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1. (a) Suppose that n -dimensional vectors a , b satisfy a + b = 3 and a b = 4. Find the value of a b .

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(b) Suppose that 3-dimensional vectors a , b satisfy a b = 2 and a b = 3. Find the value of a b .

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2. Consider a plane P containing three points (1 , 1, 3), (1, 4, 1), (2, 1, 1) in R3 . Find the point on the plane P that is the closest to

the point (0 , 1, 2).

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3. (a) Find a Cartesian(rectangular) equation and a cylindrical equation of the surface given by the spherical equation

2 cos2 = 1,and give the name of the surface.

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(b) Sketch the region in R 3 given by the following inequalities in the spherical coordinate system:

2 cos2 1 and 2.You must specify interior or exterior whenever necessary.

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4. Consider a vector function f = ( f 1 , f 2 ) : R 2 R2 dened by

f 1 (x, y ) =xy 3

x 4 + y4 if (x, y ) = (0 , 0)

0 if (x, y ) = (0 , 0),

f 2 (x, y ) = xy x + y 1.(a) Compute the derivative of f at (0 , 0).

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(b) Is f continuous at (0 , 0)? Justify your answer.

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(c) Is f differentiable at (0 , 0)? Justify your answer.

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5. Consider two surfaces in R 3 given by z = xy and z = 34 x2

y2 , respectively.

(a) Show that they intersect perpendicularly at the point (2 , 1, 2).

(b) Find a vector parametrization (path) for the line tangent to both surfaces at (2 , 1, 2).

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(c) Find a path tracing out the intersection of the two surfaces containing the point (2 , 1, 2).

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(d) Find the arclength of the path obtained in (c) from (0 , 0, 0) to (2 , 1, 2) by using the indenite integral

x 2 + a2 dx = 1

2x x 2 + a2 + a2 ln x + x 2 + a2 + C.

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6. (a) The variable z is dened implicitly as a function of x, y via the equation xy 3 + yz 3 + x 3 z = 1. Find the value of z y

at

(x,y,z ) = (1 , 1, 1).

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(b) The Laplacian 2 f of a C 2 -function f : R 3 R is dened to be

2 f = 2 f x 2

+ 2 f y 2

+ 2 f z 2

. Let g : R R be a C 2 -function

such that g(1) = g (1) = g (1) = 1. And let f (x,y,z ) = g( x2 + y2 + z 2 ). Find

2 f (0, 1, 0).

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7. For the scalar function f (x,y,z ) = xyze y + z 2 and the point a = (1 , 0, 1), answer the following.(a) Find the rate of change of the function value when ( x,y,z ) changes (innitesimally) from a in the direction (1 , 2, 2).

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(b) Find the unit vector pointing the direction in which the function decreases the most rapidly at a.

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(c) Find an equation of the plane tangent to the surface f (x,y,z ) = 1 at a .

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8. Let c : R R2 be a differentiable path such that c(2) = (1 , 1), c(3) = ( 1, 1), and c (2) = (2 , 1). For the function f : R

2

R3

dened by

f (x, y ) =x 2 + y2

2 , xy, x 2 y

2 + y ,

let p : R R3 be the composite path p (t) = f c(t).

(a) Find the tangent vector p (2).

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(b) Suppose that two particles follow the path p . If one of them ies off in the tangent direction at t = 2, what is the distancebetween the two particles at t = 3.

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9. Find all values of the constant a that make the 2-dimensional vector eld

F (x, y ) = ax

x2 + 2 y2

, y

x2 + 2 y2

irrotational.

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10. Using Maple with linalg and plots packages, we would like to verify the gradient vector eld of the scalar function

f (x, y ) = exy + sin xy

is orthogonal to level curves of f on the square 1 x, y 1. Fill in the blank(s) of each step with appropriate Maple expressionsto obtain the following result on the screen:

(I) Open linalg package to use the command grad.(i) (linalg):

(II) Dene the function f .f:= (ii) :

(III) Compute the gradient of f .Gf:= (iii) :

(IV) Open plots package to use the commands eldplot and contourplot.(iv) (plots):

(V) Save the plot data to draw the gradient eld and level curves.PGf:= (v) :LCf:= (vi) :

(VI) Display the two data simutaneously.(vii) ;

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