1. 11.1 Functions of two or more variable

2. Functions
We can study functions of two or more variables from four viewpoints:
Verbally
Numerically
Algebraically
Visually

3. The index W depends on the actual temperature T and the wind speed v
The index W depends on the actual temperature T and the wind speed v. A table of values of W(T, v). W(–5, 50) = –15

4. Find the domain and range of
The domain of g is

5. This is the disk with center (0, 0) and radius 3, as the next slide shows.
The range of g is
Since z is a positive square root, z ≥ 0. Also

6. Visual Representations
One way to visualize a function of two variables is through its graph.
For example, we sketch the graph of
Solution Squaring both sides of this equation gives x2 + y2 + z2 = 9, which is an equation of the sphere with center the origin and radius 3.

7. Since z ≥ 0, the graph of g is just the top half of this sphere.
The figure shows the graph of g:

8. Another way to see functions is a contour map on which points of constant elevation are joined to form level curves

9. A common example of level curves occurs in topographical maps of mountainous regions, as shown on the next slide.
The level curves are curves of constant elevation above sea level.
If we walk along one of these contour lines we neither ascend nor descend.

10. Example The figure shows a contour map for a function.
Use it to estimate f(1, 3) and f(4, 5).

11. The point (1, 3) lies part way between the level curves with z-values 70 and 80.
So we estimate that f (1, 3) ≈ 73.
Similarly, we estimate that f (4, 5) ≈ 56.

12. Example Sketch the level curves of the function
Solution The level curves are
This is a family of concentric circles with center (0, 0) and radius
The cases k = 0, 1, 2, 3 are shown on the next slide:

14. Three or More Variables
A function of three variables, f, is a rule that assigns to each ordered triple (x, y, z) in a domain D in space a unique real number denoted by f(x, y, z).
For instance, the temperature T at a point on the surface of the Earth depends on the longitude x and latitude y of the point and on the time t, so T = f(x, y, t).

15. Example Find the domain of f if f(x, y, z) = ln (z – y) + xy sin z.
Solution The expression for f(x, y, z) is defined as long as z – y > 0, so the domain of f is the half-space

16. Homework
P746 1,5,7,9,11,15,21,