Mathematical functions that depend on several input variables are referred to as functions of multiple variables or multivariable functions. Multivariable functions transfer a set of numbers (vectors) to another set of numbers in an contrastto single variable functions, which map one number to another.

Here are some key terms and characterstics in relation  to the functions of various variables :

1. Domain : A multivariable function's domain is made up of all potential input values or vectors that might be used to construct the function.

2. Range : The collection of all possible output values or vectors that amultivariable  function can produce is known as its Range.

3. Graph : A multivariable function's graph shows how the input variables and output values relate to one another in a higher-dimensional space. The graph is often depicted in two dimensions as a surface or a curve.

4. Partial Derivatives : While maintaining other variables constant, partial derivatives calculate the rate of change of a multivariable function with regard to each individual input variable. They reveal details about the many directions in which the function shifts.

5. Gradient : A multivvariable function's gradient is a vector pointing in the direction of the function's steepest increase at a specific point. It is calculated by using the function's partial derivatives with respect to each variable .

6. Critical point : In the domain of a multivariable ,function a critical point is any location  where the gradient is zero or undefinable. These points can be the function's saddle points, local minima, or local maxima.

7.Extrema : The extrema of a multivariable function are its highest and minimum values respectively. They can appear in crucial locations or at the domain's edge.

8. Optimization : Finding the greatest or least values of a multivariable function while keeping in mind certain restrictions is known as optimization . this is helpful in a variety of disciplines including physics, engineering and economics.

9. Level surfaces : In three dimensions, level surfaces are collections of points where the function takess a constant value. In two dimensions level curves are similar. They give the behaviour of the function a visual representation.

10.  Chain Rule : The derivatives of a composition of functions can ne calculated using the chain rule for multivariable functions. It establishes a connection between the partial derivatives of the composite function and those  of the individual functions involved.

These ideas serve as the cornerstone for the study of many variable functions in calculus and mathematical analysis. They are crucial for comprehensing and assessing how functions behave in diverse mathematical, engineering, and scientific situations.