Functions of several variables Math 200 Week 4 - Monday Functions of several variables

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

Domain for a function of two variables 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

One more 2-variable example Math 200 One more 2-variable example Find and sketch the domain for the function 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 GRAPH DOMAIN

Domain of a function of three variables 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

Level curves and contour plots (or contour maps) 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

Think of this as a topographical map of the surface f(x,y) = z 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=-1 z=3 z=1 z=1 z=3 z=0 z=2 z=2 z=-1 Think of this as a topographical map of the surface f(x,y) = z

Math 200

Math 200 z=0 z=1 z=-1 z=2 z=3

Level surfaces k = -1 -1 = x2 + y2 - z2 z2 = x2 + y2 +1 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