## Activity: Vector Differential--Rectangular

Static Fields 2022 (7 years)

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..

• This activity is used in the following sequences
• Media
• group Vector Differential--Curvilinear

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 2022 (7 years)

Integration Sequence

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 Quantifying Change (Remote)

group Small Group Activity

30 min.

##### Quantifying Change (Remote)

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
• group Visualization of Divergence

group Small Group Activity

30 min.

##### Visualization of Divergence
Vector Calculus II 2022 (7 years) Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the definition of the divergence of a vector field at a point: The divergence of a vector field at a point is flux per unit volume through an infinitesimal box surrounding that point. Optionally, students can use a Mathematica notebook to verify their predictions.
• group Navigating a Hill

group Small Group Activity

30 min.

##### Navigating a Hill
Static Fields 2022 (3 years)
• group Using $pV$ and $TS$ Plots

group Small Group Activity

30 min.

##### Using $pV$ and $TS$ Plots
Energy and Entropy 2021 (2 years)

Students work out heat and work for rectangular paths on $pV$ and $TS$ plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.
• group Vector Integrals (Contour Map)

group Small Group Activity

30 min.

##### Vector Integrals (Contour Map)

• group Number of Paths

group Small Group Activity

30 min.

##### Number of Paths

Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.
• group Work By An Electric Field (Contour Map)

group Small Group Activity

30 min.

##### Work By An Electric Field (Contour Map)

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.
• accessibility_new Using Arms to Visualize Complex Numbers (MathBits)

accessibility_new Kinesthetic

10 min.

##### Using Arms to Visualize Complex Numbers (MathBits)

Arms Sequence for Complex Numbers and Quantum States

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.
• assignment Memorize $d\vec{r}$

assignment Homework

##### Memorize $d\vec{r}$
Static Fields 2022 (2 years)

Write $\vec{dr}$ in rectangular, cylindrical, and spherical coordinates.

1. Rectangular: $$\vec{dr}=$$
2. Cylindrical: $$\vec{dr}=$$
3. Spherical: $$\vec{dr}=$$

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.

## Instructor's Guide

### Main Ideas

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.

### Introduction

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.

### Wrap-up

The only wrap-up needed is to make sure that all students have (and understand the geometry of!) the correct formulas for $d\vec{r}$.

Learning Outcomes