Multivariable Calculus

Multivariable Calculus extends the principles of differentiation and integration to functions of several variables, enabling the analysis of surfaces, solids, and fields in three or more dimensions. This field introduces new tools like partial derivatives, which measure rates of change with respect to a single variable at a time, and multiple integrals, which are used to compute volumes, masses, and surface areas. It culminates in the fundamental theorems of vector calculus, such as Stokes' Theorem and the Divergence Theorem, which relate the behavior of a function inside a region to its values on the boundary, providing a powerful framework for applications in physics, engineering, and computer graphics.

1. Foundations of Three-Dimensional Space
   1. Three-Dimensional Coordinate Systems
      1. Rectangular Coordinate System in 3D
         1. Coordinate Axes and Planes
         2. Plotting Points in 3D Space
         3. Octants in 3D Space
      2. Distance Formula in Three Dimensions
      3. Midpoint Formula in 3D
      4. Equations of Spheres
         1. Standard Form
         2. General Form
         3. Finding Center and Radius
   2. Vector Fundamentals
      1. Definition and Representation of Vectors
         1. Geometric Representation
         2. Component Form
         3. Position Vectors
      2. Vector Magnitude and Direction
         1. Computing Magnitude
         2. Direction Angles
         3. Direction Cosines
      3. Standard Basis Vectors
      4. Unit Vectors
         1. Definition and Properties
         2. Finding Unit Vectors
         3. Applications of Unit Vectors
   3. Vector Operations
      1. Vector Addition and Subtraction
         1. Geometric Interpretation
         2. Component-wise Operations
         3. Properties of Vector Addition
      2. Scalar Multiplication
         1. Definition and Properties
         2. Geometric Effects
      3. Linear Combinations of Vectors
   4. The Dot Product
      1. Definition of the Dot Product
         1. Algebraic Definition
         2. Geometric Definition
      2. Properties of the Dot Product
      3. Computing Angles Between Vectors
      4. Orthogonality and Perpendicular Vectors
      5. Vector Projections
         1. Scalar Projection
         2. Vector Projection
         3. Applications of Projections
      6. Work as a Dot Product
   5. The Cross Product
      1. Definition of the Cross Product
      2. Computing Cross Products
         1. Determinant Method
         2. Component Method
      3. Properties of the Cross Product
      4. Right-Hand Rule
      5. Geometric Interpretation
         1. Perpendicularity of Cross Product
         2. Magnitude as Area
      6. Applications of Cross Products
         1. Area of Parallelograms and Triangles
         2. Torque and Moment
      7. Scalar Triple Product
         1. Definition and Computation
         2. Volume of Parallelepipeds
         3. Coplanarity of Vectors
   6. Lines in Three-Dimensional Space
      1. Vector Equations of Lines
      2. Parametric Equations of Lines
      3. Symmetric Equations of Lines
      4. Direction Vectors
      5. Relationships Between Lines
         1. Parallel Lines
         2. Intersecting Lines
         3. Skew Lines
      6. Distance from Point to Line
      7. Distance Between Lines
   7. Planes in Three-Dimensional Space
      1. Equations of Planes
         1. Vector Form
         2. Scalar Form
         3. Point-Normal Form
      2. Normal Vectors to Planes
      3. Finding Equations of Planes
      4. Relationships Between Planes
         1. Parallel Planes
         2. Intersecting Planes
         3. Angle Between Planes
      5. Distance from Point to Plane
      6. Distance Between Parallel Planes
      7. Line-Plane Intersections
   8. Quadric Surfaces
      1. Introduction to Surfaces in 3D
      2. Traces and Cross-Sections
      3. Cylinders
         1. Circular Cylinders
         2. Elliptic Cylinders
         3. Parabolic Cylinders
         4. Hyperbolic Cylinders
      4. Quadric Surfaces
         1. Ellipsoids
         2. Paraboloids
            1. Elliptic Paraboloids
            2. Hyperbolic Paraboloids
         3. Hyperboloids
            1. Hyperboloid of One Sheet
            2. Hyperboloid of Two Sheets
         4. Cones
            1. Elliptic Cones
            2. Circular Cones
      5. Sketching and Identifying Quadric Surfaces