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Curvilinear Basis Vectors: Instructor's Guide

Main Ideas

  • basis vectors
  • cylindrical coordinates
  • spherical coordinates

Students' Task

Estimated Time: 15 min

Prerequisite Knowledge



Activity: Introduction

We usually do this activity after giving the students a brief introduction to cylindrical and spherical coordinates.

  • We highlight the different conventions between math and physics in terms of which angle is called $\theta$ and which angle is called $\phi$ (we use the physics convention that $\theta$ is the angle down from the $z$-axis and $\phi$ is the angle from the $x$-axis).
  • We emphasize surfaces of constant coordinate (e.g. in spherical coordinates, surfaces with constant $\phi$ are planes, which surfaces of constant $r$ are spherical shells).


  • Place an origin of coordinates on the floor in a back corner of the room.
  • “Close your eyes and imagine that your arm is the phi hat basis vector in spherical coordinates. Point your arm in the appropriate direction, evaluated at your right shoulder.”
  • Once everybody has committed to an answer, ask the students to open their eyes.
  • Repeat for $\hat{r}$ and $\hat{\theta}$ in spherical coordinates, and all the basis vectors in cylindrical coordinates.

Activity: Student Conversations

  • Which direction should the basis vector point? A basis vector $\widehat{coordinate}$ is the unit vector that points in the direction that $coordinate$ is changing, i.e. $\hat x$ is the unit vector that points in the direction that $x$ is changing.
  • $\hat\theta$ should point generally downward Make sure that the directions in which you pointed agree with the directions in Figure Figure \ref{Coords}.(FIXME - need figure)
  • No curved arms The basis vectors are vectors, i.e. they are straight arrows in space, even when they correspond to coordinates that are angles.
  • Basis vectors are not constant in space Although the basis vectors that correspond to rectangular coordinates $x$, $y$, and $z$ are constants, i.e. they point the same direction at each point in space; most of the basis vectors that correspond to cylindrical and spherical coordinates point in different directions at each point in space. When using basis vectors adapted to curvilinear coordinates in derivatives and integrals, it is essential to remember that these basis vectors are not constant.
  • A nice example of a field The basis vectors adapted to a single coordinate form a simple example of the geometrical notion of a vector field, i.e. a vector at every point in space. For example, the polar basis ${\hat r,\hat\phi}$ is shown in this Figure Figure \ref{polarvec}. FIXME - need figure
  • Radial basis vectors In cylindrical coordinates, the radial basis vector is always parallel to the $x$-$y$ plane (i.e. the floor of your classroom), but in spherical coordinates, the radial basis vector points directly away from the origin (i.e. is only parallel to the floor when evaluated for points on the floor).

Activity: Wrap-up

No wrap-up in needed.


Writing a vector in curvilinear coordinates.

This activity is part of a sequence of activities which address the Geometry of Vector Fields. The following activities are part of this sequence which can be used as extensions to this activity.

  • Follow-up activities:
    • Visualizing Gradient: This activity uses Mathematica to show the gradient of several scalar fields.
    • Drawing Electric Field Vectors: In direct analogy to Drawing Equipotential Surfaces, this small group activity has students sketch the electric field due to a quadrupole in the plane of the charges.
    • Visualizing Electric Flux: This computer visualization activity uses Mathematica to explore the effects of placing a point charge inside, outside, and on a cubical Gaussian surface which allows students to visualize the electric flux of a point charge through a Gaussian surface in different locations with respect to the point charge.
    • Visualizing Divergence: This computer visualization activity has students predict the sign of divergence at various points in many vector fields generated by a Mathematica notebook.
    • Visualizing Curl: Similar to Visualizing Divergence, this activity uses a Mathematica notebook of various vector fields to assist students in the geometric interpretation of the curl of a vector field by predicting the sign of the curl at various points in vector fields.

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