Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z =...

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

Examples:

f (x , y) = x2 + 2y2

f (2, 1) =?

f (1, 2) =?

f (x , y , z) = x − 2y + 3z

Examples:

f (x , y) = x2 + 2y2

f (2, 1) =?

f (1, 2) =?

f (x , y , z) = x − 2y + 3z

Examples:

f (x , y) = x2 + 2y2

f (2, 1) =?

f (1, 2) =?

f (x , y , z) = x − 2y + 3z

Examples:

f (x , y) = x2 + 2y2

f (2, 1) =?

f (1, 2) =?

f (x , y , z) = x − 2y + 3z

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.

z = f (x , y).

x and y are called the independent variables (or input variables).

z is called the dependent variable (or output variable).

z = f (x , y).

The Domain of f is the set of input variables for which f isdefined.

Check out the previous examples...

z = f (x , y).

Find the domain and range of the following:

w = x ln(z) + y ln(x).

Find the domain and range of the following:

w = x ln(z) + y ln(x).

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.

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.

Examples

x2 + y2 < 1.

x2 + y2 ≤ 1.

y < x2.

y ≥ x .

y = x3.

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

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.

f (x , y) =√x2 − y .

f (x , y) =√

1− (x2 + y2).

f (x , y) =1

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

z > 0.

z ≥ 0

x2 + y2 + z2 ≤ 0.

f (x , y) =√x2 − y .

f (x , y) =√

1− (x2 + y2).

f (x , y) =1

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).

(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).

(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).

Use Maple to graph:

f (x , y) = x2 + y2.

g(x , y) = x2 − y2.

h(x , y) = x2 sin(y).

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 .

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 .

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?

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?

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?

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?

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?

Functions of several variablesccroke/lecture1(14.1).pdfFor functions of two variables can write z =...

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