Applied Mathematics

Applied Mathematics

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation and study of mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

History
Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability. These areas of mathematics were intimately tied to the development of Newtonian physics, and in fact, the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a legacy as well: until the early 20th century, subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments. Engineering and computer science departments have traditionally made use of applied mathematics.

Applications of Derivative

Equations of normal and tangent lines
A derivative at a point in a curve can be viewed as the slope of the line tangent to that curve at that point. Given this, the natural next question is what the equation of that tangent line is. In this tutorial, well not only find equations of tangent lines, but normal ones as well !

Equations of normal and tangent lines
A derivative at a point in a curve can be viewed as the slope of the line tangent to that curve at that point. Given this, the natural next question is what the equation of that tangent line is. In this tutorial, well not only find equations of tangent lines, but normal ones as well!

Applying differentiation in different fields
The idea of a derivative being the instantaneous rate of change is useful when studying or thinking about phenomena in a whole range of fields. In this tutorial, we begin to just scratch the surface as we apply derivatives in fields as disperse as biology and economics.

check out following site for more information about Applications of Derivative
http://www.whitman.edu/mathematics/calculus/calculus_06_Applications_of_the_Derivative.pdf

Integral calculus



Indefinite integral as anti-derivative
You are very familiar with taking the derivative of a function. Now we are going to go the other way aroundif I give you a derivative of a function, can you come up with a possible original function. In other words, well be taking the anti-derivative!

Definite integrals
Until now, we have been viewing integrals as anti-derivatives. Now we explore them as the area under a curve between two boundaries (we will now construct definite integrals by defining the boundaries). This is the real meat of integral calculus!

Fundamental theorem of calculus
You get the general idea that taking a definite integral of a function is related to evaluating the antiderivative, but where did this connection come from. This tutorial focuses on the fundamental theorem of calculus which ties the ideas of integration and differentiation together. Well explain what it is, give a proof and then show examples of taking derivatives of integrals where the Fundamental Theorem is directly applicable.


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Differential equation
A differential equation is a mathematical equation that relates some function of one or more variables with its derivatives. Differential equations arise whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

The theory of differential equations is well developed and the methods used to study them vary significantly with the type of the equation.

Ordinary and partial
 An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function.

 Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. For example, Bessels differential equation

A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

Linear and non-linear
Both ordinary and partial differential equations are broadly classified as linear and nonlinear.

 1. A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products of the unknown function and its derivatives are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.

 2. There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).

Introduction of Differential equations

Probability
Many events cant be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin


When a coin is tossed, there are two possible outcomes

Heads (H) OR
Tails (T)