Multivariable Functions

$$f:{\mathbb{R}}^n\to \mathbb{R}$$

Assigns a unique number $f(x)$ to every $x$ in $(a,b)$

Visualizing

Domain

• A collection of all $(x,y,\dots )$ for which the rule makes sense.
• To visualize domain find conditions where the input to a function would not be within it’s range and solve for a quadric surface

Graphing 2 Variable Functions

• $z=f(x,y)$ graph of $f(x,y)$
• Take the planar domain and raise it along z by the value of the function there

Drawing a Dimension Lower

• Level set
  • Collection of all points in the domain on which a 2 variable function has a given value
  • $\{P\in D\mid f(P)=k\}$
  • Kind of similar to cross sections
• Contour Maps
  • A collection of level curves of a function of two variables
• Level surface
  • Collection of all points in the domain on which a 3 variable function has a given value
  • Forms a 3D surface

Limit

• Multivariable Limits

Derivatives

• Partial Derivatives
• Gradient Operator
• Directional Derivative
• Multivar Extreme Values

Integrals

• Double Integral
• Triple Integrals
• Line Integrals
• Surface Integrals
• Applications
  • Surface Area
  • Moment of Inertia
  • Center of Mass