Calculus - Study guide

1. Algebra and trigonometry review
   1. Functions
   2. Graphs
   3. Equations
   4. Inequalities
   5. Exponents
   6. Logarithms
   7. Trigonometric identities
2. Main concepts of calculus
   1. Limits
   2. Derivatives
   3. Integrals
3. Rules and techniques of calculus
   1. Product rule
   2. Quotient rule
   3. Chain rule
   4. Power rule
   5. Fundamental theorem of calculus
   6. Substitution rule
   7. Integration by parts
   8. U-substitution
   9. Trigonometric substitution
   10. Partial fractions
4. Advanced topics of calculus
   1. Multivariable calculus
      1. Partial derivatives
      2. Multiple integrals
      3. Vector fields
      4. Gradient
      5. Divergence
      6. Curl
      7. Line integrals
      8. Surface integrals
      9. Green's theorem
      10. Stokes' theorem
      11. Divergence theorem
   2. Vector calculus
      1. Vectors
      2. Matrices
      3. Dot product
      4. Cross product
      5. Vector-valued functions
      6. Arc length
      7. Arc curvature
      8. Parametric equations
      9. Polar coordinates
      10. Cylindrical coordinates
      11. Spherical coordinates
   3. Differential equations
      1. First-order differential equations
         1. Separable equations
         2. Linear equations
         3. Exact equations
         4. Integrating factors
         5. Bernoulli equations
         6. Existence theorem
         7. Uniqueness theorem
         8. Direction fields
         9. Euler's method
         10. Applications of first-order differential equations
      2. Second-order differential equations
         1. Homogeneous equations with constant coefficients
         2. Nonhomogeneous equations
         3. Method of undetermined coefficients
         4. Variation of parameters
         5. Reduction of order
         6. Euler-Cauchy equations
         7. Power series solutions
         8. Applications of second-order differential equations
      3. Systems of differential equations
         1. First-order linear systems
         2. Matrices
         3. Eigenvalues
         4. Phase plane analysis
         5. Nonlinear systems
         6. Nonlinear stability
         7. Laplace transform
         8. Applications of systems of differential equations
   4. Calculus of variations
      1. Functionals
      2. Extremals
      3. Euler-Lagrange equation
      4. Natural boundary conditions
      5. Isoperimetric problems
      6. Hamilton's principle
      7. Lagrangian mechanics