![Page 1: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/1.jpg)
Functions of several variables
Christopher Croke
University of Pennsylvania
Math 115
Christopher Croke Calculus 115
![Page 2: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/2.jpg)
Functions of several variables:
Examples:
f (x , y) = x2 + 2y2
f (2, 1) =?
f (1, 2) =?
f (x , y) = cos(x) sin(y)exy +√x − y
f (x , y , z) = x − 2y + 3z
Christopher Croke Calculus 115
![Page 3: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/3.jpg)
Functions of several variables:
Examples:
f (x , y) = x2 + 2y2
f (2, 1) =?
f (1, 2) =?
f (x , y) = cos(x) sin(y)exy +√x − y
f (x , y , z) = x − 2y + 3z
Christopher Croke Calculus 115
![Page 4: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/4.jpg)
Functions of several variables:
Examples:
f (x , y) = x2 + 2y2
f (2, 1) =?
f (1, 2) =?
f (x , y) = cos(x) sin(y)exy +√x − y
f (x , y , z) = x − 2y + 3z
Christopher Croke Calculus 115
![Page 5: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/5.jpg)
Functions of several variables:
Examples:
f (x , y) = x2 + 2y2
f (2, 1) =?
f (1, 2) =?
f (x , y) = cos(x) sin(y)exy +√x − y
f (x , y , z) = x − 2y + 3z
Christopher Croke Calculus 115
![Page 6: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/6.jpg)
Functions of several variables:
Examples:
f (x , y) = x2 + 2y2
f (2, 1) =?
f (1, 2) =?
f (x , y) = cos(x) sin(y)exy +√x − y
f (x , y , z) = x − 2y + 3z
Christopher Croke Calculus 115
![Page 7: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/7.jpg)
For functions of two variables can write
z = f (x , y).
x and y are called the independent variables (or input variables).z is called the dependent variable (or output variable).
Similar terminology applies for more variables.
The Domain of f is the set of input variables for which f isdefined.Check out the previous examples...
When a function is given by a formula assume that the domain isthe largest set where the function makes sense.
The Range of f is the set of output values. This will be a subsetof the reals.
Christopher Croke Calculus 115
![Page 8: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/8.jpg)
For functions of two variables can write
z = f (x , y).
x and y are called the independent variables (or input variables).
z is called the dependent variable (or output variable).
Similar terminology applies for more variables.
The Domain of f is the set of input variables for which f isdefined.Check out the previous examples...
When a function is given by a formula assume that the domain isthe largest set where the function makes sense.
The Range of f is the set of output values. This will be a subsetof the reals.
Christopher Croke Calculus 115
![Page 9: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/9.jpg)
For functions of two variables can write
z = f (x , y).
x and y are called the independent variables (or input variables).z is called the dependent variable (or output variable).
Similar terminology applies for more variables.
The Domain of f is the set of input variables for which f isdefined.Check out the previous examples...
When a function is given by a formula assume that the domain isthe largest set where the function makes sense.
The Range of f is the set of output values. This will be a subsetof the reals.
Christopher Croke Calculus 115
![Page 10: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/10.jpg)
For functions of two variables can write
z = f (x , y).
x and y are called the independent variables (or input variables).z is called the dependent variable (or output variable).
Similar terminology applies for more variables.
The Domain of f is the set of input variables for which f isdefined.Check out the previous examples...
When a function is given by a formula assume that the domain isthe largest set where the function makes sense.
The Range of f is the set of output values. This will be a subsetof the reals.
Christopher Croke Calculus 115
![Page 11: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/11.jpg)
For functions of two variables can write
z = f (x , y).
x and y are called the independent variables (or input variables).z is called the dependent variable (or output variable).
Similar terminology applies for more variables.
The Domain of f is the set of input variables for which f isdefined.
Check out the previous examples...
When a function is given by a formula assume that the domain isthe largest set where the function makes sense.
The Range of f is the set of output values. This will be a subsetof the reals.
Christopher Croke Calculus 115
![Page 12: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/12.jpg)
For functions of two variables can write
z = f (x , y).
x and y are called the independent variables (or input variables).z is called the dependent variable (or output variable).
Similar terminology applies for more variables.
The Domain of f is the set of input variables for which f isdefined.Check out the previous examples...
When a function is given by a formula assume that the domain isthe largest set where the function makes sense.
The Range of f is the set of output values. This will be a subsetof the reals.
Christopher Croke Calculus 115
![Page 13: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/13.jpg)
For functions of two variables can write
z = f (x , y).
x and y are called the independent variables (or input variables).z is called the dependent variable (or output variable).
Similar terminology applies for more variables.
The Domain of f is the set of input variables for which f isdefined.Check out the previous examples...
When a function is given by a formula assume that the domain isthe largest set where the function makes sense.
The Range of f is the set of output values. This will be a subsetof the reals.
Christopher Croke Calculus 115
![Page 14: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/14.jpg)
For functions of two variables can write
z = f (x , y).
x and y are called the independent variables (or input variables).z is called the dependent variable (or output variable).
Similar terminology applies for more variables.
The Domain of f is the set of input variables for which f isdefined.Check out the previous examples...
When a function is given by a formula assume that the domain isthe largest set where the function makes sense.
The Range of f is the set of output values. This will be a subsetof the reals.
Christopher Croke Calculus 115
![Page 15: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/15.jpg)
Find the domain and range of the following:
w =1
xy
w = x ln(z) + y ln(x).
Christopher Croke Calculus 115
![Page 16: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/16.jpg)
Find the domain and range of the following:
w =1
xy
w = x ln(z) + y ln(x).
Christopher Croke Calculus 115
![Page 17: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/17.jpg)
Some terminology for sets in the plane
Let R be a region in the plane.
x is an Interior point if there is a disk centered at x andcontained in the region.
Christopher Croke Calculus 115
![Page 18: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/18.jpg)
Some terminology for sets in the plane
Let R be a region in the plane.
x is an Interior point if there is a disk centered at x andcontained in the region.
Christopher Croke Calculus 115
![Page 19: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/19.jpg)
Some terminology for sets in the plane
Let R be a region in the plane.
x is an Interior point if there is a disk centered at x andcontained in the region.
Christopher Croke Calculus 115
![Page 20: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/20.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 21: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/21.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 22: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/22.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 23: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/23.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 24: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/24.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 25: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/25.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 26: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/26.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 27: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/27.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 28: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/28.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 29: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/29.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 30: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/30.jpg)
x is called a Boundary Point if every disk centered at x hits bothpoints that are in R and points that are outside.
The Interior of R is the set of all interior points.
The Boundary of R is the set of all boundary points of R.
R is called Open if all x ∈ R are interior points.
R is called Closed if all boundary points of R are in R.
Christopher Croke Calculus 115
![Page 31: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/31.jpg)
Examples
x2 + y2 < 1.
x2 + y2 ≤ 1.
y < x2.
y ≥ x .
y = x3.
Christopher Croke Calculus 115
![Page 32: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/32.jpg)
In 3-dimensions the same terminology holds except we use ballscentered at x rather than disks.
Examples:
z > 0.
z ≥ 0
x2 + y2 + z2 ≤ 0.
R is called Bounded if it lies in a (generally big) disk (or ball in3-dims)As examples consider the domains of:
f (x , y) =√x2 − y .
f (x , y) =√
1− (x2 + y2).
f (x , y) =1
xy.
Christopher Croke Calculus 115
![Page 33: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/33.jpg)
In 3-dimensions the same terminology holds except we use ballscentered at x rather than disks.Examples:
z > 0.
z ≥ 0
x2 + y2 + z2 ≤ 0.
R is called Bounded if it lies in a (generally big) disk (or ball in3-dims)As examples consider the domains of:
f (x , y) =√x2 − y .
f (x , y) =√
1− (x2 + y2).
f (x , y) =1
xy.
Christopher Croke Calculus 115
![Page 34: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/34.jpg)
In 3-dimensions the same terminology holds except we use ballscentered at x rather than disks.Examples:
z > 0.
z ≥ 0
x2 + y2 + z2 ≤ 0.
R is called Bounded if it lies in a (generally big) disk (or ball in3-dims)
As examples consider the domains of:
f (x , y) =√x2 − y .
f (x , y) =√
1− (x2 + y2).
f (x , y) =1
xy.
Christopher Croke Calculus 115
![Page 35: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/35.jpg)
In 3-dimensions the same terminology holds except we use ballscentered at x rather than disks.Examples:
z > 0.
z ≥ 0
x2 + y2 + z2 ≤ 0.
R is called Bounded if it lies in a (generally big) disk (or ball in3-dims)As examples consider the domains of:
f (x , y) =√x2 − y .
f (x , y) =√
1− (x2 + y2).
f (x , y) =1
xy.
Christopher Croke Calculus 115
![Page 36: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/36.jpg)
Graphs of functions of two variables
The Graph of f (x , y) is the set of points in 3-space of the form
(x , y , f (x , y))
where (x , y) is in the domain of f .
That is the set of points (x , y , z) where z = f (x , y).
Christopher Croke Calculus 115
![Page 37: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/37.jpg)
Graphs of functions of two variables
The Graph of f (x , y) is the set of points in 3-space of the form
(x , y , f (x , y))
where (x , y) is in the domain of f .That is the set of points (x , y , z) where z = f (x , y).
Christopher Croke Calculus 115
![Page 38: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/38.jpg)
Graphs of functions of two variables
The Graph of f (x , y) is the set of points in 3-space of the form
(x , y , f (x , y))
where (x , y) is in the domain of f .That is the set of points (x , y , z) where z = f (x , y).
Christopher Croke Calculus 115
![Page 39: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/39.jpg)
Christopher Croke Calculus 115
![Page 40: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/40.jpg)
Use Maple to graph:
f (x , y) = x2 + y2.
g(x , y) = x2 − y2.
h(x , y) = x2 sin(y).
Christopher Croke Calculus 115
![Page 41: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/41.jpg)
Level curves and contour lines
A Level Curve of a function f (x , y) is a curve of the formf (x , y) = c for a fixed number c . (Note this is a curve in theplane.)
A Contour line is the curve in 3-space gotten by raising the levelcurve f (x , y) = c to the plane z = c . In other words it is theintersection of the graph of f with the plane z = c .
Christopher Croke Calculus 115
![Page 42: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/42.jpg)
Level curves and contour lines
A Level Curve of a function f (x , y) is a curve of the formf (x , y) = c for a fixed number c . (Note this is a curve in theplane.)
A Contour line is the curve in 3-space gotten by raising the levelcurve f (x , y) = c to the plane z = c . In other words it is theintersection of the graph of f with the plane z = c .
Christopher Croke Calculus 115
![Page 43: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/43.jpg)
Level curves and contour lines
A Level Curve of a function f (x , y) is a curve of the formf (x , y) = c for a fixed number c . (Note this is a curve in theplane.)
A Contour line is the curve in 3-space gotten by raising the levelcurve f (x , y) = c to the plane z = c .
In other words it is theintersection of the graph of f with the plane z = c .
Christopher Croke Calculus 115
![Page 44: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/44.jpg)
Level curves and contour lines
A Level Curve of a function f (x , y) is a curve of the formf (x , y) = c for a fixed number c . (Note this is a curve in theplane.)
A Contour line is the curve in 3-space gotten by raising the levelcurve f (x , y) = c to the plane z = c . In other words it is theintersection of the graph of f with the plane z = c .
Christopher Croke Calculus 115
![Page 45: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/45.jpg)
Level curves and contour lines
A Level Curve of a function f (x , y) is a curve of the formf (x , y) = c for a fixed number c . (Note this is a curve in theplane.)
A Contour line is the curve in 3-space gotten by raising the levelcurve f (x , y) = c to the plane z = c . In other words it is theintersection of the graph of f with the plane z = c .
Christopher Croke Calculus 115
![Page 46: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/46.jpg)
Christopher Croke Calculus 115
![Page 47: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/47.jpg)
Christopher Croke Calculus 115
![Page 48: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/48.jpg)
Christopher Croke Calculus 115
![Page 49: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/49.jpg)
Christopher Croke Calculus 115
![Page 50: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/50.jpg)
Find level curves of f (x , y) = x2 + y2.
See what Maple can do.You have seen these before (e.g. isobars, isotherms, indifferencecurves....)For functions of 3-variables we get Level Surfaces f (x , y , z) = c .What about f (x , y , z) = x2 + y2 + z2?
Christopher Croke Calculus 115
![Page 51: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/51.jpg)
Find level curves of f (x , y) = x2 + y2.
See what Maple can do.
You have seen these before (e.g. isobars, isotherms, indifferencecurves....)For functions of 3-variables we get Level Surfaces f (x , y , z) = c .What about f (x , y , z) = x2 + y2 + z2?
Christopher Croke Calculus 115
![Page 52: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/52.jpg)
Find level curves of f (x , y) = x2 + y2.
See what Maple can do.You have seen these before (e.g. isobars, isotherms, indifferencecurves....)
For functions of 3-variables we get Level Surfaces f (x , y , z) = c .What about f (x , y , z) = x2 + y2 + z2?
Christopher Croke Calculus 115
![Page 53: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/53.jpg)
Find level curves of f (x , y) = x2 + y2.
See what Maple can do.You have seen these before (e.g. isobars, isotherms, indifferencecurves....)For functions of 3-variables we get Level Surfaces f (x , y , z) = c .
What about f (x , y , z) = x2 + y2 + z2?
Christopher Croke Calculus 115
![Page 54: Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z = f(x;y): x and y are called the independent variables (or input variables). z is](https://reader033.vdocuments.mx/reader033/viewer/2022052800/5f0f8c307e708231d444b4b6/html5/thumbnails/54.jpg)
Find level curves of f (x , y) = x2 + y2.
See what Maple can do.You have seen these before (e.g. isobars, isotherms, indifferencecurves....)For functions of 3-variables we get Level Surfaces f (x , y , z) = c .What about f (x , y , z) = x2 + y2 + z2?
Christopher Croke Calculus 115