In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.
This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..
1. Integration Sequence | Vector Differential--Curvilinear >>
Vector Differential: Rectangular Coordinates:Find the general form for \(d\vec{r}\) in rectangular coordinates by determining \(d\vec{r}\) along the specific paths in the figure below.
- Path 1: \[d\vec{r}=\hspace{35em}\]
- Path 2: \[d\vec{r}=\hspace{35em}\]
- Path 3: \[d\vec{r}=\hspace{35em}\]
If all three coordinates are allowed to change simultaneously, by an infinitesimal amount, we could write this \(d\vec{r}\) for any path as:
\[d\vec{r}=\hspace{35em}\]
This is the general line element in rectangular coordinates.
This activity allows students to derive formulas for \(d\vec{r}\) in rectangular coordinates, using purely geometric reasoning. The formula forms the basis of a unified view of all of vector calculus, so this activity is essential. For more information on this unified view, see our publications, especially: Using differentials to bridge the vector calculus gap
Using a picture as a guide, students write down an algebraic expression for the vector differential in rectangular coordinates.
Begin by drawing a curve (like a particle trajectory, but avoid "time" in the language) and an origin on the board. Show the position vector \(\vec{r}\) that points from the origin to a point on the curve and the position vector \(\vec{r}+d\vec{r}\) to a nearby point. Show the vector \(d\vec{r}\) and explain that it is tangent to the curve.
We often do this rectangular case, paths 1-3, for the students, as a mini-lecture to get them started quickly before doing activity Vector Differential--Curvilinear.
The only wrap-up needed is to make sure that all students have (and understand the geometry of!) the correct formulas for \(d\vec{r}\).
group Small Group Activity
30 min.
vector calculus coordinate systems curvilinear coordinates
In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).
Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
assignment Homework
Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
accessibility_new Kinesthetic
10 min.
arms complex numbers Argand diagram complex plane rectangular form exponential form complex conjugate math
Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.