1. CALCULUS III CHAPTER 5: Orthogonal curvilinear coordinates
(Plane) polar coordinates
Cylindrical coordinates
Spherical coordinates
General orthogonal curvilinear coordinates
VECTOR FIELDS IN CURVILINEAR COORDINATES
Scalar and vector fields in orthogonal curvilinear coordinates
The gradient operator in curvilinear coordinates
The divergence operator in curvilinear coordinates
The curl operator in curvilinear coordinates

2. ORTHOGONAL CURVILINEAR COORDINATES

3. In the previous chapters, we have mainly worked in ℝ 𝑚 using cartesian coordinates
However, in some cases it is easier to work in other systems of coordinates: if the system has for instance spherical symmetry, it is easier to describe it using spherical coordinates
We have already seen some concepts from curvilinear coordinates:
- Calculus II
D and 3D integration of scalar functions in polar, cylindrical, and spherical variables (Chapter 1)
parametrisation of curves and surfaces in polar, cylindrical, spherical variables and others (Chapter 2)
Integrals of vector fields over parametrised surfaces (chapter 4)
In this chapter we will develop a framework where vector fields calculus can be extended to generic curvilinear coordinate systems.

4. (Plane) polar coordinates
Definition
Relation between partial derivatives
Transformation from (x,y) to (u,v): area element
In polar (u,v)=(r,θ)
𝑑𝑥𝑑𝑦→𝑟𝑑𝑟𝑑𝜃
where

6. Unit vectors in polar coordinates
are orthogonal, just like i and j, and have unit norm.

7. Reminder of cylindrical coordinates
It’s like 2D polars with a third dimension described by the cartesian coordinate 𝑧:
Comment: watch out the change of notation from previous chapters in the angle 𝜃→ φ

8. Unit vectors and area and volume elements in cylindrical coordinates
Area element
Volume element
( The Jacobian determinant of the transformation is again 𝐽=𝑟)

9. Reminder of spherical coordinates

10. General orthogonal curvilinear coordinates
In the former examples, unit vectors were orthogonal. One can construct other systems of coordinates in ℝ 𝑚 , whose coordinate lines will in general be curvilinear, and whose unit vectors will be orthogonal (that is, in every point the tangent vectors to each of the 𝑚 coordinate lines will be orthogonal).
For the sake of being concrete, we focus on ℝ 3 (𝑚=3)

11. General orthogonal curvilinear coordinates
Cartesian, cylindrical, and spherical coordinates as special cases of curvilinear coordinates:

12. VECTOR FIELDS IN CURVILINEAR COORDINATES

13. Scalar fields in orthogonal curvilinear coordinates

14. Vector fields in orthogonal curvilinear coordinates projection in different basis
Coordinate systems act as basis where vector functions and vector fields can be expressed (projected).
Up to know we have expressed vector fields in cartesian coordinate system
Any vector field can be indeed expressed in other basis:
Each component can be found using the matrix transformation R

15. Vector fields in orthogonal curvilinear coordinates projection in different basis
Recipe
Express 𝐹 𝑥 , 𝐹 𝑦 , 𝐹 𝑧 in terms of 𝜃,𝜑 through
Then you have 𝑭= 𝐹 𝑥 𝜃,𝜑 𝒊 + 𝐹 𝑦 𝜃,𝜑 𝑗 + 𝐹 𝑧 𝜃,𝜑 𝑘
Use the matrix transformation 𝑹 𝑇 to express 𝒊 in terms of 𝒆 𝑟 , 𝒆 𝜃 , 𝒆 𝜑
Proceed analogously for 𝒋 and 𝒌
Substitute those expressions in 2.
Collect terms

16. Vector fields in orthogonal curvilinear coordinates scalar and vector product
Consider two vectors v and w defined in a given curvilinear system in ℝ 3
The scalar (dot) and vector (cross) products are defined as usual

17. Vector fields in orthogonal curvilinear coordinates Vector differentiation (with respect to a variable other than position)
In this particular case things are easy. For instance, consider cylindrical coordinates
and differentiate with respect to time. Applying chain’s rule,

18. Vector fields in orthogonal curvilinear coordinates Vector differentiation with respect to position: GRADIENT OPERATOR

19. Vector fields in orthogonal curvilinear coordinates Vector differentiation with respect to position: GRADIENT OPERATOR

20. Vector fields in orthogonal curvilinear coordinates Vector differentiation with respect to position: DIVERGENCE OPERATOR
Consider a vector field 𝑭= 𝐹 1 𝑢 1 , 𝑢 2 , 𝑢 3 𝒆 1 + 𝐹 2 𝑢 1 , 𝑢 2 , 𝑢 3 𝒆 2 + 𝐹 3 𝑢 1 , 𝑢 2 , 𝑢 3 𝒆 3

21. Vector fields in orthogonal curvilinear coordinates Vector differentiation with respect to position: DIVERGENCE OPERATOR

22. Vector fields in orthogonal curvilinear coordinates Vector differentiation with respect to position: CURL OPERATOR
Consider a vector field 𝑭= 𝐹 1 𝑢 1 , 𝑢 2 , 𝑢 3 𝒆 1 + 𝐹 2 𝑢 1 , 𝑢 2 , 𝑢 3 𝒆 2 + 𝐹 3 𝑢 1 , 𝑢 2 , 𝑢 3 𝒆 3

23. Vector fields in orthogonal curvilinear coordinates Vector differentiation with respect to position: CURL OPERATOR