Vector, Vector Projection, Distance, Point, Plane, Equation, Light Intensity, Function, Gradient
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[12] 1. (a) Find the scalar projection of the vector 〈 1 , π, 2 π〉 along the vector 〈 2 , 6 , 3 〉.
(b) Find the vector projection of 〈 1 , π, 2 π〉 along the vector 〈 2 , 6 , 3 〉.
(c) Find the distance of the point (2, π + 14, 2 π − 27) from the plane given by the equation 2(x − 1) + 6(y − 14) + 3(z + 27) = 0.
[12] 2. In an experiment, the light intensity is an unknown function I(x, y, z) (it depends on the point in space). Its gradient at the point (1, 2 , 3) is given:
∇I|(1, 2 ,3) = 〈 0. 1 , − 0. 2 , 0. 5 〉
(we ignore the units).
(a) Find the directional derivative DuI in the direction of the vector 〈 1 , 4 , 12 〉.
(b) A small light sensor is moving through this space, with the position function
r(t) = 〈t, 2 t^2 , 3 t^4 〉.
Find the velocity of the sensor at the time t = 1.
(c) Find
dI dt
|t=1 – the rate of change of intensity that the sensor (with the position function from part (b)) registers at the time t = 1.
[10] 4. The double integral of some function over a domain D is represented by the iterated integral as follows: (^) ∫ ∫
D
f (x, y) dA =
0
∫ 2 √x √x^ f^ (x, y)^ dy dx.
(a) Sketch the domain D.
(b) Change the order of integration, so that in the result, integration with respect to y is on the outside.
[8] 5. A wire that occupies the segment in space between the points (0, 1 , 1) and (2, 1 , 4) has the density given by the formula ρ(x, y, z) = yex^ + z,
(we ignore the units). Find the total mass of the wire.
[15] 7. The vector field F is defined on the whole space by the formula
F(x, y, z) = 〈y^3 + e^4 x
2 , −x^3 − sin(y), 3 z^4 〉.
(a) Find curl (F).
(b) Evaluate (by direct computation) the flux integral ∫ ∫
S
curl (F) · dS,
where S is the surface defined by x^2 + y^2 ≤ 1, z = 0, (the unit disc in the xy-plane), oriented upward.
(This question is continued on the next page).
(c) Now let F be the same vector field as in parts (a) and (b), and let M be the hemisphere x^2 + y^2 + z^2 = 1, z ≤ 0, oriented downward. Evaluate ∫ ∫
M
curl (F) · dS
in any way you like.
(b) Let F be the vector field defined by the formula
F(x, y, z) = 〈xz + y^2 , x^2 z^3 , z^2 − y〉.
Compute div (F).
(c) Let S be the closed surface that encloses the solid E from part (a) (so that S consists of the part of the cone that lies inside the sphere, and the spherical cap), oriented outward. Compute, in any way you like, the flux integral ∫ ∫
S
F · dS,
The End
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The University of British Columbia Sessional Examinations - December 2010
Mathematics 263 Multivariable and Vector Calculus
Closed book examination Time: 2.5 hours
Surname(s): Given Name(s):
Student Number: Instructor’s Name:
Signature: Section Number:
No books, notes, cell-phones or calculators are allowed, except for one letter-size formula sheet (you can use both sides). You must show all your work (i.e., intermediate steps) for full credit. If you need more space, use the back of the previous page, or ask for a booklet. You must turn in all the booklets you use.
Rules governing examinations
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