Functions of several variables Christopher Croke University of Pennsylvania Math 115 Christopher Croke Calculus 115
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