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\LAB{Finding $d\hbox{\boldmath$\rr$}$}{Finding \boldmath$d\rr$}
\markright{\MakeUppercase{\thesection \ Finding} $d\rr$}
\label{dr}
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%*==== Finding dr-vector ====
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\subsection{Essentials}
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%*=== Essentials ===
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\subsubsection{Main ideas}
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%*== Main ideas ==
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\Goal{
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\begin{itemize}
\item
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%*-
Introduces $d\rr$, the key to vector calculus, as a geometric object.
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\end{itemize}
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}
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\noindent
\textbf{
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%***
Don't skip this activity if you use nonrectangular basis vectors!
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}
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\footnote{
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%*((
An alternative is to present this material in lecture, rather than
as a group activity. In this case, we strongly recommend assigning the
generalizations to cylindrical and spherical coordinates as homework; see
%*Section 3.4.))
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Section~\ref{elements}.}
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\subsubsection{Prerequisites}
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%*== Prerequisites ==
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\Req{
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\begin{itemize}
\item
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Familiarity with $\rhat$ and $\phat$.
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\end{itemize}
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}
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\subsubsection{Warmup}
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%*== Warmup ==
Draw a picture on the board showing $d\rr$ as the infinitesimal change in the
position vector $\rr$ between two infinitesimally close points.
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\subsubsection{Props}
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%*== Props ==
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\begin{itemize}
\item
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whiteboards and pens
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\item
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Big arrows, perhaps made of straws, which can represent an orthonormal basis,
and which can be moved around a curve on the board.
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\end{itemize}
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\subsubsection{Wrapup}
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%*== Wrapup ==
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\begin{itemize}
\item
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Emphasize that $d\rr$ is the same geometric object regardless of how it is
expressed.
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\item
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Discuss the geometry of $ds$ as the magnitude of $d\rr$, that is, $ds=|d\rr|$.
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\item
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%*-
This is a good place to introduce the idea of ``what sort of a beast is it'';
see
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Chapter~\ref{beast}.
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%* Chapter 8.
The vector differential $d\rr$ is an infinitesimal
differential having both direction and (infinitesimal) length. When writing
an expression for $d\rr$, students should make sure that each term has these
same properties.
%*
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\end{itemize}
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\newpage
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\subsection{Details}
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%*=== Details ===
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\subsubsection{In the Classroom}
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%*== In the Classroom ==
Most groups will miss the factor of $r$ in the $\phat$ component of $d\rr$.
Watch for this as you walk around the classroom. A good thing to point out is
that $d\phi$ is not a length.
Some groups will then remember the formula for arclength and be able to figure
out the rest on their own. Other groups will need to be reminded about the
relationship between arclength and radius on a circle. A good way to do this
is to ask them for the formula for the circumference of a circle, then half a
circle, a quarter, etc. Make sure to give the angles in radians! Eventually,
they get the point.
Some students may wonder whether the top of the (Cartesian) rectangle is $\pm
dx\,\ii$. This question is ill-posed, since the sign of $dx$ itself depends
on which way you're going; you can't change your mind in the middle of a
problem. The safest way to resolve such problems is to anchor all vectors to
the same point, as shown in the figures.
For the polar rectangle, many students will realize that that there are
second-order differences between the two arcs, but few will realize that there
are also second-order differences in the radial sides, due to changes in
$\rhat$.
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\subsubsection{Subsidiary ideas}
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%*== Subsidiary ideas ==
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\Sub{
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\begin{itemize}
\item
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%*-
This is a good place to emphasize the relationship between the dot product and
the Pythagorean Theorem.
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\end{itemize}
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}
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\subsubsection{Homework}
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%*== Homework ==
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\HW{
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\begin{itemize}
\item
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%*-
Have students determine $d\rr$ in 3 dimensions in rectangular, cylindrical and
spherical coordinates; see
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Section~\ref{elements}.
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%*Section 3.4.
(Spherical coordinates are tricky; most students miss the factor of
$\sin\theta$ in the $\phat$ component.)
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\item
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Find $d\rr$ along the diagonal of a square.
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% Add picture?
% Attach worksheet?
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\end{itemize}
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}
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\subsubsection{Essay questions \None}
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%*== Essay questions ==
%* (none yet)
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\Essay{
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}
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\goodbreak
\subsubsection{Enrichment}
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%*== Enrichment ==
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\Rich{
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\begin{itemize}
\item
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%*-
Emphasize that $d\rr$ is the concept which unifies most of vector calculus.
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\item
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%*-
It may be helpful to some students to be asked to orient the arrows (see
Props) themselves at various points in the plane.
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\end{itemize}
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}
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