Calculus 3 Overview

Disclaimer: This is a brief overview and cannot cover all topics in depth.

Part 1: 3D Space, Vectors, and Surfaces

• 3D Functions: Functions exist in three-dimensional space, taking two inputs (x, y) and producing an output (z).
• Vectors: Quantities with both direction and magnitude, applicable in 3D space.

Part 2: Vector Multiplication

• Addition & Subtraction: Similar to numbers, vectors can be added and subtracted.
• Dot Product:
  • Multiply each component of one vector by the corresponding component of another and sum the results.
  • Connection to spatial relationships: Perpendicular vectors have a dot product of zero; parallel vectors' dot product equals the product of their magnitudes.
• Cross Product:
  • Expressed as a matrix determinant involving unit vectors i, j, k.
  • Results in a vector perpendicular to the two multiplied vectors.

Part 3: Limits and Derivatives of Multivariable Functions

• Limits: Approach a value on the xy-plane.
• Directional Derivatives: Infinite directions to derive at a certain point.
• Partial Derivatives:
  • With respect to x gives the derivative in the positive x direction.
  • With respect to y gives the derivative in the positive y direction.
• Gradient: Vector with components being the derivatives in the x and y directions.

Part 4: Double Integrals

• Concept: Used to find volume under a surface.
• Process:
  • Integrate in the x direction first, then in the y direction.
• Non-Rectangular Regions: Bounds can be functions.
• Polar Coordinates: Transform double integrals using polar coordinates.

Part 5: Triple Integrals and 3D Coordinate Systems

• Triple Integrals:
  • Used for integrating over 3D regions with three boundaries.
  • Applications include finding average temperature over a volume.
• Coordinate Systems:
  • Cylindrical Coordinates: Radius (r), angle (θ), and height (z).
  • Spherical Coordinates: Radius (ρ), angle from the vertical (φ), and angle (θ).

Part 6: Coordinate Transformations and the Jacobian

• Coordinate Systems: Can be defined using any functions of (u, v).
• Polar Coordinates: Example of coordinate transformation.
• Jacobian:
  • Accounts for distortion when switching coordinate systems.
  • Related to matrix determinants.

Part 7: Vector Fields, Scalar Fields, and Line Integrals

• Vector Fields: Assign a vector to each point in space.
• Scalar Fields: Assign a scalar (number) to each point.
• Scalar Line Integrals: Integrals over a bent surface in 3D space.
• Vector Line Integrals: Integrals over vector fields, analogous to work done by forces.
• Conservative Fields: Path-independent line integrals, similar to conservative forces like gravity.
• Divergence: Measure of outflow at a point in a vector field.
• Curl: Measure of rotation around a point in a vector field.

Conclusion: This overview covers major topics but leaves out many details. Further resources are suggested for deeper understanding.