# Graph of a function of multiple variables

## Definition

Suppose is a real-valued function of variables . The **graph** of is a subset of , with coordinate axes , given as follows:

where denotes the domain of .

Alternatively, it is given by the equation:

For nice enough functions, this graph looks like a *hypersurface* of codimension one (and dimension ) inside .

## 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 -hyperplane |

range | project the entire graph on the -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 -coordinate is in the domain and in *no* point if the -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 -axis, the intersection with the graph has size one (if the intersection with the -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 is a subset of \{ 1,2,\dots,n\}</math>. Suppose we fix the values of the coordinates in , so we get for .

This is now a function of the remaining variables, which is a total of variables.

The graph of this function is obtained by intersecting the original graph with the affine subspace given by . Note that this subspace has dimension , and the intersection of the graph with this is expected to have dimension .

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

The graph of this is obtained by intersecting the original graph with the plane given by equations for all . There are such equations. The plane itself is parallel to the -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 |
---|---|

is continuous in at the point | Consider the graph restricted to the plane . This graph is continuous at . |

is separately continuous in all variables at the point . | The above holds for all . |

is continuous in everywhere. | The restrictions of the graph to all planes parallel to the -plane are continuous functions. |

is separately continuous in all variables everywhere. | The above holds for all . |

### Partial derivatives in graphical terms

`For further information, refer: partial derivative`

Suppose is a function of variables and suppose is a point in the domain of . Consider the graph of in given by:

For any , we define the partial derivative , also denoted , as follows:

- First, consider the intersection of the graph of with the plane given by the set of equations for all . This is a plane parallel to the -plane.
- In this plane, consider the slope of the tangent line at . This is the value of the partial derivative.

### Gradient vector in graphical terms

`For further information, refer: gradient vector`

Suppose is a function of multiple variables and suppose is a point in the domain of . We say that is differentiable at if the gradient vector exists. This is equivalent to the graph of the function having a well defined tangent hyperplane at the point . The equation of the tangent hyperplane is given by: