Prerequisites

Activity: Scalar Surface and Volume Elements

Using the expressions for the infinitesimal elements of length derived and discussed in § {Activity: Infinitesimal Distance in Cylindrical and Spherical Coordinates} and § {Wrap-Up: Infinitesimal Distance in Cylindrical and Spherical Coordinates}, you should find explicit formulas for the surface or volume elements in each of the following cases:

  • the scalar surface elements for the three surfaces (top, bottom, and curved side) of a cylinder with finite length;
  • the scalar surface element for a horizontal plane through the $z$-axis;
  • the scalar surface element for the surface of a sphere.
  • the volume element of a cylinder;
  • the volume element for a sphere.

You may find the diagrams in § {Infinitesimal Distance in Cylindrical and Spherical Coordinates} to be helpful.