Multivariable Calculus explores functions of multiple variables, including partial derivatives, multiple integrals, and vector calculus.
A function of several variables is a rule that assigns a single output to each combination of inputs from two or more variables. Unlike single-variable calculus, where inputs are just points on a line, multivariable functions map points from planes or higher-dimensional spaces.
Imagine the temperature across a metal plate. The temperature at any point depends on both its \(x\) and \(y\) coordinates. This can be described by a function \(T(x, y)\). In three dimensions, functions like \(f(x, y, z)\) can represent things like air pressure in a room.
Graphing these functions often involves 3D plots or contour maps. Level curves are lines where the function has the same value, like the rings on a topographic map.
Functions of several variables model phenomena like weather, economics, and engineering systems.
The elevation at each point on a hiking trail is a function \(z = f(x, y)\), where \((x, y)\) is your location.
A bakery's profit depending on the number of cupcakes and cookies sold can be modeled with \(P(c, k)\).
A function of several variables takes two or more inputs and gives one output, mapping across planes and higher dimensions.