Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

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The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla



to the nabla for another coordinate system, say… cylindrical coordinates.



What we’ll need:

1. The Cartesian Nabla:



2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:



3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:



How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of other variables that the first variable depends on. This is exactly what we need because we want to start out with a differential operator in terms of x,




in our Cartesian to cylindrical example and convert it to differential operators in terms of the cylindrical variables, . For x, the chain rule gives us:



NOTE: More generally, if you’re given a differential operator in terms of a variable, u, that can be expressed in terms of n other variables, v, then to convert the differential operator in terms of u to differential operators in terms of the v’s the chain rule gives:



This is explained very nicely and in more detail here and here.

OK, so let’s get to work converting the three differential operators of the Cartesian nabla into differential operators in terms of cylindrical coordinates. We start with x,



using,



We can construct the first differential term above as:



but, we really need to get all the x’s and y’s out of the mix and just have cylindrical coordinates remaining. So, using a few substitutions we get:



Now, follow the same procedure for the next differential term in the chain rule.



Remember though that r depends on x and y, so when we differentiate, we need to expand r and use the simple one-variable chain rule. Using a table of derivates for the arccosine, we get:



The final derivative is taken using the quotient rule:



We already know from our work above that



so



Now, we start the lengthy process of simplifying:



Because z is the same z as in the Cartesian system, it doesn’t depend on x, (exactly as in the Cartesian system), so we have



and our complete conversion for the x derivative term is:



Now use the same techniques to convert the differential for y:



Starting in a similar manner:



Then, evaluating the second term of the chain rule we get:



Remember again that r depends on x and y, so when we differentiate, we need to expand r and use the one variable chain rule.



The final derivative is taken using the quotient rule:



We already know from our work above that



so



Now, we simplify again:



So,



Finally, since z is not transformed between coordinate systems:



Now that we have each differential converted, we can write the nabla operator as



Which is kind of a mess! We know that when we’re finished we expect to see nicely grouped cylindrical differential terms with cylindrical unit vectors, so let’s group on the cylindrical differentials



Now remembering our original definition of cylindrical unit vectors,



We can make two more substitutions and arrive at the final result for nabla in cylindrical coordinates!