Calculus

1. Multivariable Calculus
   1. Functions of Several Variables
      1. Introduction
         1. Definition and examples of functions of several variables
         2. Notation and domain considerations
      2. Limits and Continuity
         1. Concept of limits for functions of two or more variables
         2. Evaluating multivariable limits
         3. Criteria for continuity in higher dimensions
      3. Partial Derivatives
         1. Definition and interpretation of partial derivatives
         2. Computing partial derivatives
         3. Higher-order partial derivatives
         4. Mixed partial derivatives and Clairaut's theorem
      4. Gradients and Directional Derivatives
         1. Definition of the gradient vector
         2. Properties and applications of gradients
         3. Calculating directional derivatives
         4. Tangent planes and linear approximation in higher dimensions
      5. Level Curves and Surfaces
         1. Understanding level curves for functions of two variables
         2. Level surfaces for functions of three variables
         3. Applications in optimization and constraint handling
   2. Multiple Integrals
      1. Double Integrals
         1. Definition and notation of double integrals
         2. Iterated integrals concept and computation
         3. Applications in computing areas and volumes
         4. Change of variables and Jacobian determinant
         5. Polar coordinates transformation
      2. Triple Integrals
         1. Understanding and computing triple integrals
         2. Applications in volume and mass determination
         3. Change of variables in three dimensions
         4. Use of cylindrical and spherical coordinates
   3. Applications of Multiple Integrals
      1. Center of Mass and Centroid determination
      2. Moments of inertia and applications in physics
      3. Probability and expectation value in continuous random variables
   4. Vector Calculus
      1. Vector Fields
         1. Definition and examples of vector fields
         2. Applications in fluid flow and electromagnetism
      2. Line Integrals
         1. Concepts and computation of line integrals
         2. Applications in work done by a force field
         3. Path independence and conservative vector fields
      3. Surface Integrals
         1. Definition and calculation of surface integrals
         2. Applications in flux through a surface
      4. Divergence and Curl
         1. Understanding and computing divergence
         2. Concept and computation of curl
         3. Physical interpretation in fluid dynamics
      5. Integral Theorems of Vector Calculus
         1. Green's Theorem
            1. Relationship between line integrals and double integrals
            2. Applications in planar regions
         2. Stokes' Theorem
            1. Generalization of Green’s Theorem
            2. Applications in three-dimensional fields
         3. Divergence Theorem (Gauss’s Theorem)
            1. Relationship between surface integrals and triple integrals
            2. Applications in electromagnetism and fluid dynamics