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

- Limit of a function
- One-sided limit
- Limit of a sequence
- Indeterminate form
- Orders of approximation
- (ε, δ)-Definition of a limit

### Differential calculus

- Derivative
- Notations
- Simplest rules
- Derivative Examples
- Derivatives of trigonometric functions
- Inverse functions and differentiation
- Implicit differentiation
- Higher Order Derivatives
- Stationary point
- Maxima and minima
- First derivative test
- Second derivative test
- Extreme value theorem
- Differential equation
- Differential operator
- Newton's method
- Taylor's theorem
- L'Hôpital's rule
- Leibniz's rule
- Mean value theorem
- Logarithmic derivative
- Differential
- Related rates
- Regiomontanus' angle maximization problem

### Integral calculus

- Antiderivative, Indefinite integral
- Simplest rules
- Sum rule in integration
- Constant factor rule in integration
- Linearity of integration
- Arbitrary constant of integration
- Fundamental theorem of calculus
- Integration by parts
- Inverse chain rule method
- Substitution rule
- Differentiation under the integral sign
- Trigonometric substitution
- Partial fractions in integration
- Quadratic integral
- Trapezium rule
- Integral of secant cubed
- Arclength

### Special functions and numbers

- Natural logarithm
- e (mathematical constant)
- Exponential function
- Stirling's approximation
- Bernoulli numbers

### Numerical integration

### Lists and tables

- Table of common limits
- Table of derivatives
- Table of integrals
- List of integrals
- List of integrals of rational functions
- List of integrals of irrational functions
- List of integrals of trigonometric functions
- List of integrals of inverse trigonometric functions
- List of integrals of hyperbolic functions
- List of integrals of exponential functions
- List of integrals of logarithmic functions
- List of integrals of area functions

- Table of mathematical symbols

### Multivariable Calculus

- Partial derivative
- Disk integration
- Shell integration
- Gabriel's horn
- Jacobian matrix
- Hessian matrix
- Curvature
- Green's theorem
- Divergence theorem
- Stokes' theorem