Calculus

• Single Variable Calculus
  • Metric Topology
    • Definitions
      • Open and Closed Balls and Intervals
      • Limit Point and Closure
  • Sequence and Convergence
    • Accumulation Point
    • Limit Point
    • Important Theorems
    • Cauchy convergence
    • Monotonic Sequences
  • Functional Limit and Continuity
    • Continuity
      • Sequential Continuity
      • Properties
      • Continuous Everywhere
      • Uniform Continuity
      • Lipschitz Continuity
  • Differentiation
    • Differentiation as a rate of change
    • Differentiation as a linear approximation
    • Properties
    • Important Theorems
      • Boundedness theorem
      • EVT: Extreme value theorem
      • Bolzano’s theorem
      • IVT: Intermediate value theorem
      • MVT: Mean value theorem
      • Rolle’s theorem
    • Application: Local extremum
      • Critical Point
      • First derivative test
        • For critical points
        • For monotonic functions
      • Second derivative test
        • For critical points
  • Integration
    • Integration of step functions
      • Properties
    • Integration of general function
      • Properties
    • Indefinite Integral
    • Fundamental theorem of calculus
    • Integration Strategies
      • Integration by parts
      • Feynman’s Trick
      • Integration Bee
  • Series
    • Series with Positive Terms
      • Comparison Tests
      • Ratio Test
      • Integral Test
    • Series with Mixed Terms
      • Absolute Convergence
      • Convergence of Alternating Series
    • Power Series
      • Root Test
  • Useful Resources
• Multivariable Calculus
  • Different Forms of Multivariable Functions
    • Parametric Surface
    • Scalar field
    • Vector field
  • Continuity
  • Differentiation
    • Directional Derivative as a rate of change in scalar fields
      • Partial Derivative
      • Directional Derivative isn’t sufficient
        • Example
    • Total Derivative as a linear approximation in general
      • Gradient
      • Jacobian
    • Differentiability : Continuously Differentiable Functions
    • Properties
    • Higher Order Derivative
      • Higher Order Partial Derivative
      • Higher Order Total Derivative
        • Hessian
        • Laplacian
    • Application
      • Normal vector to level sets
        • Level sets
      • Local extremum
        • First Derivative Test
        • Second Derivative Test
  • Matrix Calculus: Tricks and Useful Results
    • Useful Derivatives
  • Integration
    • Fubini’s Theorem
    • Gaussian Integral using Polar Substitute
    • Useful Resources