Calculating Line Elements in Cylindrical and Spherical Coordinates ...

Calculating Line Elements inCylindrical and Spherical Coordinatesby Corinne Manogue and Katherine Meyerc○1997 Corinne A. ManogueRectangular Coordinates:The arbitrary infinitesimal displacement vector in Cartesian coordinates is:d⃗r = dx î + dy ĵ + dz ˆkGiven the cube shown below, find d⃗r on each of the three paths, leading froma to b.zPath 1: d⃗r =Path 2: d⃗r =Path 3: d⃗r =path 1abpath 3yxpath 2The expression above for d⃗r is valid for any path in rectangular coordinates.Find the appropriate expression for d⃗r for the path which goes directly froma to c as drawn below.Path 4: d⃗r =zapath 4cyxHowever, Cartesian coordinates would be a poor choice to describe a path ona cylindrically or spherically shaped surface. Next we will find an appropriateexpression in these cases.

Cylindrical Coordinates:SEE DIAGRAM ON NEXT PAGEYou will now derive the general form for d⃗r in cylindrical coordinates bydetermining d⃗r along the specific paths below.Note that an infinitesmial element of length in the ˆr direction is simply dr,just as an infinitesimal element of length in the î direction is dx. But, aninfinitesimal element of length in the ˆφ direction is not just dφ, since thiswould be an angle and does not even have the units of length.Geometrically determine the length of the three paths leading from a to band write these lengths in the corresponding boxes on the diagram.Now, remembering that d⃗r has both magnitude and direction, write downbelow the infintesimal displacement vector d⃗r along the three paths from ato b. Notice that, along any of these three paths, only one coordinate r, φ, orz is changing at a time. (i.e. along path 1, dr ≠ 0, but dφ = 0 and dz = 0).Path 1: d⃗r =Path 2: d⃗r =Path 3: d⃗r =If all three coordinates are allowed to change simultaneously, by an infinitesimalamount, we could write this d⃗r for any path as:d⃗r=This is the general line element in cylindrical coordinates.

zpath 3length=(r+dr, φ+ d φ,z+dz)=ba= (r, φ,z)path 1length=path 2length=yxφdφLine Elements in Cylindrical Coordinates

zθlength=path 1(r, θ, φ)=apath 2path 3length=dθlength=b=(r+dr, θ+d θ, φ +d φ)xφdφLine Elements in Spherical CoordinatesSpherical Coordinates:You will now derive the general form for d⃗r in spherical coordinates by determiningd⃗r along the specific paths below. As in the cylindrical case, notethat an infinitesmial element of length in the ˆθ or ˆφ direction is not justdθ or dφ. You will need to be more careful. Geometrically determine thelength of the three paths leading from a to b and write these lengths in thecorresponding boxes on the diagram. Now, remembering that d⃗r has bothmagnitude and direction, write down below the infintesimal displacementvector d⃗r along the three paths from a to b. Notice that, along any of thesethree paths, only one coordinate r, θ, or φ is changing at a time. (i.e. alongpath 1, dr ≠ 0, but dθ = 0 and dφ = 0).Path 1: d⃗r =Path 2: d⃗r =Path 3: d⃗r =y(Be careful, this is the tricky one.)If all 3 coordinates are allowed to change simultaneously, by an infinitesimalamount, we could write this d⃗r for any path as:d⃗r=This is the general line element in spherical coordinates.