Activity: Vector Differentials

Prerequisite concepts

Differential Calculus Difference / Change

Understanding both difference (how far apart two values are at one time) and change (how far apart the value of a single parameter is at two different times) is necessary for understanding derivatives.

Vector Calculus $d \vec r$ is a small displacement vector

$d \vec r$ represents a small displacement vector (i.e., it points along the direction of a step with a magnitude equal to the length of that step).

Instructor guide vfdrvectorcurvi

Vector Differentials

We use the vector differential to create a unified view of calculus. Students have a lot of difficulty writing down exact differentials and writing down equations for vectors in curvilinear coordinates. This activity allows students to use geometric reasoning to practice both of these skills. In particular, students are provided with a variety of alternate tangible representations for rectangular, cylindrical, and spherical coordinates, including oatmeal cans, globes, pineapples, and pumpkins.

Representations used

$d \vec r$

3D computer model

Concepts taught

Differential form of $\vec r$ in spherical and cylindrical coordinates

FIXME

The magnitude of $d\vec r$ is the distance between two nearby points

FIXME: What's the point of this concept?

The direction of $d\vec r$ is tangent to a path between two nearby points

FIXME: Tangent to what path? How is this useful?