Another service from Omega

Vector Functions: Limits, Continuity


A vector function is a function that produces a vector. The typical example is the trajectory of a particle in space. At each time "t" the particle has position vector "r(t)". In a given coordinate system we can write the componets of "r(t)" in terms of the basis vectors i,j and k. Here is an example:

> i := vector([1,0,0]): j:=vector([0,1,0]): k:=vector([0,0,1]):
> r := t -> cos(t)*i + sin(t)*j + t*k;

                      r := t -> cos(t) i + sin(t) j + t k

Hence, the positions of the particle at times t=0, t=Pi/2 and t=Pi are given by the vectors:

> r(0); r(Pi/2); r(Pi);


                                  j + 1/2 Pi k

                                   - i + Pi k

Here is a picture of this curve

> with(plots):
> spacecurve(evalm(r(t)), t=0..4*Pi, color=YELOW,axes=NORMAL);

picture a picture here


The fundamental concept of Calculus is the concept of "limit". Once you know how to compute limits, the definitions of "continuity", "derivative", and "integral" will follow easily. If you revise your Calc I notes you will soon discover that the definition of limit depends only on the concept of distance between two points. It then follows that if you know how to compute the distance between two points then you also know how to compute the limit of a function that takes values (or that it is defined) in that space of points.

The definition of limit, in words, is:
We say that the limit of the function f(x) when x approaches x0 is y0 iff we can get f(x) as close as we want from y0 by taking x sufficiently close to x0.

The equivalent epsilon (e), delta (d), definition is:
For each e > 0 there is d > 0, such that
| x - x0 | < d ===> | f(x) - y0 | < e

It suffices to replace: |z| with the magnitude of the vector z (i.e. sqrt(innerprod(z,z))) to obtain the general definition of "limit" in any number of dimensions. In particular it folows (prove it) that if the function f takes a real number t into an n-dimensional vector y = (y1,y2,...,yn) then each of the coordinates of y are real-valued functions of the real variable t or,

f(t) = y = (f1(t),f2(t),...,fn(t))


                 Limit   f(t) = z = (z1,z2,...,zn)
		 t -> t0
if and only if,
                        Limit   f1(t) = z1
                        t -> t0

                        Limit   f2(t) = z2
                        t -> t0

	                Limit   fn(t) = zn
                        t -> t0

So in this case there is nothing new to be learned. To compute the limit of a vector-valued function of a single variable t, all we need to do is to compute several limits of real-valued functions of the variable t. In this case the difference between Calc I and Calc III is that in Calc I you need to do it once but in Calc III you need to do it 3 times! So,... Just do it!

By the way the (meta)-equation:

(Calc III) = 3*(Calc I)
will appear to be true several times in this course. But watch out for exceptions to the rule!

Let's work out an example of limits with Maple. Let

> f := t -> expand(r(t)/t): `f(t)` = f(t);

                                cos(t) i   sin(t) j
                         f(t) = -------- + -------- + k
                                    t          t

and to compute the limit of f(t) when t approaches 0 all we (well in this case, Maple) need to do is to compute the limits when t goes to 0 for each of the three coordinate functions of f.

> Limit(`f(t)`,t=t0) = expand(limit(f(t),t=t0));

                                   cos(t0) i   sin(t0) j
                    Limit   f(t) = --------- + --------- + k
                    t -> t0            t0          t0

But when t0 = 0,

> Limit(`f(t)`,t=0) = limit(f(t),t=0);

                            Limit  f(t) = undefined
                            t -> 0

Which is the correct answer since the limit of the first coordinate function does not exist at 0. Here are other limits,

> Limit(`f(t)`,t=Pi/4) = expand(limit(f(t),t=Pi/4));

                                          1/2        1/2
                                         2    i     2    j
                  Limit         f(t) = 2 ------ + 2 ------ + k
                  t -> (1/4 Pi)            Pi         Pi

> evalf(",3);
                    Limit         f(t) = .896 i + .896 j + k
                    t -> (1/4 Pi)

One more limit, now at t=135 degrees,

> Limit(`f(t)`,t=3*Pi/4) = expand(simplify(limit(f(t),t=3*Pi/4)));

                                           1/2          1/2
                                          2    i       2    j
               Limit         f(t) = - 2/3 ------ + 2/3 ------ + k
               t -> (3/4 Pi)                Pi           Pi

Easy ha?

Do you remember the old tune (that you used to sing in Calc I)? it goes.... the limit of a sum is the sum of the limits... the limit of a prod.. Try to sing it in 3D! How does it sound?

Look at this,

> Limit(a*`f(t)` + b*`r(t)`,t=s) = limit(a*f(t)+b*r(t),t=s);

                                  (cos(s) i + sin(s) j + k s) (a + b s)
         Limit  a f(t) + b r(t) = -------------------------------------
         t -> s                                     s

so what do you say? Does it sound like the old tune?


We say that a vector valued function r is continuous at t=a if,
                               Limit  r(t) = r(a)
                               t -> a

Thus, the helix is continuous at t=0 but the function f defined above is not.

Link to the commands in this file
Carlos Rodriguez <>
Last modified: Wed Oct 23 13:50:44 EDT 1996