## Activity: Vector Differential--Rectangular

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

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