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# Surface and Volume Elements in Cylindrical and Spherical Coordinates

**Keywords:**Vector Calculus, Symmetry, Coordinate Systems, Curvilinear Coordinates, Differential Elements, Small Group Activity

### Highlights of the activity

- This small group activity is designed to help upper-division undergraduate students find differential area and volume elements in cylindrical and spherical coordinates.
- Students use the cross product definition for differential area and volume to find expressions for various surface and volume elements using rectangular, cylindrical, and spherical coordinates (e.g. area element of the side of a cylinder).
- The compare and contrast discussion highlights different groups’ geometric thinking and strategies to determine surface and volume elements in curvilinear coordinates.

### Reasons to spend class time on the activity

Many problems in E&M ask students to integrate using either cylindrical or spherical coordinates. In other problems, students may have to integrate using some other generalized coordinates. In this activity, students practice a general procedure from finding the differential area and volume elements in non-rectangular coordinates. The examples chosen (cylindrical and spherical coordinates) will be useful in many standard problems.