Glossary of Mathematical Terms - The Story of Mathematics
In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns that The expressions on the two sides of the equals sign are called the "left-hand. algebraic equation: a combination of numbers and letters equivalent to a sentence (the more advanced manipulation of numbers is usually known as number theory) correlation: a measure of relationship between two variables or sets of data, decimal number: a real number which expresses fractions on the base Fill in each blank so that the resulting statement is true. An equation that expresses a relationship between two or more variables, such as H = ( − a) is called.
Ordinary differential equations[ edit ] Main article: Ordinary differential equation An ordinary differential equation or ODE is an equation containing a function of one independent variable and its derivatives.
The term "ordinary" is used in contrast with the term partial differential equationwhich may be with respect to more than one independent variable. Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form.
Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.
Partial differential equations[ edit ] Main article: Partial differential equation A partial differential equation PDE is a differential equation that contains unknown multivariable functions and their partial derivatives. This is in contrast to ordinary differential equationswhich deal with functions of a single variable and their derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model.
Dependent and independent variables review
PDEs can be used to describe a wide variety of phenomena such as soundheatelectrostaticselectrodynamicsfluid flowelasticityor quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Remember that variables are items which can assume different values. A function tries to explain one variable in terms of another.
Consider the above example where the amount you choose to spend depends on your salary. Here there are two variables: Independent variables are those which do not depend on other variables. Dependent variables are those which are changed by the independent variables. The change is caused by the independent variable. In our example salary is the independent variable and the amount you spend is the dependent variable. To continue with the same example what if the amount you choose to spend depends not only on your salary but also on the income you receive from investments in the stock market.
Now there are three variables: A function is a mathematical relationship in which the values of a single dependent variable are determined by the values of one or more independent variables. Function means the dependent variable is determined by the independent variable s.
A goal of economic analysis is to determine the independent variable s which explain certain dependent variables. For example what explains changes in employment, in consumer spending, in business investment etc.? Functions with a single independent variable are called univariate functions. There is a one to one correspondence.
Variables, Functions and Equations
Functions with more than one independent variable are called multivariate functions. The independent variable is often designated by x. The dependent variable is often designated by y.
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We say y is a function of x. This means y depends on or is determined by x. If we know the value of x, then we can find the value of y. In pronunciation we say " y is f of x. In other words the parenthesis does not mean that f is multiplied by x.
It is not necessary to use the letter f.
Variables, Functions and Equations
We may look at functions algebraically or graphically. If we use algebra we look at equations. If we use geometry we use graphs. There is one dependent variable, the price of pizza and there are three independent variables, the prices of tomato sauce, cheese, and pizza dough. It says that the quantity of pizza demanded depends on the price of pizza and the number of potential pizza eaters.
There is one dependent variable, the quantity of pizza demanded, and there are two independent variables, the price of pizza and the number of potential pizza eaters. This is a very general form of the consumption function. In order to use it economists must put it into a more precise mathematical form. The use of functional notation: The independent variable, x, can have different values. When x changes y also changes. This means find the value of y when x equals 0.What Is an Algebraic Expression That Describes a Relationship Between Several Variables?
This means find the value of y when x equals 1.