Functions of Several Variables - Mathematics Departmentdp399/math200/Slides/...MATH 200 DEFINITIONS A function of two variables, x and y, is a rule that assigns to each ordered pair,

FUNCTIONS OF SEVERAL VARIABLES

MATH 200 WEEK 4 - MONDAY

MATH 200

MAIN GOALS FOR TODAY

▸ Be able to describe and sketch the domain of a function of two or more variables

▸ Domains will be 2D or 3D regions

▸ Know how to evaluate a function of two or more variables

▸ Be able to compute and sketch level curves & surfaces

▸ These are just traces of the form z = k

MATH 200

EXAMPLES AND NOTATION▸ The following are all

functions of several variables

▸ f(x,y) = sin(x) + cos(y)

▸ g(x,y,z) = xyz

▸ z = ln(x2+y2)

▸ A = bh/2

▸ V = lwh

MATH 200

DEFINITIONS▸ A function of two variables, x and y, is a rule that assigns

to each ordered pair, (x,y), exactly one real number.

▸ We assign the value of f(x,y) to z to get a surface

▸ The domain of a function of two variables is the set of ordered pairs (x,y) for which f is defined

▸ A function of three variables, x, y, and z, is a rule that assigns to each ordered triple, (x,y,z), exactly one real number.

▸ The domain of a function of three variables is the set of ordered triples (x,y,z) for which f is defined

MATH 200

DOMAIN FOR A FUNCTION OF TWO VARIABLES▸ Consider the function

f(x,y) = ln(1-x2-y2)

▸ We know that the input for ln() must be positive

▸ 1 - x2 - y2 > 0

▸ x2 + y2 < 1

▸ Let’s sketch the domain along with the graph of f

MATH 200

ONE MORE 2-VARIABLE EXAMPLE▸ Find and sketch the domain for the function

f(x, y) =!x2 + y2 − 1

▸ We need the argument of the square root to be greater than or equal to zero

▸ x2 + y2 - 1 ≥ 0

▸ x2 + y2 ≥ 1

▸ All points on and outside the unit circle

MATH 200

DOMAIN GRAPH

MATH 200

DOMAIN OF A FUNCTION OF THREE VARIABLES▸ Consider the function

f(x,y,z) = arcsin(x2+y2+z2)

▸ Notice that there’s no graph of f - it would be 4D!

▸ But we can still find the domain:

▸ -1 ≤ x2+y2+z2 ≤ 1

▸ x2+y2+z2 ≤ 1

▸ Every point on and inside the unit sphere

MATH 200

LEVEL CURVES AND CONTOUR PLOTS (OR CONTOUR MAPS)▸ A level curve for a function

f(x,y) is a trace of the form z=constant

▸ It’s often useful to graph and label several level curves together on one set of axes

▸ We call this a contour plot

▸ E.g. consider the function z=x2-y2

▸ z=0: 0 = x2 - y2

▸ x2 = y2

▸ |x| = |y|

▸ z=1: 1 = x2 - y2

▸ x2 = y2 + 1

▸ z=2: x2 = y2 + 2

▸ z=-1: y2 = x2 + 1

▸ z=-2: y2 = x2 + 2

MATH 200

▸ z=0: |x| = |y|

▸ z=1: x2 = y2 + 1

▸ z=2: x2 = y2 + 2

▸ z=-1: y2 = x2 + 1

▸ z=-2: y2 = x2 + 2

z=0z=1 z=1

z=-1

z=-1

z=2 z=2z=3 z=3

THINK OF THIS AS A TOPOGRAPHICAL MAP OF THE SURFACE f(x,y) = z

MATH 200

MATH 200

z=0z=1 z=1

z=-1

z=-1

z=2 z=2z=3 z=3

MATH 200

LEVEL SURFACES▸ While we can’t graph

functions of three variables, we can plot their level surfaces

▸ Level surfaces: given f(x,y,z), setting the function equal to a constant yields a level surface

▸ E.g. consider the function f(x,y,z) = x2 + y2 - z2

▸ Set f(x,y,z) = k (const.)

▸ k = -1 ▸ -1 = x2 + y2 - z2

▸ z2 = x2 + y2 +1 ▸ Hyperboloid of 2 sheets

▸ k = 0 ▸ = x2 + y2 - z2

▸ z2 = x2 + y2

▸ Double Cone

▸ k = 1 ▸ 1 = x2 + y2 - z2

▸ z2 = x2 + y2 -1 ▸ Hyperboloid of 1 sheet

MATH 200

▸ k = -1 ▸ -1 = x2 + y2 - z2

▸ z2 = x2 + y2 +1 ▸ Hyperboloid of 2 sheets

▸ k = 0 ▸ = x2 + y2 - z2

▸ z2 = x2 + y2

▸ Double Cone

▸ k = 1 ▸ 1 = x2 + y2 - z2

▸ z2 = x2 + y2 -1 ▸ Hyperboloid of 1 sheet

WE CAN THINK OF THESE LEVEL SURFACES AS 3D CROSS-SECTIONS OF A 4D OBJECT