Calculus 3 Overview

Disclaimer

• Cannot cover everything in 8 minutes
• Inspired by Fuzzy Penguin Ams
• Linked video on Calc 2 in the description

Part 1: 3D Space, Vectors, and Surfaces

• Functions in 3D: Take two inputs (X and Y) and produce an output (Z)
• Vectors: Quantities with direction and magnitude, applied to 3D vectors

Part 2: Vector Multiplication

• Addition and Subtraction: Similar to regular numbers
• Dot Product: Multiply corresponding elements, sum results
  • Perpendicular vectors: Dot product = 0
  • Parallel vectors: Dot product = product of magnitudes
  • Formula: Involves angle θ between vectors, and magnitudes
• Cross Product: Determinant of matrix involving unit vectors i, j, k
  • Always generates a vector orthogonal to the two multiplied vectors

Part 3: Limits and Derivatives of Multivariable Functions

• Limits: Approach a value on the XY plane
• Derivatives: Defined similarly to 2D
  • Directional Derivatives: Infinitely many directions, each having its own derivative
  • Partial Derivatives: With respect to X (positive X direction) and Y (positive Y direction)
  • Gradient: Vector with components of partial derivatives in X and Y directions

Part 4: Double Integrals

• Double Integral: Finds volume under a function
  • First integral in X direction, then in Y direction
  • Integrate over non-rectangular regions with functions as bounds
  • Use polar coordinates for double integrals: drdθ

Part 5: Triple Integrals and 3D Coordinate Systems

• Triple Integrals: Used for volume, can define with three boundaries
  • Example: X from 0 to 2, Y from 0 to X (triangular base), Z from 0 to Y² (parabolic shape)
  • Applications: Average temperature over a 3D surface
• 3D Coordinate Systems
  • Cylindrical Coordinates: r (radius), z (height), θ (angle)
  • Spherical Coordinates: ρ (radius from origin), φ (angle from vertical), θ (angle)

Part 6: Coordinate Transformations and the Jacobian

• Infinite Coordinate Systems: Can be defined as x = g(u,v) and y = h(u,v)
  • Example: Polar coordinates x = r cosθ and y = r sinθ
• Jacobian: Function added to the integral to account for coordinate system distortion
  • Example: Double integral conversion to polar coordinates,
  • Involves matrix determinants

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

• Vector Field: Vector assigned to each point in a plane or space
• Scalar Field: Regular numbers assigned to points, similar to 3D functions
• Scalar Line Integral: Integral over a bent sheet in 3D space
• Vector Line Integral: Integral over vector fields, analogous to work done by forces
• Conservative Vector Fields: Path independent integrals, similar to conservative forces (e.g., gravity)
• Properties of Vector Fields
  • Divergence: Amount of outflow from a point
  • Curl: Amount of rotation around a point

Conclusion

• Overview not exhaustive
• Potential future videos on specific topics
• Additional resources linked in the description for in-depth learning