A multivariable function is a function with several independent variables. Functions with more than one variable are essential in order to mathematically model complicated physical phenomena, engineering, chemical, agricultural, and economic systems.
In what follows, \( C \) represents the constant of integration where applicable.
Examples of Multivariable Functions
Explore how different geometric and physical quantities depend on multiple variables.
Example 1: Area of a Rectangle
A rectangle has a width \( W \) and a length \( L \). The area \( A \) of the rectangle is given by:
\[ A = W \cdot L \]It is clear that if \( W \) and \( L \) vary, area \( A \) depends on two variables: width \( W \) and length \( L \). Thus, \( A \) is said to be a function of two variables, written as \( A(W, L) \).
Example 2: Volume of a Rectangular Solid
A rectangular solid has width \( W \), length \( L \), and height \( H \). The volume \( V \) of the rectangular solid is given by:
\[ V = W \cdot L \cdot H \]If \( W \), \( L \), and \( H \) vary, volume \( V \) depends on 3 variables: width \( W \), length \( L \), and height \( H \), making it a function of three variables \( V(W, L, H) \).
Example 3: Volume of a Circular Cylinder
The volume \( V \) of a circular cylinder of radius \( r \) and height \( h \) is given by:
\[ V = \pi r^2 h \]If \( r \) and \( h \) vary, we can say that volume \( V \) is a function of two variables \( r \) and \( h \), denoted as \( V(r, h) \).
Example 4: Room Temperature Over Space and Time
Let \( T \) be the temperature in a room. Using a rectangular coordinate system of axes \( (x, y, z) \), temperature \( T \) can vary with spatial position \( (x, y, z) \) and time \( t \). It can be expressed as:
\[ T(x, y, z, t) \]This represents a function of 4 variables (three spatial coordinates and one temporal coordinate).