contents 1 vector functions and space curves 2 · department of mathematical sciences @iupui 26100...

Contents

1 Vector Functions and Space Curves 21.1 Limits, Derivatives, and Integrals of Vector Functions . . . . . . . . . . . . . . . . . . . . . . 21.2 Arc Length and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Motion in Space: Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Functions of Several Variables 52.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Tangent Planes and Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Directional Derivatives and the Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Minimum and Maximum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Multiple Integrals 143.1 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Double Integrals over General Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Applications of Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 Triple Integrals in Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 Triple Integrals in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.8 Change of Variables in Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Vector Calculus 234.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Fundamental Theorem of Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 Curl and Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.6 Parametric Surfaces and their Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.7 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.8 Stoke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.9 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Department of Mathematical Sciences @IUPUI 26100 Exam Jam

1 Vector Functions and Space Curves

1.1 Limits, Derivatives, and Integrals of Vector Functions

Consider the vector function

~r(t) =⟨√

4− t2, e−3t, ln(t + 1)⟩.

a. Find the domain of ~r(t).

b. Compute limt→−1+

~r(t). Is ~r(t) continuous at t = −1?

c. Find ~r ′(t).

d. Find parametric equations for the tangent line to ~r(t) at t = 0.

1.2 Arc Length and Curvature

~r(t) =⟨√

2t, et, e−t⟩, 0 ≤ t ≤ 1.

a. Compute the length of the curve.

b. Find the unit tangent and unit normal vectors to the curve.

c. Compute the curvature of the function at t = 0.

1.3 Motion in Space: Velocity and Acceleration

~r(t) =

⟨−1

2t2, t

a. Find the velocity, acceleration, and speed of the particle with the position function ~r(t).

b. Find the normal and tangent components of the acceleration at t = 1.

2 Functions of Several Variables

2.1 Domain and Range

Find and sketch the domain of the function f(x, y, z).

f(x, y, z) =√

1− x2 − y2 − z2

2.2 Limits and Continuity

Is the function g(x, y) defined by

g(x, y) =

x2 − y2

x2 + y2if (x, y) 6= (0, 0)

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

continuous at (x, y) = (0, 0)?

2.3 Partial Derivatives

Find all first order partial derivatives of the following functions.

a. z = (2x + 3y)10

c. t =ev

u + v2

b. w = zexyz

d. u = xy/z

2.4 Tangent Planes and Linear Approximation

Consider the function

f(x, y) = 1 + x ln(xy − 5).

a. Explain why f(x, y) is differentiable at (x, y) = (2, 3).

b. Find the linearization L(x, y) of f(x, y) at the point (2, 3) and use it to approximate (2.01, 2.99).

2.5 The Chain Rule

Use the chain rule to find the partial derivatives∂z

∂s,∂z

∂t, and

∂uwith z(x, y), x(s, t, u) and y(s, t, u) defined

below.

z = x4 + x2y, x = s + 2t− u, y = s + u2

2.6 Directional Derivatives and the Gradient

Consider the function

f(x, y) = sin(2x + 3y).

a. Find the gradient of f(x, y).

b. Find the maximal rate of change of f(x, y) at the point (−6, 4) and the direction in which it occurs.

c. Find the rate of change of f(x, y) at the point (−6, 4) in the direction of 〈√

3,−1〉.

2.7 Minimum and Maximum Values

Find the local maxima, minima, and saddle points of the following function.

f(x, y) = ey(y2 − x2)

Find the absolute maximum and minimum values of the function

f(x, y) = x2 + y2 − 2x

on the closed, triangular region D with vertices (2, 0), (0, 2), and (0,−2).

2.8 Lagrange Multipliers

Find the extreme values of the function f(x, y) = x2 + 2y2 on the circle x2 + y2 = 1.

3 Multiple Integrals

3.1 Iterated Integrals

Calculate the following iterated integrals.

12x2y3 dx

xydy dx

y2e2x dy dx

x2 + 1dy dx

3.2 Double Integrals over General Regions

Evaluate the integrals by reversing the oder of integration.

dx dy b.

3.3 Double Integrals in Polar Coordinates

Evaluate the given integral by changing to polar coordinates.∫∫R

sin(x2 + y2) dA

Where R is the region in the first quadrant between the circles of radii 1 and 3 centered at the origin.

3.4 Applications of Iterated Integrals

Find the center of mass of the lamina whose boundary consists of the semi-circle y =√

1− x2 and y =√4− x2 together with the portions of the x-axis that join them and whose density at any point is inversely

proportional to its distance from the origin.

Find the area of the part of the surface z = xy that lies within the cylinder x2 + y2 = 1.

3.5 Triple Integrals

Evaluate the triple integrals.

∫∫∫E

y dV, E = {(x, y, z)|0 ≤ x ≤ 3, 0 ≤ y ≤ x, x− y ≤ z ≤ x + y}

∫∫∫E

x2 + z2dV, E = {(x, y, z)|1 ≤ y ≤ 4, y ≤ z ≤ 4, 0 ≤ x ≤ z}

3.6 Triple Integrals in Cylindrical Coordinates

Evaluate the triple integral ∫∫∫E

where E is the area enclosed by z = x2 + y2 and z = 4.

3.7 Triple Integrals in Spherical Coordinates

Find the volume of the solid that lies below z =√

4− x2 − y2 and z =√x2 + y2, and above z = 0.

3.8 Change of Variables in Iterated Integrals

Evaluate the double integral by making an appropriate change of variables.∫∫R

(y − x

Where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, 2), and (0, 1).

4 Vector Calculus

4.1 Vector Fields

Sketch the vector fields.

a. ~F (x, y) = −~i +~j b. ~F (x, y) =

4.2 Line Integrals

Calculate the following line integrals.

xyz2 ds where C is the line segment from (−1, 5, 0) to (1, 6, 4).

~F · d~r where ~F (x, y) = 〈y, x〉 and C is the upper half of the circle of radius 1 centered at the origin

oriented counter-clockwise.

4.3 Fundamental Theorem of Line Integrals

Prove that the vector field ~F (x, y) is conservative and find a function f such that ~∇f = ~F (x, y).

~F (x, y) = 〈x2, y2〉

Use your result to calculate ∫C

~F · d~r

where C is the arc of the parabola y = 2x2 from (−1, 2) to (2, 8).

4.4 Green’s Theorem

Use Green’s Theorem to evaluate the line integral∫C

y3 dx− x3 dy

where C is the circle x2 + y2 = 4 oriented counterclockwise.

Use Green’s Theorem to evaluate ∫C

~F · d~r

~F (x, y) = 〈y cosx− xy sinx, xy + x cosx〉

and C is the triangle with vertices (0, 0), (2, 0), and (0, 4).

4.5 Curl and Divergence

Compute the curl and divergence of the vector field.

~F (x, y, z) = 〈ex sin y, ey sin z, ez sinx〉

4.6 Parametric Surfaces and their Areas

Find the surface area of the part of the plane 3x + 2y + z = 6 that lies in the first octant.

Find the surface area of the part of the cone z =√x2 + y2 that lies between the plane y = x and the cylinder

y = x2.

4.7 Surface Integrals

Evaluate the surface integrals.

∫∫S

x2yz dS where S is the part of the plane z = 1 + 2x+ 3y that lies above the rectangle [0, 3]× [0, 2].

∫∫S

~F · d~S where ~F (x, y, z) = 〈x,−z, y〉 and S is the part of the sphere x2 + y2 + z2 = 4 in the first

octant with orientation towards the origin.

4.8 Stoke’s Theorem

Use Stoke’s Theorem to evaluate ∫∫S

curl ~F · d~S

where ~F (x, y, z) = 〈x2z2, y2z2, xyz〉 and S is the part of the surface z = x2 + y2 that lies inside the cylinderx2 + y2 = 4 oriented upwards.

4.9 The Divergence Theorem

Use the Divergence Theorem to calculate the surface integral∫∫S

~F · d~S

where ~F (x, y, z) = 〈x3 + y3, y3 + z3, x3 + z3〉 and S is the sphere of radius two centered at the origin.

contents 1 vector functions and space curves 2 · department of mathematical sciences @iupui 26100...

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