lecture 1 january 23, 2018 -...

Lecture 1Course introduction and vector basics

Dan [email protected]

MATH 233, Spring 2018University of Massachusetts

January 23, 2018

(2) Outline

1. Overview of multivariable calculus

2. Course policies (very brief summary)

3. Calculus I/II review

4. Section 12.2: Vectors

(3) Calculus of one variable

One variable that depends on one other variable, e.g. y = f(x).

y = sinx+ cosx

x

y

(4) Multivariable Calculus

One variable that depends on multiplevariables, e.g. z = f(x, y)

z = x2 + y2

−1 −0.5 00.5 1−1

0

1

0

2

x

y

z

Multiple variables which depend on onevariable, e.g. F(t) = 〈x(t), y(t), z(t)〉

F(t) =

⟨2 + cos t, 2 + sin t,

t2

6π2

⟩

x

y

z

(5) Multivariate Calculus

• What kinds of things will we learn?• Geometry of 3D space, curves and surfaces• Rates of change in a quantity that depends on multiple variables• Integrals over an area or along a curve

• Why study any of this?• 3D geometry is essential if you want to design 3D objects, either real (engineering)

or virtual (computer graphics)• Very often we want to describe a system too complicated for just one dependent

variable and one independent variable• Path of an object moving in 3D space• Financial models with many dependencies• . . . and many more examples

(6) Important stuff

See the syllabus for more details on all of this. Please read the whole thing carefully.

• Grading: Homework 25% (two types), three exams 25% each

• Add/Drop deadline Feb. 5

• Exams: Feb. 21, Apr. 5, May 7

• Website: www.math.umass.edu/~nichols/math233.html

• Has the syllabus, schedule, lecture notes, etc. . .

• Email me ASAP if you need to miss an exam or if you need specialaccommodations

www.math.umass.edu/~nichols/math233.html

(7) Homework

• Paper homework• Conceptual questions or exam practice problems• Assigned at the end of each lecture, due at the beginning of the next lecture

• I can’t accept your paper HW if you arrive more than 10 min. after class begins

• Easy grading

• WebAssign homework• Usually due at 11:59 PM on Wednesday nights• Develop skills by solving more complicated problems

(8) Ways to get help

• email: [email protected]

• I check this often and I usually respond quickly• Instead of using WebAssign’s “ask your instructor” button, please just email me.

(Or post in the Moodle forum)

• Office hours:• Weds 1:00-4:00 PM in LGRT 1117• If I have to change these, I’ll announce it via email• You can always email me to make an appointment for some other time

• Tutoring center:• LGRT 140• Open Mon-Thurs, 10:00 AM - 3:00 PM starting next week• This room has plenty of computers and helpful* TAs, so it’s a great place to work on

homework

* usually

[email protected]

(9) Quick review of calc I & II

• We will rely on skills from Calculus I and II, so it’s very important that youremember the main ideas of those classes

• We’ll quickly review some topics today

(10) Calculus I

• Derivatives• Product rule, quotient rule, chain rule• Linearization, tangent line

• Interpretation of first, second derivatives• When is a graph increasing/decreasing? concave up/down?

• Maximization/minimization• Critical points, first/second derivative test

(11) Calculus I

Example 1: Suppose y = 3eu/2 and u = log x+ 1. What is dydx?

dy

du= 3eu/2 · 1

2=

3

2eu/2

du

dx=

1

x

Now we use the chain rule:

dy

dx=dy

du

du

dx

=

(3

2eu/2

)· 1x=

(3

2e

log x+12

)· 1x

=3

2

(elog x

)1/2e1/2

1

x=

3

2

√e

x.

(12) Calculus I

Example 2: Find the local maxima/minima of the function f(x) = 7 + 9x− 4x3

x

y

0−1 1

Critical points: f ′(x) = 0

f ′(x) = 9− 12x2 = 0

x2 =9

12=

3

4

x = ±√3

2

(13) Calculus I

Example 2: (cont.)

• First derivative test: at a CP, does f ′(x) change sign from + to − or vice versa?

• Second derivative test: at a CP, is f ′′(x) is positive or negative?

The second derivative is f ′′(x) = −24x.

f ′′

(√3

2

)= −24

√3

2= −12

√3

So f has a maximum at x =√32 .

f ′′

(−√3

2

)= −24 · −

√3

2= 12

√3

And f has a minimum at x = −√32 .

(14) Calculus II

• Integrals• Substitution and other integration techniques• Using an integral to calculate the area under a curve

• Parametric equations• Parametric plane curves• Eliminate the parameter• We’ll review this a bit next week

• Polar coordinates• Converting between polar and Cartesian coordinates• Polar curves• We’ll review this later in the semester when we do section 5.4

• Sequences and series, power series• We won’t use these in this course (but you’ll need them for MATH 331 and other

courses)

(15) Calculus II

Example 3: Compute the definite integralˆ e

1

(log t)3

tdt

• Remember that ddt log t =

1t . So if we substitute u = log t, we can rewrite

(log t)3 = u3.

• Furthermore, dudt = 1

t , so dt = t du.

• Rewrite the whole integral in terms of u and evaluate:ˆ 1

0

u3

tt du =

ˆ 1

0u3 du

=u4

4

∣∣∣∣10

=1

4.

• Notice we changed the endpoints of integration when we switched to integratingwith respect to u.

(16) Vectors

A number is a mathematical object used for counting, measuring, and labelling.

In Calculus we work with the real numbers, which are used to measure continuousquantities.

Examples:

• Concentration of a drug in patient’s bloodstream

• Amount of money in a bank account

• Distance between Earth and the Moon

These quantities are all magnitudes – they are answers to a question of the form “howmuch?”

(17) Vectors

But sometimes we want to model things that are more complicated than just anumber.

• Force of air resistance on a flying object

• Velocity (speed and direction) of a car driving in a parking lot

• Position of the Earth relative to the sun

Each of these things can be represented as a vector.

Vectors have both magnitude and direction. They answer two questions at the sametime: “how much?” and “in what direction?”

(18) Vectors

So think of vectors as a replacement for ordinary numbers• We can use them to model a problem we want to solve• We have a set of abstract rules for doing arithmetic with them (e.g. addition)• Once we describe a problem using vectors, we can use these rules to quickly

perform useful computations• We can write one vector quantity as a function of another, and then use Calculus

to see how it’s changing

Description of

problem in words

Description of problem

in math objectsResult

Answer

Natural language

Math objects/symbols translate/model

calculate

interpret

(19) Vectors: definition

DefinitionA vector is a mathematical object which has both magnitude and direction.

• Think of vectors as arrows pointing from the originin space.

• Length and direction of the arrow represent themagnitude and direction of the vector

• Compare to numbers, which we visualize as pointson a number line

• These are 2-dimensional vectors;later we’ll see vectors in higher dimensions.

• We write vectors in bold (v)or with an arrow above (~v)

x

y

vu

(20) Vectors: definition

Whenever we have an initial point and a terminal point, we can draw a vector betweenthem

A

B

−−→AB

We use vectors in this sense to represent the displacement (change in position) of amoving object between two points in time.

If an object starts at point A and moves to point B, its displacement vector is−−→AB.

(21) Vectors: components

We can describe a vector v by giving its components.• Put the initial point of v at the origin and label the terminal point with

coordinates (a, b).• Now we can write the vector as v = 〈a, b〉.

• a is the x-component• b is the y-component

x

y

(0, 0)

(3, 2)

v

3

2

In this case, v = 〈3, 2〉.

(22) Vectors: components

• Usually the best way to describe a vector is by giving its components.

• A vector in 2D space has 2 components

• A vector in 3D space has 3 components (more about this in the next lecture)

(23) Vectors: magnitude and direction

Let v = 〈a, b〉

• Magnitude: distance formula

‖v‖ =√a2 + b2

• Sometimes written |v|• This is the ‘length’ of the vector

• Direction: inverse trigonometry(or unit vector) x

y

(0, 0)

(a, b)

v

(a, 0)a

(0, b)

b

This should seem familiar – polar coordinates

(24) The zero vector

There’s one special vector that’s different from the others

• The zero vector, written as 0 or ~0

• Magnitude is 0, has no direction• This is the only vector with no direction

• Components: 〈0, 0〉 (all zero)

x

y

0

(25) Vector arithmetic: addition

To add vectors, we place them “tip-to-tail” and then draw a new vector from thebeginning of the chain to the end.Suppose we want to add u+ v:

u

v

v

u+ v

u

It’s the same in either order: u+ v = v + u.


We can also just add the components

• u = 〈1, 2〉• v = 〈3, 1〉• u+ v = 〈1 + 3, 2 + 1〉

= 〈4, 3〉

y

x

u

v

uv

u+ v

The x-component of u+ v is the x-component of u plus the x-component of v...

(27) Vector arithmetic: addition: example

Example 4: Draw the vectors u = 〈0, 3〉 and v = 〈3,−1〉 at the origin. Draw the sumu+ v and find its components.

x

y

u

v

u+ v

So u+ v = 〈3, 2〉.


Remember the diagram from when we talked about the components of a vector:

x

y

(0, 0)

(a, b)

v

(a, 0)a

(0, b)

b

What we really did was break v into a sum of two perpendicular vectors:

• One parallel to the x-axis. The x-component is the magnitude of this vector.

• One parallel to the y-axis. The y-component is the magnitude of this vector.

(29) Vectors

• A vector doesn’t have a fixed location; it’s just an arrow• To figure out the components, we put the initial point at the origin• To add u+ v, we place v starting at the end of u

• But when we use vectors to represent real-world quantities, we may draw a vectorat a certain place in a diagram

• Example: the Earth in orbit around the Sun

ra

v

Sun

Earth

• r is the position of Earth relative to the Sun

• v is the velocity of the Earth

• a is the acceleration of the Earth

(30) Vector arithmetic: scalars

DefinitionA scalar has only magnitude, no direction.

• These are the ordinary numbers you’re familiar with,like −5, 0, 1.3, π, . . .

• We give them a new name to contrast them with vectors

• We write scalars in normal type: a is a scalar, u is a vector

(31) Vector arithmetic: scalar multiplication

There’s a natural rule for multiplying a vector by a scalar.

• Must be consistent with addition,e.g. 2v should be the same as v + v


If v is a vector and c is a scalar, the vector cv

• has magnitude equal to |c| times the magnitude of v

• points in the same direction as v if c > 0

• points in the opposite direction from v if c < 0

• is the zero vector if c = 0

v 2v 0.5v −0.5v

Scalar multiplication can’t change the direction of a vector (other than reversing it). Itonly ‘stretches’ the magnitude.


• With components: c 〈a, b〉 = 〈ca, cb〉.• Multiply both components by the scalar

• Magnitude: ‖cv‖ = |c|‖v‖

(34) Vector arithmetic: subtraction

We can also define subtraction: u− v = u+ (−1)v.

u

v

(−1)v

(−1)v

u− v

u− v

• First, draw (−1)v, which is just theopposite of v

• Place it ‘tip-to-tail’ with u

• Add them to getu+ (−1)v

= u− v

• Notice v + (u− v) = u,as expected(just like subtracting numbers)

Shortcut: to subtract v from u, just draw the vector from the terminal point of v tothe terminal point of u.

(35) Vector arithmetic: algebraic properties

Mostly it’s the same as arithmetic with numbers

u+ v = v + u Order doesn’t matter for addition

u+ 0 = u Anything plus zero is itself

1u = u Anything times 1 is itself

u+−u = 0 A vector plus its opposite is 0

(cd)u = c(du) Associative law for multiplication

u+ (v +w) = (u+ v) +w Associative law for addition

c(u+ v) = cu+ cv Distributive law

(c+ d)u = cu+ du Distributive law

• You can add two scalars or two vectors,but you cannot add a scalar and a vector

• You can multiply two scalars, or a vector and a scalar,but you cannot multiply two vectors (until next week...)

(36) Vector arithmetic: example

Example 5: Let u = 〈1, 2〉, v = 〈5,−1〉, and w = 〈0,−1〉. Calculate the followingvectors:

3u

=3 〈1, 2〉= 〈3, 6〉

2u+ v

=2 〈1, 2〉+ 〈5,−1〉= 〈2, 4〉+ 〈5,−1〉= 〈2 + 5, 4 +−1〉= 〈7, 3〉

v −w

= 〈5,−1〉 − 〈0,−1〉= 〈5− 0,−1−−1〉= 〈5, 0〉

v − 5u− 11w

= 〈5,−1〉 − 5 〈1, 2〉 − 11 〈0,−1〉= 〈5,−1〉 − 〈5, 10〉 − 〈0,−11〉= 〈5− 5 + 0,−1− 10−−11〉= 〈0, 0〉

(37) Unit vectors: definition

DefinitionA unit vector is a vector whose length (magnitude) is 1 (unity).

Examples:

• 〈0, 1〉• 〈−1, 0〉

•⟨√

22 ,−

√22

⟩ x

y

〈0, 1〉

〈−1, 0〉 ⟨√22 ,−

√22

⟩

A unit vector is a good way to give a direction when we don’t care about magnitude.Sometimes we write unit vectors with a ‘hat’ over the symbol like this: v̂.

(38) Unit vectors

Often we want to find the unit vector that points in a specific direction. We write theunit vector in the direction of v as v̂.

Let v = 〈2, 4〉. Suppose we want to find the unit vector v̂ inthe direction of v.

• Need a vector that points in the same direction, butwith length 1

• Length of v is ‖v‖ =√22 + 42 = 2

√5

• Multiplying by a positive scalar ‘stretches’ or ‘shrinks’the vector

• Let v̂ = 12√5v = 1

2√5〈2, 4〉 =

⟨1√5, 2√

5

⟩• Then ‖v̂‖ = 1

2√5‖v‖ = 1

2√52√5 = 1.

x

y

v

v̂

(39) Unit vectors

So for any (nonzero) vector v, the unit vector along v is always

v̂ =

(1

‖v‖

)v.

Or: multiply v by the inverse of its magnitude (a scalar) to get v̂.

(40) Unit vectors: standard basis vectors

DefinitionThe (2D) standard basis vectors are i = 〈1, 0〉 and j = 〈0, 1〉.

• We can write any vector in the formv = ai+ bj where a, b are scalars.

• Example:

v = 〈1.7, 2〉= 〈1.7, 0〉+ 〈0, 2〉= 1.7 〈1, 0〉+ 2 〈0, 1〉= 1.7i+ 2j

x

y

i

j

v = 〈1.7, 2〉

〈1.7, 0〉

〈0, 2〉

1.7i

2j

So instead of using components, we can write a vector in this form instead.

(41) Unit vectors: standard basis vectors

• When we express a vector v in terms of i and j,the coefficients of i, j are the components of v. So v = 〈a, b〉 = ai+ bj always.

• In 3D space, there are three standard basis vectors:i = 〈1, 0, 0〉, j = 〈0, 1, 0〉, and k = 〈0, 0, 1〉.

(42) Unit vectors: examples

Example 6: Write each of the following as a sum of the standard basis vectors i and j.

〈−3, 12〉=− 3i+ 12j

〈0, 0.5〉=0i+ 0.5j

=0.5j

〈7, 1〉+ 〈−3, 2〉=7i+ j+−3i+ 2j

=4i+ 3j

Example 7: Find the unit vector in the direction of the vector v = 〈3,−4〉.• The magnitude is ‖v‖ =

√32 + (−4)2 =

√25 = 5

• So v̂ = 1‖v‖v = 1

5v = 15 〈3,−4〉 = 〈3/5,−4/5〉.

(43) Applications: position

When we need to specify a location in two or more dimensions, vectors make thingseasier.

An object’s position vector gives the location of the object relative to something else.

• u is the Earth’s position relative tothe Sun

• v is the Moon’s position relative tothe Earth

• u+ v is the Moon’s position relativeto the Sun

Sun

Earth

Moon

u

vu+ v

(44) Applications: motion

We can use a vector to represent the velocity of an object, v.

• The magnitude of v is the speed of the object

• The direction of v is the direction the object is moving

• An object at rest has v = 0.

We can do the same thing with acceleration. The acceleration vector a

• Points in the direction in which the object is accelerating

• Has magnitude equal to the strength of the acceleration

v

a

(45) Applications: motion

Example: a projectile fired from a cannon

y

x

v

a

v

a

va

a(t) = 〈0,−g〉v(t) = v(0) +

´ t0 a(s) ds = v(0) + t 〈0,−g〉

(46) Applications: force diagrams

Newton’s second law of motion says: F = ma

• An object’s acceleration is equal to its mass times the net force on the object

• The mass m is a scalar. The net force F is a vector. Acceleration a is a vector.

• Each force acting on the object is a vector

• The net force is the sum of all these vectors.

w

n

F

• w: weight (force of gravity)

• n: normal force (surface pushing back againstthe block)

• F = w + n: net force

(47) Vectors: summary

• Like numbers, vectors are abstract objects that can be used to represent things

• Vectors have direction and magnitude

• We express vectors using their components 〈a, b〉 or as a sum of standard basisvectors ai+ bj

• Vector arithmetic is similar to number arithmetic, but with key differences

• Unit vectors have magnitude 1. Often used to indicate direction only

(48) Homework

• Paper homework 1 due at the beginning of class Thursday

• Paper homework 0 (Calc I/II Review) won’t be collected or graded, but please doit and check your answers against the answer key

• First WebAssign homework due next week• Try to log in ASAP, talk to me if you have any problems

lecture 1 january 23, 2018 -...

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