Calculus 3 Final Exam Review: Key Formulas and Concepts (Ch. 12.1 - 16.9)
Chapter 12: Vectors and the Geometry of Space
- 3D Coordinate Systems
- Vectors: magnitude, unit vectors, vector addition, scalar multiplication
- Dot Product: angle between vectors, projections
- Cross Product: orthogonal vector, area of parallelograms
- Lines and Planes: vector equations, normal vectors
- Surfaces: cylinders, quadric surfaces
Chapter 13: Vector Functions
- Vector-Valued Functions: curves in space
- Derivatives and Integrals of Vectors
- Arc Length and Curvature
- Motion in Space: velocity, acceleration, speed, tangential and normal components
Chapter 14: Partial Derivatives
- Functions of Several Variables
- Limits and Continuity in 3D
- Partial Derivatives: first and higher-order
- Tangent Planes and Linear Approximations
- Chain Rule
- Directional Derivatives and Gradient Vector
- Optimization: max/min values, second derivative test
- Lagrange Multipliers
Chapter 15: Multiple Integrals
- Double Integrals (Rectangular & General Regions)
- Polar Coordinates in Double Integrals
- Applications: area, volume, mass, center of mass
- Triple Integrals (Rectangular, Cylindrical, Spherical Coordinates)
- Change of Variables and Jacobians
Chapter 16: Vector Calculus
- Vector Fields
- Line Integrals
- Fundamental Theorem for Line Integrals
- Greens Theorem
- Curl and Divergence
- Parametric Surfaces and Surface Area
- Surface Integrals
- Stokes Theorem
- Divergence Theorem
Key Theorems & Tools to Know
- Gradient (f), Divergence (F), Curl (F)
- The Fundamental Theorem for Line Integrals
- Greens Theorem
- Stokes Theorem
- The Divergence Theorem