Graph of a function of two variables
Definition
Suppose is a function of two variables
, with domain a subset
of
. The graph of
is a subset of three-dimensional Euclidean space
with coordinates
, given by the equation:
Equivalently, it is the set of points:
Pictorially, this graph looks like a surface for a nice enough function .
Another way of defining the graph is that for every point , there is precisely one point of the graph on the line
, namely the point with
.
The - and
-axes are the independent variable axes and the
-axis, also called the
-axis, is the dependent variable axis.
Aspects
Domain and range
Aspect of the function | How it can be deduced from the graph |
---|---|
domain | project the entire graph on the ![]() |
range | project the entire graph on the ![]() |
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 two variables. This says that any line parallel to the
-axis (the function value axis) intersects the graph in exactly one point if the
-pair for the line is in the domain, and intersects the graph in no point if the
-pair for the line is not in the domain. In particular, any line parallel to the
-axis must intersect the graph of a function in at most one point.
If a particular subset of violates this condition, it cannot be realized as the graph of a function.
Restriction to one variable
Graph of the restriction
Suppose we fix but allow
to vary. On the domain side, this is equivalent to considering the horizontal line
in the
-plane. Suppose we are interested in studying the restriction of the function to this line (or more precisely, the intersection of this line with
). In other words, we are interested in studying the function:
This is a function of one variable, namely . Further, the graph of this function can be obtained by intersecting the graph of the original function with the plane
in
. Note that this plane is parallel to the
-plane. In this plane, we treat
as the independent variable and
as the dependent variable.
Analogously, suppose we fix but allow
to vary. On the domain side, this is equivalent to considering the vertical line
in the
-plane. Suppose we are interested in studying the restriction of the function to this line (or more precisely, the intersection of this line with
). In other words, we are interested in studying the function:
This is a function of one variable, namely . Further, the graph of this function can be obtained by intersecting the graph of the original function with the plane
in
. Note that this plane is parallel to the
-plane. In this plane, we treat
as the independent variable and
as the dependent variable.
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 |
---|---|
![]() ![]() ![]() |
Consider the graph restricted to the plane ![]() ![]() |
![]() ![]() ![]() |
Consider the graph restricted to the plane ![]() ![]() |
![]() ![]() |
Both the above conditions. |
![]() ![]() |
The restrictions of the graph to all planes parallel to the ![]() |
![]() ![]() |
The restrictions of the graph to all planes parallel to the ![]() |
![]() |
Both the above conditions, i.e., the restrictions of the graph to all planes parallel to either the ![]() ![]() |
Partial derivatives in graphical terms
For further information, refer: partial derivative
We have the following:
Partial derivative | Graphical interpretation |
---|---|
The partial derivative ![]() ![]() |
The slope of the tangent line at ![]() ![]() ![]() |
The partial derivative ![]() ![]() |
The slope of the tangent line at ![]() ![]() ![]() |
Directional derivatives in graphical terms
For further information, refer: directional derivative
The directional derivative in the direction of a unit vector
at a point
can be determined as follows: first, intersect the graph of the function with the plane
. This plane is perpendicular to the
-plane and its intersection with the
-plane is the line through
in the direction of the unit vector
.
This intersection can be thought of as the graph of a function of one variable, where the point is treated as the origin, the direction
is the independent variable axis, and the
-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
We say that is differentiable at a point
if the gradient vector exists at the point. This is equivalent to the graph of the function having a well defined tangent plane at
. Further, the equation of this tangent plane is given by:
Another way of putting this is:
Note that it is possible that the partial derivatives both exist but the function is not differentiable. In this case, the surface does not have a well defined tangent plane at the point. Even though we can define a plane by the equation above, this is not the tangent plane, because the tangent plane does not exist.