Final answer:
The question involves determining the correct formula for a surface area element in various coordinate systems. For small patches in Cartesian coordinates, dS = dy * dz or dS = dx * dy, while in polar coordinates, the correct formula is dS = r * dr * dθ for an infinitesimal sector of a circle.
Step-by-step explanation:
The question is pertaining to the calculation of surface area elements in various coordinate systems. When we talk about a surface area element, typically denoted as dS, we are referring to an infinitesimally small patch on the surface of a three-dimensional object. The correct formula for a surface area element depends on the coordinate system you are using.
- dS = dy * dz is typically used when dealing with a small rectangular patch on a surface in Cartesian coordinates where the patch is parallel to the yz-plane.
- dS = dx * dy is another Cartesian example, for a patch parallel to the xy-plane.
- dS = r * dr * dθ is the formula for polar coordinates, which would represent an infinitesimal sector of a circle, and it applies when you are integrating over a circular area.
- dS = dr * dθ is not complete for a surface area element as it lacks a radial component that would provide the correct dimensional area.
For a sphere of radius r, which was mentioned in the student's query, the surface area is indeed 4 * pi * r², and the volume formula is (4/3) * pi * r³. These should not be confused with the infinitesimal surface area elements as they represent the total surface area and volume of the sphere respectively.