Functions of several variables: Basic concepts and visualization
Basic concepts
Functions of a single variable operate like rules that produce a new number out a given number using a mathematical formula. But you can also imagine rules and formulas that produce a new number out of two given numbers. Then we speak of functions of two unknowns, also known as functions of two variables.
Simple examples
- The area \(O\) of a triangle with base \(b\) and height \(h\): \(O(b,h)=\tfrac{1}{2}\cdot b\cdot h\)
- The disistance \(s\) travelled in a uniform motion with speed \(v\) and duration \(t\): \(s(v,t) = v\cdot t\).
The terminology of function of a single variable is also used for functions of several variables. The second example is written in functional form, but yopu will probably encounter it more in the form of a relationship between variables: \(s=v\cdot t\). The speed \(v\) and duration \(t\) are the independent variables and the distance traveled \(s\) is the dependent variable, because its value depends on the values of the dependent variables. The function definition is the expression \(v\cdot t\). The dependent variables are explicitly written, isolated from the independent variables. Hoewever many relationships are not explicitly given in the form of a function. An example:
The thin lens formula for a thin lens with focal length \(f\) is \[\frac{1}{b}+\frac{1}{v}=\frac{1}{f}\text{,}\] where \(v\) is the distance from the object to the lens and \(b\) is the distance from the image to the lens.
This is called an implicit relationship between quantities. Such a relationship is sometimes a function, but not always: one example is the equation \(x^2+y^2+z^2=1\) for the sphere with centre in the origin \((0,0,0)\) and radius \(1\), for which \(z\) cannot be defined by a single function of \(x\) and \(y\).
A relationship between the three variables \(x\), \(y\) and \(z\), where \(x\) and \(y\) are taken as independent variables, is a function \(z=z(x,y)\) if all acceptable values of \(x\) and \(y\) lead to exactly one value for \(z\). Every pair of acceptable values of \(x\) and \(y\) is called an original; every corresponding value of \(z\) is then called the function value of the original. A single-valued function relates a given original to exactly one output value. All originals together form the domain of the function and and all possible function values together form the range of the function.
