Lecture Notes: Multivariable Functions and Calculus

Introduction to Multivariable Functions

• Definition: Functions with more than one independent variable.
• Key Question: What is a derivative with more than one independent variable?
• Basic Concepts:
  • Domain and Range: Extend similar concepts from single-variable functions.
  • Graphing: Requires one more dimension than the number of independent variables.
  • Notation: Function notation always lists independent variables.

Graphing and Dimensions

• Graphing Rules:
  • To graph a function with n independent variables, use n+1 dimensions.
• Examples:
  • Single variable (x): 2D graph (x and y plane).
  • Two variables (x, y): 3D graph (x, y, and height).
  • Three variables (x, y, z): 4D, which can't be graphically visualized in traditional means.
• Conceptual Understanding:
  • 2D for two variables involves creating a surface.
  • 3D for three variables would involve a 4D surface, conceptualized but not visualized.

Domain and Range in Multivariable Functions

• Domain: Describes allowable inputs (independent variables).
  • Use notation such as x, y in plane for 2D or x, y, z space for 3D.
  • Constraints from square roots, denominators, etc.
• Range: Describes possible outputs (values of the function), typically along a dependent variable axis (z or analogous).

Examples of Domains and Ranges

• Simple Case: f(x) = x^2, domain all real numbers greater than zero if inside a square root.
• Complex Case: Functions with square roots or denominators have restrictions.
• Example Calculation:
  • x^2 + y^2 < 4 implies a circle, with restrictions on z values for valid outputs.

Graphing Domains

• 1 Variable: Graph on a number line.
• 2 Variables: Use an xy-plane, shade valid regions.
• 3 Variables: Use xyz space, consider volumes and surfaces.

Domain and Function Graphing Steps

1. Set Equal to a Dependable Variable:
   • Convert function notation to a format involving z.
2. Identify Known Surfaces:
   • Recognize shapes like planes, paraboloids, etc.
3. Graph Planes Using Intercepts:
   • Cover-up method to find x, y, z intercepts for planes.
4. Graph Domains with Circle, Hyperbolas, etc.

Example Graphs

• Plane: Found using intercepts.
• Hyperbolas: Identified by x/y interactions.
• Upper Ellipsoids: Derived from squared terms and constants.

Level Curves and Contour Plots

• Level Curves: Traces of the surface at specific heights.
• Contour Plots: Collection of level curves projected downward.
• Applications: Topographical maps resemble contour plots.

Practical Example

• Ellipsoid and Hyperbola: Describe what shapes surfaces take at various z-values.
• Graphical Representation: Visualize using a software tool or detailed manual calculation.

Final Notes

• Multivariable calculus involves a vast overlap with geometry.
• Visualization plays a crucial role in understanding these concepts.
• Practice with sketching simple surfaces helps solidify understanding for more complex functions.