lecture 1 january 23, 2018 -...
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Lecture 1Course introduction and vector basics
Dan Nicholsnichols@math.umass.edu
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
(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: nichols@math.umass.edu
• 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
(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.
(26) Vector arithmetic: addition
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〉.
(28) Vector arithmetic: addition
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
(32) Vector arithmetic: scalar multiplication
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.
(33) Vector arithmetic: scalar multiplication
• 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
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