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Curvilinear Coordinates

Main Ideas

This activity allows students to derive formulas for $d\rr$ in rectangular, cylindrical, and spherical coordinates, using purely geometric reasoning. These formulas form the basis of our unified view of all of vector calculus, so this activity is essential. For more information on this unified view, see our publications, especially:

Students' Task

Estimated Time: 20 min.

Using a picture as a guide, students write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Prerequisite Knowledge

  • Vectors
  • Curvilinear coordinates


Activity: Introduction

It is helpful to 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 $\rr$ that points from the origin to a point on the curve and the position vector $\rr+d\rr$ to a nearby point. Show the vector $d\rr$ and explain that it is tangent to the curve.

We often do the rectangular case, paths 1–3, for the students, to get them started quickly.

Activity: Student Conversations

Path 4 is troublesome for many students. It helps to tell the students that the strategy is always to write down $\Vec dr$ in the relevant coordinate system (rectangular, in this case). And then to “use what you know”. So, on path 4, they should write $d\rr = dx\,\ii + dy\,\jj + dz\,\kk$ and then to use the following information for the path:

  1. $z=0 \Rightarrow dz=0$
  2. $y=x \Rightarrow dy=dz$

Therefore, on path 4, $d\rr = dx\,(\ii+\jj)$

For the case of cylindrical coordinates, students who are pattern-matching will write $d\rr = dr\,\rhat + d\phi\,\phat + dz\,\zhat$. Point out that $\phi$ is dimensionless and that path two is an arc with arclength $r\,d\phi$.

Some students will remember the formula for arclength, but many will not. The following sequence of prompts can be helpful.

  1. What is the circumference of a circle?
  2. What is the arclength for a half circle?
  3. What is the arclength for the angle $\pi\over 2$?
  4. What is the arclength for the angle $\phi$?
  5. What is the arclength for the angle $d\phi$?

For the spherical case, students who are pattern matching will now write $d\rr = dr\,\rhat + d\phi\,\phat + d\theta\,\that$. It helps to draw a picture in cross-section so that they can see that the circle whose arclength gives the coefficient of $\hat \theta$ has radius $r\,\sin\theta$. It can also help to carry around a basketball to write on to talk about the three dimensional geometry of this problem.

Activity: Wrap-up

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


It is helpful if students quickly move on to using these vector differential formulas to calculate line, surface, and volume integrals. See for example the activity: Surface and Volume Elements in Cylindrical and Spherical Coordinates.

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