Functions of several variables

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

2. Math 200 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 Main Goals for Today

3. 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 Examples and Notation

4. Math 200 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 Definitions

5. 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 Domain for a function of two variables

6. Math 200 Find and sketch the domain for the function One more 2-variable example • 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

7. Math 200 GRAPH DOMAIN

8. 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 Domain of a function of three variables

9. 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 Math 200 Level curves and contour plots (or contour maps) • 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

10. 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

11. Math 200

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

13. 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.) Math 200 Level surfaces • 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

14. 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