Graph of a function of multiple variables

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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)) \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 \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_1x_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_1x_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 \ne i. There are n - 1 such equations. The plane itself is parallel to the x_ix_{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 \ne 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_ix_{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 \ne i. This is a plane parallel to the x_ix_{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)