2.1 Domains and ranges of multivariable functions

Functions of several variables expand our mathematical toolkit, letting us model complex systems with multiple inputs. This topic introduces multivariable, vector-valued, and scalar-valued functions, exploring their domains and ranges.

We'll dive into how these functions work in different coordinate systems, particularly Cartesian coordinates. We'll also look at projections and coordinate planes, which help us visualize and analyze these functions in 3D space.

Function Types

Multivariable and Vector-Valued Functions

Top images from around the web for Multivariable and Vector-Valued Functions

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

• 

  Vector-Valued Functions and Space Curves · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

1 of 3

Top images from around the web for Multivariable and Vector-Valued Functions

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

• 

  Vector-Valued Functions and Space Curves · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

1 of 3

• Multivariable function maps input from multiple variables to a single output value
• Vector-valued function takes one or more input variables and returns a vector
  • Components of the output vector can be functions of the input variable(s)
  • Example: $\vec{r}(t) = \langle \cos(t), \sin(t), t \rangle$ maps a single variable $t$ to a 3D vector
• Both multivariable and vector-valued functions can be used to model complex systems or phenomena (fluid dynamics, electromagnetic fields)

Scalar-Valued Functions

• Scalar-valued function maps one or more input variables to a single scalar output value
  • Example: $[f(x, y)](https://www.fiveableKeyTerm:f(x,_y)) = x^2 + y^2$ maps two variables $(x, y)$ to a single scalar value
• Can be represented graphically as a surface in three-dimensional space when there are two input variables
• Scalar-valued functions are used in optimization problems (finding maxima/minima) and in modeling physical quantities (temperature, pressure)

Domain and Range

Domain of Multivariable Functions

• Domain is the set of all possible input values for which a function is defined
• For a function $f(x, y)$, the domain is a subset of the $xy$-plane
  • Example: $f(x, y) = \sqrt{x^2 + y^2}$ has domain $\mathbb{R}^2$ (all real $x$ and $y$ values)
• Domains can be restricted by the context of the problem or the nature of the function
  • Example: A function modeling the height of a physical object may have a domain restricted to non-negative values

Range and Codomain

• Range is the set of all possible output values of a function
  • For scalar-valued functions, the range is a subset of the real numbers
  • For vector-valued functions, the range is a subset of the codomain (target space)
• Codomain is the set of all possible output values, which may include values not actually produced by the function
• Range is determined by applying the function to all values in its domain and collecting the results

Coordinate Systems

Cartesian Coordinates

• Cartesian coordinates $(x, y, z)$ represent points in three-dimensional space using perpendicular axes
• Each coordinate represents the signed distance from the origin along the corresponding axis
• Cartesian coordinates are the most common system for representing multivariable functions
  • Example: $f(x, y) = x^2 - y^2$ is easily visualized in the Cartesian plane
• Useful for problems involving linear relationships or when axes have distinct meanings (time, position)

Projection and Coordinate Planes

• Projection is the mapping of points in higher-dimensional space onto a lower-dimensional subspace
• Coordinate planes are the two-dimensional projections of 3D space onto the $xy$, $xz$, or $yz$ planes
  • Example: The $xy$-plane is the projection of 3D space onto the plane where $z = 0$
• Projections are used to visualize and analyze cross-sections or slices of multivariable functions
• Coordinate planes help in understanding the behavior of functions by examining their traces (cross-sections parallel to coordinate axes)