The previous section developed definitions for limits, derivatives, and integrals of vector valued functions. These definitions were developed for curves which can be parameterized in one variable. In the notation of analysis these are functions \(f:\R \to \R^n\) for \(n \ge 1\) some integer. These functions are curves (topologically one dimensional). This section will develop definitions for limits, derivatives, and integrals on functions of multiple variables (\(f:\R^n \to \R\)). These functions are surfaces for the case \(f:\R^2 \to \R\) (topologically two dimensional).
Subsection2.1.1Limits for Higher Dimensional Functions
As preparation for the limit questions in this section we consider again how we can build surfaces. You can review Figure 1.3.1 and Activity 15 as needed.
Activity25.
Again, we are plotting by hand to emphasize the nature of surfaces, rather than as a skill to master. Here we analyze a surface that is not a plane and consider what the curves in it are.
We analyze the surface \(z=x^2+y^3.\) Use Geogebra’s 3D Calculator 2
(a)
Let \(x=0\text{.}\) Substitute this into the surface equation.
(i)
What shape is this equation?
(ii)
Graph the shape in Geogebra. Type \((0,t,z)\) replacing z with the equation.
(b)
Let \(x=1\text{.}\) Substitute this into the surface equation.
(i)
What shape is this equation?
(ii)
Graph this shape in Geogebra also.
(c)
Let \(x=-1\text{.}\) Substitute this into the surface equation.
(i)
What shape is this equation?
(ii)
Graph this shape in Geogebra also.
(d)
Let \(y=0\text{.}\) Substitute this into the surface equation.
(i)
What shape is this equation?
(ii)
Graph the shape in Geogebra. Type \((t,0,z)\) replacing z with the equation.
(e)
Let \(y=1\text{.}\) Substitute this into the surface equation.
(i)
What shape is this equation?
(ii)
Graph this shape in Geogebra also.
(f)
Let \(y=-1\text{.}\) Substitute this into the surface equation.
(i)
What shape is this equation?
(ii)
Graph this shape in Geogebra also.
(g)
Graph the rest of the surface by typing \(x^2+y^3\) in Geogebra.
Activity26.
Suppose Guido is romping on the surface \(z=x^2+y^2.\)
(a)
If his path (path’s shadow in the \(xy\)-plane) is \(x=t, y=1\) what is the equation \(P(t)\) of the curve he is following?
(b)
If he never steps off this path, for all practical purposes is Guido romping on a surface or a curve?
(c)
Calculate \(\lim_{t \to 0} P(t).\)
(d)
If his path is \(x=0, y=t+1\) what is the equation \(P(t)\) of the curve he is following?
(e)
Calculate \(\lim_{t \to 0} P(t).\)
(f)
If his path is \(x=t, y=t+1\) what is the equation \(P(t)\) of the curve he is following?
(g)
Calculate \(\lim_{t \to 0} P(t).\)
(h)
If his path is \(x=t^2, y=t+1\) what is the equation \(P(t)\) of the curve he is following?
(i)
Calculate \(\lim_{t \to 0} P(t).\)
Subsection2.1.2Derivation
Activity27.
These examples from single variable calculus remind us of conditions under which a limit does not exist.
(a)
Recall that the floor function \(\lfloor x \rfloor\) gives the greatest integer less than or equal to its input. \(\lfloor 7.3 \rfloor=7\) and \(\lfloor 7 \rfloor = 7\text{.}\) Evaluate the following limits.
(i)
\(\lim_{x \to 2.5} \lfloor x \rfloor\)
(ii)
\(\lim_{x \to 3.3} \lfloor x \rfloor\)
(iii)
\(\lim_{x \to 3} \lfloor x \rfloor\)
(b)
\(\lim_{x \to 0} \sin\left(\frac{1}{x}\right)\) Examining the graph may be helpful for this limit.
(c)
What two conditions cause a limit not to exist?
Activity28.
Extending the definition of limit to higher dimensions requires considering the conditions on the current definition of limit.
(a)
For functions \(f:\R \to \R\) the domain is one dimensional. We check the limit from the left and right. For a limit to exist what must be true about these directional limits?
(b)
For functions \(f:\R^2 \to \R\) the domain is two dimensional. From how many directions can a limit approach a point?
(c)
For a limit to exist what must be true about these directional limits?
Figure2.1.1.Paths for Limits on a Surface
ExercisesExercises
1.
Suppose Guido is romping on the surface \(f(x,y)=\frac{x^2-y^2}{x^2+y^2}.\)
(a)
If Guido’s route (2D) is along the path \(p(t)=(t,0),\) what is the curve (3D) he is following?
(b)
As \(t\) approaches 0 (which is \((x,y) \to (0,0)\)), what is the limit?
(c)
What is the limit as \(t\) approaches 0, if the path is \(p(t)=(0,t)\text{?}\)
(d)
What is the limit as \(t\) approaches 0, if the path is \(p(t)=(t,t)\text{?}\)
(e)
What is the limit as \(t\) approaches 0, if the path is \(p(t)=(t,mt)\) for any real number \(m\text{?}\)
(f)
What is true about \(\lim_{(x,y) \to (0,0)} f(x)\text{?}\)
2.
Suppose Guido is romping on the surface\(f(x,y)=\frac{x^2y}{x^4+y^2}.\)
(a)
What is the limit as \(t\) approaches 0, if the path is \(p(t)=(t,0)\text{?}\)
(b)
What is the limit as \(t\) approaches 0, if the path is \(p(t)=(0,t)\text{?}\)
(c)
What is the limit as \(t\) approaches 0, if the path is \(p(t)=(t,t)\text{?}\)
(d)
What is the limit as \(t\) approaches 0, if the path is \(p(t)=(t,mt)\) for any real number \(m\text{?}\)
(e)
What is the limit as \(t\) approaches 0, if the path is \(p(t)=(t,t^2)\text{?}\)
(f)
What is true about \(\lim_{(x,y) \to (0,0)} f(x)\text{?}\)
