## Calculus

- Limits and Continuity
- Intuitive definition of limits
- limit laws and elementary methods
- sandwich theorem and a famous limit related to trigonometric functions
- Definition of Limits at Infinity
- continuity and discontinuous points
- properties of continuous functions and intermediate value theorem
- formal (precise) definition of a limit

- Derivatives

## Linear Algebra

## Multivariable Calculus

- Vectors and Analytic Geometry
- Concepts of Vectors
- Linear Operations of Vectors
- Three Dimensional Coordinates
- Projection, Direction Angle and Direction Cosine
- Dot Product
- Cross Product
- Triple Scalar Product
- Equation of Lines
- Equations of Planes
- Distance from a Point to a Line or to a Plane
- Distance Between Lines or Between Planes
- Equations of Cylinders and Surfaces of Revolution
- Quadric Surfaces
- Space Curves

- Partial Derivatives and Their Applications
- Functions of Several Variables
- Limit and Continuity of Several Variable Functions
- Partial Derivatives
- Higher Partial Derivatives
- Proof of Clairaut’s Theorem Equqlity of Mixed Derivative
- Differentiability and Total Differential
- Chain Rule for Multivariable Functions
- Implicit Differential for Multivariable Functions
- Directional Derivatives
- Gradient Vector
- Tangent Planes and Normal Lines
- Maximum and Minimum of Multivariable Functions
- Lagrange Multiplier
- Lagrange Multiplier with Two Constraints

- Multiple Integration
- Definition of Multiple Integrals
- How to Evaluate Double Integrals in Rectangular Coordinates
- More Examples of Double Integrals in Rectangular Coordinates
- Properties of Double Integrals
- Polar Coordinates
- Evaluate Double Integrals in Polar Coordinates
- Evaluate Surface Area Using Double Integrals
- Evaluate Surface Area for Parametric Surfaces
- Triple Integrals in Rectangular Coordinates
- Evaluate Triple Integrals in Cylindrical Coordinates
- Evaluate Triple Integrals in Spherical Coordinates