Graph of a function of multiple variables

Definition

Suppose $f$ is a real-valued function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ . The graph of $f$ is a subset of $R^{n + 1}$ , with coordinate axes $x_{1}, x_{2}, \dots, x_{n}, x_{n + 1}$ , given as follows:

${(x_{1}, x_{2}, \dots, x_{n}, f (x_{1}, x_{2}, \dots, x_{n})) ∣ (x_{1}, x_{2}, \dots, x_{n}) \in d o m (f)}$

where $d o m (f)$ denotes the domain of $f$ .

Alternatively, it is given by the equation:

$x_{n + 1} = f (x_{1}, x_{2}, \dots, x_{n})$

For nice enough functions, this graph looks like a hypersurface of codimension one (and dimension $n$ ) inside $R^{n + 1}$ .

Particular cases

Aspects

Domain and range

Aspect of the function	How it can be deduced from the graph
domain	project the entire graph on the $x_{1} x_{2} \dots x_{n}$ -hyperplane
range	project the entire graph on the $x_{n + 1}$ -axis

Vertical line test

The vertical line test for a function of one variable says that every vertical line intersects the graph in exactly one point if the $x$ -coordinate is in the domain and in no point if the $x$ -coordinate is not in the domain. There is an analogous test for a function of multiple variables. This says that for any line parallel to the $x_{n + 1}$ -axis, the intersection with the graph has size one (if the intersection with the $x_{1} x_{2} \dots x_{n}$ -hyperplane is in the domain) or zero (if it isn't).

Restriction to one variable

Graph of the restriction

We can restrict the function to a function of fewer variables by fixing the value of some of the variables. Suppose $A$ is a subset of \{ 1,2,\dots,n\}</math>. Suppose we fix the values of the coordinates in $A$ , so we get $x_{j} = a_{j}$ for $j \in A$ .

This is now a function of the remaining variables, which is a total of $n - | A |$ variables.

The graph of this function is obtained by intersecting the original graph with the affine subspace given by $x_{j} = a_{j}, j \in A$ . Note that this subspace has dimension $n + 1 - | A |$ , and the intersection of the graph with this is expected to have dimension $n - | A |$ .

Here's an extreme case: $A$ has size $n - 1$ , and the only variable omitted is $i$ . Then, the function we obtain is a function of one variable:

$x_{i} \mapsto f (a_{1}, a_{2}, \dot{,} a_{i - 1}, x_{i}, a_{i + 1}, \dots, a_{n})$

The graph of this is obtained by intersecting the original graph with the plane given by equations $x_{j} = a_{j}$ for all $j \neq i$ . There are $n - 1$ such equations. The plane itself is parallel to the $x_{i} x_{n + 1}$ -plane.

Continuity in each variable and separate continuity in graphical terms

We have the following:

Assertion about continuity	How we can verify it from the graph
$f$ is continuous in $x_{i}$ at the point $(a_{1}, a_{2}, \dots, a_{n})$	Consider the graph restricted to the plane $x_{j} = a_{j}, j \neq i$ . This graph is continuous at $x_{i} = a_{i}$ .
$f$ is separately continuous in all variables at the point $(a_{1}, a_{2}, \dots, a_{n})$ .	The above holds for all $i \in {1, 2, 3, \dots, n}$ .
$f$ is continuous in $x_{i}$ everywhere.	The restrictions of the graph to all planes parallel to the $x_{i} x_{n + 1}$ -plane are continuous functions.
$f$ is separately continuous in all variables everywhere.	The above holds for all $i \in {1, 2, 3, \dots, n}$ .

Partial derivatives in graphical terms

For further information, refer: partial derivative

Suppose $f$ is a function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ and suppose $(a_{1}, a_{2}, \dots, a_{n})$ is a point in the domain of $f$ . Consider the graph of $f$ in $R^{n + 1}$ given by:

$x_{n + 1} = f (x_{1}, x_{2}, \dots, x_{n})$

For any $i \in {1, 2, \dots, n}$ , we define the partial derivative $f_{x_{i}} (a_{1}, a_{2}, \dots, a_{n})$ , also denoted $f_{i} (a_{1}, a_{2}, \dots, a_{n})$ , as follows:

First, consider the intersection of the graph of $f$ with the plane given by the set of $n - 1$ equations $x_{j} = a_{j}$ for all $j \neq i$ . This is a plane parallel to the $x_{i} x_{n + 1}$ -plane.
In this plane, consider the slope of the tangent line at $x_{i} = a_{i}$ . This is the value of the partial derivative.

Directional derivatives in graphical terms

For further information, refer: directional derivative

The directional derivative $D_{⟨ u, v ⟩} f (x_{0}, y_{0})$ in the direction of a unit vector $⟨ u, v ⟩$ at a point $(x_{0}, y_{0})$ can be determined as follows: first, intersect the graph of the function with the plane $v (x - x_{0}) = u (y - y_{0})$ . This plane is perpendicular to the $x y$ -plane and its intersection with the $x y$ -plane is the line through $(x_{0}, y_{0})$ in the direction of the unit vector $⟨ u, v ⟩$ .

This intersection can be thought of as the graph of a function of one variable, where the point $(x_{0}, y_{0}, 0)$ is treated as the origin, the direction $(u, v, 0)$ is the independent variable axis, and the $z$ -axis direction is the dependent variable axis. Now, the directional derivative is the slope of this graph for dependent variable value of 0.

Gradient vector in graphical terms

For further information, refer: gradient vector

Suppose $f$ is a function of multiple variables $x_{1}, x_{2}, \dots, x_{n}$ and suppose $(a_{1}, a_{2}, \dots, a_{n})$ is a point in the domain of $f$ . We say that $f$ is differentiable at $(a_{1}, a_{2}, \dots, a_{n})$ if the gradient vector $(\nabla f) (a_{1}, a_{2}, \dots, a_{n})$ exists. This is equivalent to the graph of the function having a well defined tangent hyperplane at the point $(a_{1}, a_{2}, \dots, a_{n}, f (a_{1}, a_{2}, \dot{,} a_{n}))$ . The equation of the tangent hyperplane is given by:

$x_{n + 1} - f (a_{1}, a_{2}, \dots, a_{n}) = (\nabla f) (a_{1}, a_{2}, \dots, a_{n}) \cdot (⟨ x_{1}, x_{2}, \dot{,} x_{n} ⟩ - ⟨ a_{1}, a_{2}, \dot{,} a_{n} ⟩)$