Chapter 9: Vector Calculus
Section 9.3: Differential Operators
Example 9.3.10
Derive the expression for the curl in cylindrical coordinates.
Solution
Mathematical Solution
An extension of the results in Example 9.2.9 gives
and
so that the Cartesian form of becomes
The curl of the Cartesian vector G is then
where, by the chain rule,
The Cartesian vector becomes the cylindrical vector
=
where
so .
Maple Solution - Interactive
The Context Panel can only be invoked on the displayed form of objects. The displays in this interactive derivation are very large; a few small adjustments to notation help alleviate this problem. But the computation can seem overwhelming because of the visual clutter.
Define a sequence of the cylindrical coordinate variables.
Suppress the appearance of the arguments .
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Tools≻Tasks≻Browse: Calculus - Vector≻Vector Algebra and Settings≻Display Format for Vectors
Press the Access Settings button and select "Display as Column Vector"
Display Format for Vectors
Define F, a vector field in cylindrical coordinates
Write the free vector. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻ Conversions≻Apply Co-ordinate System≻ Conversions≻To Vector Field≻Assign to a Name≻F
Convert the cylindrical vector field F to a Cartesian vector field G
Write F. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻ Conversions≻Change Co-ordinate System
Context Panel: Assign to a Name≻G
Compute the curl of G, which possible since the form of the curl in Cartesian coordinates is known.
Convert this curl from Cartesian to cylindrical coordinates.
Common Symbols palette: Apply to G and press the Enter key.
Context Panel: Student Vector Calculus≻Conversions≻Change Co-ordinate System (See Figure 9.3.10(a).)
Context Panel: Simplify≻Assuming Positive
Context Panel: Simplify≻With Side Relations (See Figure 9.3.10(b).)
Context Panel: Apply a Command≻convert, diff (See Figure 9.3.10(c).)
Context Panel: Expand
Figure 9.3.10(a) Change coordinate system
Figure 9.3.10(b) Simplify with side relations
Figure 9.3.10(c) Apply a command
The side-relation in Figure 9.3.10(b) is , where is spelled out as "theta".
Obtain the curl of a cylindrical vector field in cylindrical coordinates, and compare
Maple Solution - Coded
Initialize
Load the Student VectorCalculus package and execute the BasisFormat command.
Install a notation-improving device
Implement the declare command in the PDEtools package.
At the present time, the Typesetting tools designed to simplify notation do not work in either of the two VectorCalculus packages. Hence, the resort to the alternate, and older, notational device that suppresses the display of the independent variables and displays partial derivatives as subscripts.
Use the VectorField command to define F as a vector field in cylindrical coordinates
Use the MapToBasis command to change to Cartesian coordinates
Use the Curl command to obtain the curl of this Cartesian vector field
Express the components of this Cartesian vector in cylindrical coordinates
Use the eval command to make the appropriate substitutions, and then apply the simplify command.
Use the MapToBasis and simplify commands to change this vector to cylindrical coordinates
Use the convert command to change to partial-derivative notation
Apply the Curl command directly to the vector field F
<< Previous Example Section 9.3 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2025. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document