Subject: Calculus

Calculus

This is Calculus:

In its modern sense, calculus includes several mathematical disciplines. Most commonly, it means the detailed analysis of rates of change in functions and is of utmost importance in all sciences. It is usually divided into two (closely related through the fundamental theorem of calculus) branches: differential calculus and integral calculus.

The first, differential calculus, is concerned with finding the instantaneous rate of change (or derivative) of a function's value with respect to changes in its argument (roughly speaking, how much the value of a function changes with a small change in its argument). This derivative can also be interpreted as the slope of the function's graph at a specific point.

The second branch of calculus, integral calculus, studies methods for finding the integral of a function. An integral may be defined as the limit of a sum of terms which correspond to areas under the graph of a function. Considered as such, integration allows us to calculate the area under a curve and the surface area and volume of solids such as spheres and cones.

So, where do I start?

Calculus is a massive subject, from modern economics, to structural engineering, to fluid dynamics, to space travel, to everyday problems in life calculus was a large step towards uniting life and math. Therefore the following list of items may seem intense, and even worthy of biting down your nails even a little further, but every journey starts with that initial step forward. So get started, click on the limits link below and see why it has been said that "Among all of the mathematical disciplines the theory of differential equations is the most important... It furnishes the explanation of all those elementary manifestations of nature which involve time. (~Sophus Lie)."

Limits

  1. Limit of a function
  2. One-sided limit
  3. Limit of a sequence
  4. Indeterminate form
  5. Orders of approximation
  6. (ε, δ)-Definition of a limit

Differential calculus

  1. Derivative
  2. Notations
    1. Newton's notation for differentiation
    2. Leibniz's notation for differentiation
  3. Simplest rules
    1. Derivative of a constant
    2. Sum rule in differentiation
    3. Constant factor rule in differentiation
    4. Linearity of differentiation
    5. Calculus with polynomials
  4. Derivative Examples
    1. Chain rule
    2. Product rule
    3. Quotient rule
  5. Derivatives of trigonometric functions
  6. Inverse functions and differentiation
  7. Implicit differentiation
  8. Higher Order Derivatives
  9. Stationary point
  10. Maxima and minima
  11. First derivative test
  12. Second derivative test
  13. Extreme value theorem
  14. Differential equation
  15. Differential operator
  16. Newton's method
  17. Taylor's theorem
  18. L'Hôpital's rule
  19. Leibniz's rule
  20. Mean value theorem
  21. Logarithmic derivative
  22. Differential
  23. Related rates
  24. Regiomontanus' angle maximization problem

Integral calculus

  1. Antiderivative, Indefinite integral
  2. Simplest rules
  3. Sum rule in integration
  4. Constant factor rule in integration
  5. Linearity of integration
  6. Arbitrary constant of integration
  7. Fundamental theorem of calculus
  8. Integration by parts
  9. Inverse chain rule method
  10. Substitution rule
  11. Differentiation under the integral sign
  12. Trigonometric substitution
  13. Partial fractions in integration
  14. Quadratic integral
  15. Trapezium rule
  16. Integral of secant cubed
  17. Arclength

Special functions and numbers

  1. Natural logarithm
  2. e (mathematical constant)
  3. Exponential function
  4. Stirling's approximation
  5. Bernoulli numbers

Numerical integration

  1. Rectangle method
  2. Trapezium rule
  3. Simpson's rule
  4. Newton–Cotes formulas
  5. Gaussian quadrature

Lists and tables

  1. Table of common limits
  2. Table of derivatives
  3. Table of integrals
  4. List of integrals
    1. List of integrals of rational functions
    2. List of integrals of irrational functions
      1. Forms containing: sqrt(a+bx)
      2. Forms containing: a2 ± x2
      3. Forms containing: sqrt(x2 ± a2)
      4. Forms containing: sqrt(a2 - x2)
      5. Forms containing: sqrt(2ax - x2)
    3. List of integrals of trigonometric functions
    4. List of integrals of inverse trigonometric functions
    5. List of integrals of hyperbolic functions
    6. List of integrals of exponential functions
    7. List of integrals of logarithmic functions
    8. List of integrals of area functions
  5. Table of mathematical symbols

Multivariable Calculus

  1. Partial derivative
  2. Disk integration
  3. Shell integration
  4. Gabriel's horn
  5. Jacobian matrix
  6. Hessian matrix
  7. Curvature
  8. Green's theorem
  9. Divergence theorem
  10. Stokes' theorem

Series

  1. Infinite series
  2. Maclaurin series
  3. Taylor series
  4. Fourier series
  5. Euler–Maclaurin formula