## Activity: Vector Differential--Rectangular

Vector Calculus II 23 (8 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

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 $d\vec{r}$.

• group Vector Differential--Curvilinear

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 23 (9 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 The Hill

group Small Group Activity

30 min.

##### The Hill
Vector Calculus II 23 (4 years)

In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
• group Visualization of Divergence

group Small Group Activity

30 min.

##### Visualization of Divergence
Vector Calculus II 23 (9 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.
• assignment The Path

assignment Homework

##### The Path

Vector Calculus I 2022 You are climbing a hill along the steepest path, whose slope at your current location is $1\over5$. There is another path branching off at an angle of $30^\circ$ ($\pi\over6$). How steep is it?
• group Quantifying Change

group Small Group Activity

30 min.

##### Quantifying Change

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 Directional Derivatives

group Small Group Activity

30 min.

##### Directional Derivatives
Vector Calculus I 2022

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.
• assignment Circle Trig Complex

assignment Homework

##### Circle Trig Complex
Complex Numbers Exponential Form Rectangular Form Polar Form Unit Circle Quantum Fundamentals 2023 (2 years)

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 DELETE Navigating a Hill

group Small Group Activity

30 min.

##### DELETE Navigating a Hill
Static Fields 2023 (5 years) In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
• 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)

Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.

Learning Outcomes