IntermediateTopic 44 of 104Engineering Mathematics
Coordinate systems
Cartesian, cylindrical and spherical coordinates match the symmetry of the problem.
In depth
Cartesian (x,y,z), cylindrical (r,θ,z) and spherical (ρ,θ,φ) systems describe positions. Choose the one matching symmetry: cylinders for pipes and shafts, spheres for point sources. Volume elements are r dr dθ dz and ρ² sinφ dρ dθ dφ.
Key points
- Convert with trig
- Unit vectors may change direction
- Scale factors appear in gradient and divergence
ExampleThe volume of a sphere is ∫ρ²sinφ over all angles = 4πR³/3.
What is Coordinate systems?
Cartesian, cylindrical and spherical coordinates match the symmetry of the problem.
Why is it important to understand coordinate systems?
The right coordinate system removes complicated geometry from a problem, so cylindrical or spherical forms are chosen to match symmetry.
When should I use cylindrical coordinates?
Use them for problems with an axis of symmetry, such as pipes, shafts and wires, where the radius and the axial position are natural variables.
How do I convert between coordinate systems?
Use the standard relations, for example x equals r cos theta and y equals r sin theta, and include the Jacobian factor when changing an integral.
How is Coordinate systems used in practice?
Typical applications include analysing stress in a pressure vessel in cylindrical coordinates, modelling heat flow from a spherical tank and describing the field of an antenna in spherical coordinates.
What should I learn before Coordinate systems?
Study "Multiple integrals" first (topic 43 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Coordinate systems?
Convert with trig. Unit vectors may change direction. Scale factors appear in gradient and divergence.
Can you give an example of Coordinate systems?
The volume of a sphere is ∫ρ²sinφ over all angles = 4πR³/3.
What comes after Coordinate systems?
The next topic is "Line and surface integrals", which builds directly on this one.
How difficult is Coordinate systems?
It is classed as Intermediate in Engineering Mathematics (topic 44 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.