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 $\mathbb {R} ^{n+1}$, with coordinate axes $x_{1},x_{2},\dots ,x_{n},x_{n+1}$, given as follows:

$\{(x_{1},x_{2},\dots ,x_{n},f(x_{1},x_{2},\dots ,x_{n}))\mid (x_{1},x_{2},\dots ,x_{n})\in \operatorname {dom} (f)\}$

where $\operatorname {dom} (f)$ 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 $\mathbb {R} ^{n+1}$.

Particular cases

• Graph of a function of one variable
• Graph of a function of two variables

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 $\mathbb {R} ^{n+1}$ 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:

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