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.

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:

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