I. Introduction to Multivariable Functions
A. Motivational example of a circular cylinder
1. Variables: radius (r) and height (h)
2. Volume of the cylinder depends on r and h
B. Bivariate functions
1. Definition: Functions depending on two variables
2. Mapping from a subset of R^2 to the real line
3. Domain and range of a bivariate function
4. Illustration of a bivariate function
5. Independent variables (x and y) vs. dependent variable (z)
C. Multivariable functions
1. Definition: Functions mapping R^n to R
2. Rule assigning n-tuples to real numbers
3. Domain of a multivariable function
4. Example of a function with three variables
5. Visualization of the domain using software
II. Example Functions
A. Example 1: f(x, y) = 2x + 3y – 5
1. Domain: R^2
2. Range: R (all real numbers)
B. Example 2: g(x, y) = √(9 – x^2 – y^2)
1. Domain: x^2 + y^2 ≤ 9 (inside a circle of radius 3)
2. Range: [0, 3] (closed interval)
III. Plotting Bivariate Functions
A. Using math3d.org website
B. Example plot using the function f(x, y) = √(x^2 + y^2)
IV. Function of Three Variables
A. Definition: Rule assigning order triples to real numbers
B. Example: f(x, y, z) = ln(z – y) + e^(xy) * sin(z)
1. Domain: z > y (half-plane above z = y plane)
2. Visualization using software (e.g., Mathematica)
********** Part 2 ************
I. Introduction to level curves
A. Definition of a level curve
B. Relationship to functions of two variables
C. Level curve as the intersection of a plane and a surface
II. Examples of level curves
A. Example 1: z = f(x, y) = xy
1. Level curve for k = 0: x = 0, y = 0
2. Level curve for k = 1: hyperbola y = 1/x
3. Level curve for k = -1: reflection of hyperbola y = -1/x
B. Example 2: z = f(x, y) = x^2 + y^2
1. Level curve for k = 0: origin (0, 0)
2. Level curve for k = 1: unit circle centered at (0, 0)
3. General level curves for x^2 + y^2 = k: circles centered at (0, 0) with radius √k
III. Contour maps and their significance
A. Contour map as a collection of level curves
B. Height representation in contour maps
C. Reconstruction of the surface using contour maps
D. Advantages of contour maps in representing surfaces
IV. Applications of contour maps
A. Geology and topographic mapping
B. Temperature mapping and analysis
1. Contour maps for temperature functions
2. Interpretation of constant temperature along a curve
V. Conclusion