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Math 241: Multivariable calculus, Lecture 1Introduction, Rn, vectors.
Sections 12.1, 12.2
go.illinois.edu/math241fa17
Monday, August 28th, 2017
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.
• Worksheets.• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.
• Worksheets.• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.• Worksheets.
• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.• Worksheets.• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.• Worksheets.• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.• Worksheets.• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.• Worksheets.• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.• Worksheets.• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.• Worksheets.• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Components of the course
• Lectures will be Monday, Wednesday, Friday in 314 AltgeldHall.
• Discussion sections Tuesday and Thursday.• Worksheets.• Quiz
• Information about the course will be on the Wiki:go.illinois.edu/math241fa17.
• Webassign homework,https://www.webassign.net/uiuc/login.html
• Piazza, https://piazza.com
• Tutoring room
• Moodle for the grades
• Exams and Final
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2, r(x) = 2x2+xx3−5x+20
, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2, r(x) = 2x2+xx3−5x+20
, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2, r(x) = 2x2+xx3−5x+20
, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2,
r(x) = 2x2+xx3−5x+20
, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2,
r(x) = 2x2+xx3−5x+20
, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2, r(x) = 2x2+xx3−5x+20
,
h(θ) = sin(θ) + cos(2θ),g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2, r(x) = 2x2+xx3−5x+20
, h(θ) = sin(θ) + cos(2θ),
g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2, r(x) = 2x2+xx3−5x+20
, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2, r(x) = 2x2+xx3−5x+20
, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Calculus of 1 variable
In Calculus I and II you study real valued functions
y = f (x)
of a single real variable, x .
Examples:
• f (x) = x2, r(x) = 2x2+xx3−5x+20
, h(θ) = sin(θ) + cos(2θ),g(u) = eu,...
• T (t) = temperature in Champaign-Urbana, t hours aftermidnight on August 27.
• ρ(d) = density of a piece of wire at distance d from one end.
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.
• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
Three key concepts from Calculus I, II.
f (x), a function of one variable.
1 The derivative: f ′(x) = dfdx = d
dx f (x) = dydx .
• Rate of change.• Slope of the tangent line to the graph.
2 The integral:∫ ba f (x) dx .
• Signed area under graph.
• Average value 1b−a
∫ b
af (x) dx .
3 Fundamental Theorem of Calculus: Relates the two.
• f (b)− f (a) =∫ b
af ′(x) dx .
go.illinois.edu/math241fa17.
1 variable is too limiting
Functions of a single variable are insufficient for modeling morecomplicated situations.
Examples:
• The temperature depends on location as well as time. Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:
T (x , y , t) = temperature at time t in location (x , y).
• Density of a flat sheet of metal can depends on the point inthe sheet, specified by x and y coordinates
δ(x , y) = density of point at (x , y) in a sheet of metal
go.illinois.edu/math241fa17.
1 variable is too limiting
Functions of a single variable are insufficient for modeling morecomplicated situations.
Examples:
• The temperature depends on location as well as time. Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:
T (x , y , t) = temperature at time t in location (x , y).
• Density of a flat sheet of metal can depends on the point inthe sheet, specified by x and y coordinates
δ(x , y) = density of point at (x , y) in a sheet of metal
go.illinois.edu/math241fa17.
1 variable is too limiting
Functions of a single variable are insufficient for modeling morecomplicated situations.
Examples:
• The temperature depends on location as well as time.
Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:
T (x , y , t) = temperature at time t in location (x , y).
• Density of a flat sheet of metal can depends on the point inthe sheet, specified by x and y coordinates
δ(x , y) = density of point at (x , y) in a sheet of metal
go.illinois.edu/math241fa17.
1 variable is too limiting
Functions of a single variable are insufficient for modeling morecomplicated situations.
Examples:
• The temperature depends on location as well as time.
Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:
T (x , y , t) = temperature at time t in location (x , y).
• Density of a flat sheet of metal can depends on the point inthe sheet, specified by x and y coordinates
δ(x , y) = density of point at (x , y) in a sheet of metal
go.illinois.edu/math241fa17.
1 variable is too limiting
Functions of a single variable are insufficient for modeling morecomplicated situations.
Examples:
• The temperature depends on location as well as time. Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:
T (x , y , t) = temperature at time t in location (x , y).
• Density of a flat sheet of metal can depends on the point inthe sheet, specified by x and y coordinates
δ(x , y) = density of point at (x , y) in a sheet of metal
go.illinois.edu/math241fa17.
1 variable is too limiting
Functions of a single variable are insufficient for modeling morecomplicated situations.
Examples:
• The temperature depends on location as well as time. Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:
T (x , y , t) = temperature at time t in location (x , y).
• Density of a flat sheet of metal can depends on the point inthe sheet, specified by x and y coordinates
δ(x , y) = density of point at (x , y) in a sheet of metal
go.illinois.edu/math241fa17.
1 variable is too limiting
Functions of a single variable are insufficient for modeling morecomplicated situations.
Examples:
• The temperature depends on location as well as time. Needto specify location, e.g. by latitude x and longitude y , andtime, e.g. t hours after midnight:
T (x , y , t) = temperature at time t in location (x , y).
• Density of a flat sheet of metal can depends on the point inthe sheet, specified by x and y coordinates
δ(x , y) = density of point at (x , y) in a sheet of metal
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
• R1 = 1− dimensional space = R = real line
• R2 = 2− dimensional space= Cartesian plane= {(x , y) | x , y ∈ R}= ordered pairs of real numbers .
• R3 = 3− dimensional space= {(x , y , z) | x , y , z ∈ R}= ordered triples of real numbers.
The numbers x , y in R2 or x , y , z in R3 are the coordinates of thepoint. If P is a point in R2 it has coordinates (x , y). We write
P(x , y) in this case. same for more dimensions.
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
• R1 = 1− dimensional space = R = real line
• R2 = 2− dimensional space= Cartesian plane= {(x , y) | x , y ∈ R}= ordered pairs of real numbers .
• R3 = 3− dimensional space= {(x , y , z) | x , y , z ∈ R}= ordered triples of real numbers.
The numbers x , y in R2 or x , y , z in R3 are the coordinates of thepoint. If P is a point in R2 it has coordinates (x , y). We write
P(x , y) in this case. same for more dimensions.
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
• R1 = 1− dimensional space = R = real line
• R2 = 2− dimensional space= Cartesian plane= {(x , y) | x , y ∈ R}= ordered pairs of real numbers .
• R3 = 3− dimensional space= {(x , y , z) | x , y , z ∈ R}= ordered triples of real numbers.
The numbers x , y in R2 or x , y , z in R3 are the coordinates of thepoint. If P is a point in R2 it has coordinates (x , y). We write
P(x , y) in this case. same for more dimensions.
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
• R1 = 1− dimensional space = R = real line
• R2 = 2− dimensional space= Cartesian plane= {(x , y) | x , y ∈ R}= ordered pairs of real numbers .
• R3 = 3− dimensional space= {(x , y , z) | x , y , z ∈ R}= ordered triples of real numbers.
The numbers x , y in R2 or x , y , z in R3 are the coordinates of thepoint. If P is a point in R2 it has coordinates (x , y). We write
P(x , y) in this case. same for more dimensions.
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
• R1 = 1− dimensional space = R = real line
• R2 = 2− dimensional space= Cartesian plane= {(x , y) | x , y ∈ R}= ordered pairs of real numbers .
• R3 = 3− dimensional space= {(x , y , z) | x , y , z ∈ R}= ordered triples of real numbers.
The numbers x , y in R2 or x , y , z in R3 are the coordinates of thepoint.
If P is a point in R2 it has coordinates (x , y). We write
P(x , y) in this case. same for more dimensions.
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
• R1 = 1− dimensional space = R = real line
• R2 = 2− dimensional space= Cartesian plane= {(x , y) | x , y ∈ R}= ordered pairs of real numbers .
• R3 = 3− dimensional space= {(x , y , z) | x , y , z ∈ R}= ordered triples of real numbers.
The numbers x , y in R2 or x , y , z in R3 are the coordinates of thepoint. If P is a point in R2 it has coordinates (x , y). We write
P(x , y) in this case. same for more dimensions.
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
For any n = 1, 2, 3, 4, ..., we have
Rn = n − dimensional space= {(x1, x2, . . . , xn) | xi ∈ R}= ordered n–tuples real numbers.
x1, . . . , xn are the coordinates of the point.
We will study functions whose domain (and range) is a subset ofRn.
Dimensions 1,2 and 3 will serve as motivation and provideintuition, though much of the theory works for all n (but not all!)
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
For any n = 1, 2, 3, 4, ..., we haveRn = n − dimensional space
= {(x1, x2, . . . , xn) | xi ∈ R}= ordered n–tuples real numbers.
x1, . . . , xn are the coordinates of the point.
We will study functions whose domain (and range) is a subset ofRn.
Dimensions 1,2 and 3 will serve as motivation and provideintuition, though much of the theory works for all n (but not all!)
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
For any n = 1, 2, 3, 4, ..., we haveRn = n − dimensional space
= {(x1, x2, . . . , xn) | xi ∈ R}= ordered n–tuples real numbers.
x1, . . . , xn are the coordinates of the point.
We will study functions whose domain (and range) is a subset ofRn.
Dimensions 1,2 and 3 will serve as motivation and provideintuition, though much of the theory works for all n (but not all!)
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
For any n = 1, 2, 3, 4, ..., we haveRn = n − dimensional space
= {(x1, x2, . . . , xn) | xi ∈ R}= ordered n–tuples real numbers.
x1, . . . , xn are the coordinates of the point.
We will study functions whose domain (and range) is a subset ofRn.
Dimensions 1,2 and 3 will serve as motivation and provideintuition, though much of the theory works for all n (but not all!)
go.illinois.edu/math241fa17.
The setting: n–dimensional space, Rn.
For any n = 1, 2, 3, 4, ..., we haveRn = n − dimensional space
= {(x1, x2, . . . , xn) | xi ∈ R}= ordered n–tuples real numbers.
x1, . . . , xn are the coordinates of the point.
We will study functions whose domain (and range) is a subset ofRn.
Dimensions 1,2 and 3 will serve as motivation and provideintuition, though much of the theory works for all n (but not all!)
go.illinois.edu/math241fa17.
Plan for this course
Develop calculus to study functions of several variables.
1 Derivatives: Chapter 14 (and 13)
2 Integrals: Chapter 15 (and 13)
3 “Fundamental Theorems of Calculus” : Chapter 16
go.illinois.edu/math241fa17.
Plan for this course
Develop calculus to study functions of several variables.
1 Derivatives: Chapter 14 (and 13)
2 Integrals: Chapter 15 (and 13)
3 “Fundamental Theorems of Calculus” : Chapter 16
go.illinois.edu/math241fa17.
Plan for this course
Develop calculus to study functions of several variables.
1 Derivatives: Chapter 14 (and 13)
2 Integrals: Chapter 15 (and 13)
3 “Fundamental Theorems of Calculus” : Chapter 16
go.illinois.edu/math241fa17.
Plan for this course
Develop calculus to study functions of several variables.
1 Derivatives: Chapter 14 (and 13)
2 Integrals: Chapter 15 (and 13)
3 “Fundamental Theorems of Calculus” : Chapter 16
go.illinois.edu/math241fa17.
What do we need in order to do calculus?
Question: What sets calculus apart from algebra, trigonometry,pre-calculus?
Answer: Limits! In one variable:
limx→a
f (x) = L means
“as x approaches a, f (x) approaches L”
This requires a notion of “proximity” and hence of distance.
go.illinois.edu/math241fa17.
What do we need in order to do calculus?
Question: What sets calculus apart from algebra, trigonometry,pre-calculus?
Answer: Limits! In one variable:
limx→a
f (x) = L means
“as x approaches a, f (x) approaches L”
This requires a notion of “proximity” and hence of distance.
go.illinois.edu/math241fa17.
What do we need in order to do calculus?
Question: What sets calculus apart from algebra, trigonometry,pre-calculus?
Answer: Limits!
In one variable:
limx→a
f (x) = L means
“as x approaches a, f (x) approaches L”
This requires a notion of “proximity” and hence of distance.
go.illinois.edu/math241fa17.
What do we need in order to do calculus?
Question: What sets calculus apart from algebra, trigonometry,pre-calculus?
Answer: Limits! In one variable:
limx→a
f (x) = L means
“as x approaches a, f (x) approaches L”
This requires a notion of “proximity” and hence of distance.
go.illinois.edu/math241fa17.
What do we need in order to do calculus?
Question: What sets calculus apart from algebra, trigonometry,pre-calculus?
Answer: Limits! In one variable:
limx→a
f (x) = L means
“as x approaches a, f (x) approaches L”
This requires a notion of “proximity” and hence of distance.
go.illinois.edu/math241fa17.
What do we need in order to do calculus?
Question: What sets calculus apart from algebra, trigonometry,pre-calculus?
Answer: Limits! In one variable:
limx→a
f (x) = L means
“as x approaches a, f (x) approaches L”
This requires a notion of “proximity” and hence of distance.
go.illinois.edu/math241fa17.
Distance
Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be
|PQ| = distance from P to Q
=√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
n = 2 (Pythagorean Theorem)
Distance from (a, b) to (c, d)
is√
(a− c)2 + (b − d)2
n = 3
Distance from (a, b, c) to (p, q, r)
is√
(a− p)2 + (b − q)2 + (c − r)2
(a,b,c)
x
y
z
(p,q,r)
go.illinois.edu/math241fa17.
Distance
Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be
|PQ| = distance from P to Q
=√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
n = 2 (Pythagorean Theorem)
Distance from (a, b) to (c, d)
is√
(a− c)2 + (b − d)2
n = 3
Distance from (a, b, c) to (p, q, r)
is√
(a− p)2 + (b − q)2 + (c − r)2
(a,b,c)
x
y
z
(p,q,r)
go.illinois.edu/math241fa17.
Distance
Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be
|PQ| = distance from P to Q
=√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
n = 2 (Pythagorean Theorem)
Distance from (a, b) to (c, d)
is√
(a− c)2 + (b − d)2
n = 3
Distance from (a, b, c) to (p, q, r)
is√
(a− p)2 + (b − q)2 + (c − r)2
(a,b,c)
x
y
z
(p,q,r)
go.illinois.edu/math241fa17.
Distance
Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be
|PQ| = distance from P to Q
=√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
n = 2 (Pythagorean Theorem)
Distance from (a, b) to (c, d)
is√
(a− c)2 + (b − d)2
n = 3
Distance from (a, b, c) to (p, q, r)
is√
(a− p)2 + (b − q)2 + (c − r)2
(a,b,c)
x
y
z
(p,q,r)
go.illinois.edu/math241fa17.
Distance
Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be
|PQ| = distance from P to Q
=√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
n = 2 (Pythagorean Theorem)
Distance from (a, b) to (c, d)
is√
(a− c)2 + (b − d)2
n = 3
Distance from (a, b, c) to (p, q, r)
is√
(a− p)2 + (b − q)2 + (c − r)2
(a,b,c)
x
y
z
(p,q,r)
go.illinois.edu/math241fa17.
Distance
Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be
|PQ| = distance from P to Q
=√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
n = 2 (Pythagorean Theorem)
Distance from (a, b) to (c, d)
is√
(a− c)2 + (b − d)2
n = 3
Distance from (a, b, c) to (p, q, r)
is√
(a− p)2 + (b − q)2 + (c − r)2
(a,b,c)
x
y
z
(p,q,r)
go.illinois.edu/math241fa17.
Distance
Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be
|PQ| = distance from P to Q
=√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
n = 2 (Pythagorean Theorem)
Distance from (a, b) to (c, d)
is√
(a− c)2 + (b − d)2
n = 3
Distance from (a, b, c) to (p, q, r)
is√
(a− p)2 + (b − q)2 + (c − r)2
(a,b,c)
x
y
z
(p,q,r)
go.illinois.edu/math241fa17.
Distance
Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be
|PQ| = distance from P to Q
=√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
n = 2 (Pythagorean Theorem)
Distance from (a, b) to (c, d)
is√
(a− c)2 + (b − d)2
n = 3
Distance from (a, b, c) to (p, q, r)
is√
(a− p)2 + (b − q)2 + (c − r)2
(a,b,c)
x
y
z
(p,q,r)
go.illinois.edu/math241fa17.
Distance
Given P(x1, . . . , xn),Q(y1, . . . , yn) ∈ Rn, define the distancebetween these points to be
|PQ| = distance from P to Q
=√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
n = 2 (Pythagorean Theorem)
Distance from (a, b) to (c, d)
is√
(a− c)2 + (b − d)2
n = 3
Distance from (a, b, c) to (p, q, r)
is√
(a− p)2 + (b − q)2 + (c − r)2
(a,b,c)
x
y
z
(p,q,r)
go.illinois.edu/math241fa17.
Spheres: application of distance formula
For C (a, b, c) ∈ R3 and r > 0, the sphere of radius r withcenter C is the set
{P ∈ R3 | |PC | = r}
Then, if P has coordinates (x , y , z),
|PC | = r ⇔√
(x − a)2 + (y − b)2 + (z − c)2 = r
⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2
This is equation for sphere: set of points (x , y , z) satisfying thisequation is a sphere.
Compare equation of a circle (x − a)2 + (y − b)2 = r2.
go.illinois.edu/math241fa17.
Spheres: application of distance formula
For C (a, b, c) ∈ R3 and r > 0, the sphere of radius r withcenter C is the set
{P ∈ R3 | |PC | = r}
Then, if P has coordinates (x , y , z),
|PC | = r ⇔√
(x − a)2 + (y − b)2 + (z − c)2 = r
⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2
This is equation for sphere: set of points (x , y , z) satisfying thisequation is a sphere.
Compare equation of a circle (x − a)2 + (y − b)2 = r2.
go.illinois.edu/math241fa17.
Spheres: application of distance formula
For C (a, b, c) ∈ R3 and r > 0, the sphere of radius r withcenter C is the set
{P ∈ R3 | |PC | = r}
Then, if P has coordinates (x , y , z),
|PC | = r ⇔
√(x − a)2 + (y − b)2 + (z − c)2 = r
⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2
This is equation for sphere: set of points (x , y , z) satisfying thisequation is a sphere.
Compare equation of a circle (x − a)2 + (y − b)2 = r2.
go.illinois.edu/math241fa17.
Spheres: application of distance formula
For C (a, b, c) ∈ R3 and r > 0, the sphere of radius r withcenter C is the set
{P ∈ R3 | |PC | = r}
Then, if P has coordinates (x , y , z),
|PC | = r ⇔√
(x − a)2 + (y − b)2 + (z − c)2 = r
⇔
(x − a)2 + (y − b)2 + (z − c)2 = r2
This is equation for sphere: set of points (x , y , z) satisfying thisequation is a sphere.
Compare equation of a circle (x − a)2 + (y − b)2 = r2.
go.illinois.edu/math241fa17.
Spheres: application of distance formula
For C (a, b, c) ∈ R3 and r > 0, the sphere of radius r withcenter C is the set
{P ∈ R3 | |PC | = r}
Then, if P has coordinates (x , y , z),
|PC | = r ⇔√
(x − a)2 + (y − b)2 + (z − c)2 = r
⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2
This is equation for sphere: set of points (x , y , z) satisfying thisequation is a sphere.
Compare equation of a circle (x − a)2 + (y − b)2 = r2.
go.illinois.edu/math241fa17.
Spheres: application of distance formula
For C (a, b, c) ∈ R3 and r > 0, the sphere of radius r withcenter C is the set
{P ∈ R3 | |PC | = r}
Then, if P has coordinates (x , y , z),
|PC | = r ⇔√
(x − a)2 + (y − b)2 + (z − c)2 = r
⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2
This is equation for sphere: set of points (x , y , z) satisfying thisequation is a sphere.
Compare equation of a circle (x − a)2 + (y − b)2 = r2.
go.illinois.edu/math241fa17.
Spheres: application of distance formula
For C (a, b, c) ∈ R3 and r > 0, the sphere of radius r withcenter C is the set
{P ∈ R3 | |PC | = r}
Then, if P has coordinates (x , y , z),
|PC | = r ⇔√
(x − a)2 + (y − b)2 + (z − c)2 = r
⇔ (x − a)2 + (y − b)2 + (z − c)2 = r2
This is equation for sphere: set of points (x , y , z) satisfying thisequation is a sphere.
Compare equation of a circle (x − a)2 + (y − b)2 = r2.
go.illinois.edu/math241fa17.
What else do we need to do calculus?
Derivatives (and integrals) require limit and displacement:
f ′(x) = limh→0
f (x + h)− f (x)
h.
Need displacement from x to x + h (and from f (x) to f (x + h)).This requires vectors....
go.illinois.edu/math241fa17.
What else do we need to do calculus?
Derivatives (and integrals) require limit and displacement:
f ′(x) = limh→0
f (x + h)− f (x)
h.
Need displacement from x to x + h (and from f (x) to f (x + h)).
This requires vectors....
go.illinois.edu/math241fa17.
What else do we need to do calculus?
Derivatives (and integrals) require limit and displacement:
f ′(x) = limh→0
f (x + h)− f (x)
h.
Need displacement from x to x + h (and from f (x) to f (x + h)).This requires vectors....
go.illinois.edu/math241fa17.
Vectors in R2.
A vector in R2 is an arrow. It represents a quantity with bothdirection and magnitude.
Two vectors are equal if they have the same direction andmagnitude.
Notation: book v (bold face) or written ~v (arrow over).
go.illinois.edu/math241fa17.
Vectors in R2.
A vector in R2 is an arrow. It represents a quantity with bothdirection and magnitude.
Two vectors are equal if they have the same direction andmagnitude.
Notation: book v (bold face) or written ~v (arrow over).
go.illinois.edu/math241fa17.
Vectors in R2.
A vector in R2 is an arrow. It represents a quantity with bothdirection and magnitude.
Two vectors are equal if they have the same direction andmagnitude.
Notation: book v (bold face) or written ~v (arrow over).
go.illinois.edu/math241fa17.
Vectors in R2.
A vector in R2 is an arrow. It represents a quantity with bothdirection and magnitude.
Two vectors are equal if they have the same direction andmagnitude.
Notation: book v (bold face) or written ~v (arrow over).
go.illinois.edu/math241fa17.
Operations and constructions
Addition: Parallelogram rule. This defines the sum of two vectors:v + w.
Scalar multiplication: Scale magnitude, and reverse direction ifnegative. What is cv if c = 1, 0, 2,−1?
Displacement vector from A to B is−→AB, tail at A and tip at B.
go.illinois.edu/math241fa17.
Operations and constructions
Addition: Parallelogram rule.
This defines the sum of two vectors:v + w.
Scalar multiplication: Scale magnitude, and reverse direction ifnegative. What is cv if c = 1, 0, 2,−1?
Displacement vector from A to B is−→AB, tail at A and tip at B.
go.illinois.edu/math241fa17.
Operations and constructions
Addition: Parallelogram rule. This defines the sum of two vectors:v + w.
Scalar multiplication: Scale magnitude, and reverse direction ifnegative. What is cv if c = 1, 0, 2,−1?
Displacement vector from A to B is−→AB, tail at A and tip at B.
go.illinois.edu/math241fa17.
Operations and constructions
Addition: Parallelogram rule. This defines the sum of two vectors:v + w.
Scalar multiplication: Scale magnitude, and reverse direction ifnegative.
What is cv if c = 1, 0, 2,−1?
Displacement vector from A to B is−→AB, tail at A and tip at B.
go.illinois.edu/math241fa17.
Operations and constructions
Addition: Parallelogram rule. This defines the sum of two vectors:v + w.
Scalar multiplication: Scale magnitude, and reverse direction ifnegative. What is cv if c = 1, 0, 2,−1?
Displacement vector from A to B is−→AB, tail at A and tip at B.
go.illinois.edu/math241fa17.
Operations and constructions
Addition: Parallelogram rule. This defines the sum of two vectors:v + w.
Scalar multiplication: Scale magnitude, and reverse direction ifnegative. What is cv if c = 1, 0, 2,−1?
Displacement vector from A to B is−→AB, tail at A and tip at B.
go.illinois.edu/math241fa17.
Position vectors
Every vector is equal to one with tail at the origin O = (0, 0):
Vectors in R2 ←→ R2
−→OP ←→ P
−→OP = position vector of the point P. (vectors and points aredifferent objects)
Write ~v = 〈v1, v2〉 (or sometimes just (v1, v2)).
The numbers v1, v2 are the components of ~v .
go.illinois.edu/math241fa17.
Position vectors
Every vector is equal to one with tail at the origin O = (0, 0):
Vectors in R2 ←→ R2
−→OP ←→ P
−→OP = position vector of the point P. (vectors and points aredifferent objects)
Write ~v = 〈v1, v2〉 (or sometimes just (v1, v2)).
The numbers v1, v2 are the components of ~v .
go.illinois.edu/math241fa17.
Position vectors
Every vector is equal to one with tail at the origin O = (0, 0):
Vectors in R2 ←→ R2
−→OP ←→ P
−→OP = position vector of the point P. (vectors and points aredifferent objects)
Write ~v = 〈v1, v2〉 (or sometimes just (v1, v2)).
The numbers v1, v2 are the components of ~v .
go.illinois.edu/math241fa17.
Position vectors
Every vector is equal to one with tail at the origin O = (0, 0):
Vectors in R2 ←→ R2
−→OP ←→ P
−→OP = position vector of the point P. (vectors and points aredifferent objects)
Write ~v = 〈v1, v2〉 (or sometimes just (v1, v2)).
The numbers v1, v2 are the components of ~v .
go.illinois.edu/math241fa17.
Position vectors
Every vector is equal to one with tail at the origin O = (0, 0):
Vectors in R2 ←→ R2
−→OP ←→ P
−→OP = position vector of the point P. (vectors and points aredifferent objects)
Write ~v = 〈v1, v2〉 (or sometimes just (v1, v2)).
The numbers v1, v2 are the components of ~v .
go.illinois.edu/math241fa17.
Calculations in terms of components
For ~u = 〈u1, u2〉, ~v = 〈v1, v2〉, and c ∈ R we have
~u + ~v = 〈u1 + v1, u2 + v2〉
c~u = 〈cu1, cu2〉
‖~u‖ =√
u21 + u22 = magnitude of ~u.
Sometimes write |~u| instead. Also call it the norm or length of ~u.
A(a1, a2) and B(b1, b2), the displacement vector is−→AB = 〈b1 − a1, b2 − a2〉.
go.illinois.edu/math241fa17.
Calculations in terms of components
For ~u = 〈u1, u2〉, ~v = 〈v1, v2〉, and c ∈ R we have
~u + ~v = 〈u1 + v1, u2 + v2〉
c~u = 〈cu1, cu2〉
‖~u‖ =√
u21 + u22 = magnitude of ~u.
Sometimes write |~u| instead. Also call it the norm or length of ~u.
A(a1, a2) and B(b1, b2), the displacement vector is−→AB = 〈b1 − a1, b2 − a2〉.
go.illinois.edu/math241fa17.
Calculations in terms of components
For ~u = 〈u1, u2〉, ~v = 〈v1, v2〉, and c ∈ R we have
~u + ~v = 〈u1 + v1, u2 + v2〉
c~u = 〈cu1, cu2〉
‖~u‖ =√
u21 + u22 = magnitude of ~u.
Sometimes write |~u| instead. Also call it the norm or length of ~u.
A(a1, a2) and B(b1, b2), the displacement vector is−→AB = 〈b1 − a1, b2 − a2〉.
go.illinois.edu/math241fa17.
Calculations in terms of components
For ~u = 〈u1, u2〉, ~v = 〈v1, v2〉, and c ∈ R we have
~u + ~v = 〈u1 + v1, u2 + v2〉
c~u = 〈cu1, cu2〉
‖~u‖ =√
u21 + u22 = magnitude of ~u.
Sometimes write |~u| instead. Also call it the norm or length of ~u.
A(a1, a2) and B(b1, b2), the displacement vector is−→AB = 〈b1 − a1, b2 − a2〉.
go.illinois.edu/math241fa17.
Calculations in terms of components
For ~u = 〈u1, u2〉, ~v = 〈v1, v2〉, and c ∈ R we have
~u + ~v = 〈u1 + v1, u2 + v2〉
c~u = 〈cu1, cu2〉
‖~u‖ =√
u21 + u22 = magnitude of ~u.
Sometimes write |~u| instead. Also call it the norm or length of ~u.
A(a1, a2) and B(b1, b2), the displacement vector is−→AB = 〈b1 − a1, b2 − a2〉.
go.illinois.edu/math241fa17.
Calculations in terms of components
For ~u = 〈u1, u2〉, ~v = 〈v1, v2〉, and c ∈ R we have
~u + ~v = 〈u1 + v1, u2 + v2〉
c~u = 〈cu1, cu2〉
‖~u‖ =√
u21 + u22 = magnitude of ~u.
Sometimes write |~u| instead. Also call it the norm or length of ~u.
A(a1, a2) and B(b1, b2), the displacement vector is−→AB = 〈b1 − a1, b2 − a2〉.
go.illinois.edu/math241fa17.
Vectors in Rn
Vectors in Rn ←→ Rn
−→OP ←→ P
−→OP = position vector of the point P.
~v = 〈v1, . . . , vn〉 = position vector of the point (v1, . . . , vn).
v1, . . . , vn are the components.
Displacement vector from point A(a1, . . . , an) to B(b1, . . . , bn):
−→AB = 〈b1 − a1, . . . , bn − an〉.
“Arrow from A to B”
go.illinois.edu/math241fa17.
Vectors in Rn
Vectors in Rn ←→ Rn
−→OP ←→ P
−→OP = position vector of the point P.
~v = 〈v1, . . . , vn〉 = position vector of the point (v1, . . . , vn).
v1, . . . , vn are the components.
Displacement vector from point A(a1, . . . , an) to B(b1, . . . , bn):
−→AB = 〈b1 − a1, . . . , bn − an〉.
“Arrow from A to B”
go.illinois.edu/math241fa17.
Vectors in Rn
Vectors in Rn ←→ Rn
−→OP ←→ P
−→OP = position vector of the point P.
~v = 〈v1, . . . , vn〉 = position vector of the point (v1, . . . , vn).
v1, . . . , vn are the components.
Displacement vector from point A(a1, . . . , an) to B(b1, . . . , bn):
−→AB = 〈b1 − a1, . . . , bn − an〉.
“Arrow from A to B”
go.illinois.edu/math241fa17.
Vectors in Rn
Vectors in Rn ←→ Rn
−→OP ←→ P
−→OP = position vector of the point P.
~v = 〈v1, . . . , vn〉 = position vector of the point (v1, . . . , vn).
v1, . . . , vn are the components.
Displacement vector from point A(a1, . . . , an) to B(b1, . . . , bn):
−→AB = 〈b1 − a1, . . . , bn − an〉.
“Arrow from A to B”
go.illinois.edu/math241fa17.
Vectors in Rn
Vectors in Rn ←→ Rn
−→OP ←→ P
−→OP = position vector of the point P.
~v = 〈v1, . . . , vn〉 = position vector of the point (v1, . . . , vn).
v1, . . . , vn are the components.
Displacement vector from point A(a1, . . . , an) to B(b1, . . . , bn):
−→AB = 〈b1 − a1, . . . , bn − an〉.
“Arrow from A to B”
go.illinois.edu/math241fa17.
Vectors in Rn
Vectors in Rn ←→ Rn
−→OP ←→ P
−→OP = position vector of the point P.
~v = 〈v1, . . . , vn〉 = position vector of the point (v1, . . . , vn).
v1, . . . , vn are the components.
Displacement vector from point A(a1, . . . , an) to B(b1, . . . , bn):
−→AB = 〈b1 − a1, . . . , bn − an〉.
“Arrow from A to B”
go.illinois.edu/math241fa17.
Vectors in Rn
Vectors in Rn ←→ Rn
−→OP ←→ P
−→OP = position vector of the point P.
~v = 〈v1, . . . , vn〉 = position vector of the point (v1, . . . , vn).
v1, . . . , vn are the components.
Displacement vector from point A(a1, . . . , an) to B(b1, . . . , bn):
−→AB = 〈b1 − a1, . . . , bn − an〉.
“Arrow from A to B”
go.illinois.edu/math241fa17.
Vectors in Rn
Vectors in Rn ←→ Rn
−→OP ←→ P
−→OP = position vector of the point P.
~v = 〈v1, . . . , vn〉 = position vector of the point (v1, . . . , vn).
v1, . . . , vn are the components.
Displacement vector from point A(a1, . . . , an) to B(b1, . . . , bn):
−→AB = 〈b1 − a1, . . . , bn − an〉.
“Arrow from A to B”
go.illinois.edu/math241fa17.
Physical quantities
We will also use vectors to denote physical quantities:
Example: Forces have a direction and magnitude, so we canrepresent them with vectorsMultiple forces acting on an object, then the net force on object(or resultant force) is the sum of the forces.
Example: Wind has speed and direction, so can represent it with avector.
go.illinois.edu/math241fa17.
Physical quantities
We will also use vectors to denote physical quantities:
Example: Forces have a direction and magnitude, so we canrepresent them with vectorsMultiple forces acting on an object, then the net force on object(or resultant force) is the sum of the forces.
Example: Wind has speed and direction, so can represent it with avector.
go.illinois.edu/math241fa17.
Physical quantities
We will also use vectors to denote physical quantities:
Example: Forces have a direction and magnitude, so we canrepresent them with vectors
Multiple forces acting on an object, then the net force on object(or resultant force) is the sum of the forces.
Example: Wind has speed and direction, so can represent it with avector.
go.illinois.edu/math241fa17.
Physical quantities
We will also use vectors to denote physical quantities:
Example: Forces have a direction and magnitude, so we canrepresent them with vectorsMultiple forces acting on an object, then the net force on object(or resultant force) is the sum of the forces.
Example: Wind has speed and direction, so can represent it with avector.
go.illinois.edu/math241fa17.
Physical quantities
We will also use vectors to denote physical quantities:
Example: Forces have a direction and magnitude, so we canrepresent them with vectorsMultiple forces acting on an object, then the net force on object(or resultant force) is the sum of the forces.
Example: Wind has speed and direction, so can represent it with avector.
go.illinois.edu/math241fa17.
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