Functions of several variables
Christopher Croke
University of Pennsylvania
Math 115
Christopher Croke Calculus 115
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
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
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
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
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
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
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
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
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
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
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
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
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
Find the domain and range of the following:
w =1
xy
w = x ln(z) + y ln(x).
Christopher Croke Calculus 115
Find the domain and range of the following:
w =1
xy
w = x ln(z) + y ln(x).
Christopher Croke Calculus 115
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Examples
x2 + y2 < 1.
x2 + y2 ≤ 1.
y < x2.
y ≥ x .
y = x3.
Christopher Croke Calculus 115
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
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
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
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
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
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
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
Christopher Croke Calculus 115
Use Maple to graph:
f (x , y) = x2 + y2.
g(x , y) = x2 − y2.
h(x , y) = x2 sin(y).
Christopher Croke Calculus 115
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
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
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
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
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
Christopher Croke Calculus 115
Christopher Croke Calculus 115
Christopher Croke Calculus 115
Christopher Croke Calculus 115
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
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
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
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
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