electro C1

Chapter 1: Coordinate Systems

Introduction

A coordinate system is a mathematical system used to describe the position of a point in space.
Space is represented in 3D, using three axes that are perpendicular to each other.
In a coordinate system, basis vectors are used to describe the direction and scale of the axes, denoted as ⃗e_i.
- The index i indicates the direction of the unit vector (a synonym for basis vector).
There are three main types of coordinate systems for locating a point or solid in space:
- Cartesian coordinates
- Cylindrical coordinates
- Spherical coordinates

1. Cartesian Coordinates

A. Definition

The Cartesian coordinates are defined in 3D space with coordinates represented as M(x, y, z), where:
- x-axis: represented by vector ⃗e_x
- y-axis: represented by vector ⃗e_y
- z-axis: represented by vector ⃗e_z
- Vectors (⃗e_x, ⃗e_y, ⃗e_z) form an orthonormal basis:
- Meaning any vector in space can be expressed as a linear combination of the basis vectors.
- The unit vectors are:
  - ⃗e_x = (1, 0, 0)
  - ⃗e_y = (0, 1, 0)
  - ⃗e_z = (0, 0, 1)
A unit vector is defined as a vector with a magnitude of 1 oriented in the direction of a given vector.

B. Elementary Displacement

An elementary displacement refers to an infinitesimal (very small) change in position from a point M to M'.
It can be described by a vector −→dℓ = −−→MM', where each component of the vector represents a small variation in each direction:
- M' = M + dM
For an elementary displacement between M(x, y, z) and M'(x', y', z'), the relationship is:
- −→dℓ = dx ⃗e_x + dy ⃗e_y + dz ⃗e_z
- Where:
- x' = x + dx
- y' = y + dy
- z' = z + dz

C. Elementary Surface and Volume

An elementary surface, denoted as dS, and an elementary volume, denoted as dV, can be defined:
Consider an infinitely small cube of volume dV in space with edges of lengths dx, dy, and dz:
- This elementary volume is expressed as:
- dV = dx dy dz
The total volume can be calculated by summing all these elementary volumes:
- **V =

‬ dV =

‬ dx dy dz =
dx
dy
dz**

A surface can be obtained as the sum of elementary surfaces on each face of a cube:
- For example, if z is constant:
- **S =
dS =
dx
dy**

2. Cylindrical Coordinates

A. Definition

The 3D cylindrical coordinate system is used to describe the position of a point M in space using:
- Radial distance r
- Angle θ
- Vertical coordinate z
This system is an extension of the 2D polar coordinate system.
The basis vectors in cylindrical coordinates are {⃗e_r, ⃗e_θ, ⃗e_z}.
- Note: These vectors are not fixed in space like ( ⃗e_x, ⃗e_y, ⃗e_z); they are defined as mobile.
A point M in space is represented in coordinates (r, θ, z) with:
- OM = OM' + M'M = r⃗e_r + z⃗e_z
The basis vectors for cylindrical coordinates are projected onto the Cartesian axes:
- ⃗e_r = cos θ ⃗e_x + sin θ ⃗e_y
- ⃗e_θ = −sin θ ⃗e_x + cos θ ⃗e_y

B. Elementary Displacement

Consider a point M'(r + dr, θ + dθ, z + dz), displaced relative to M:
The basis vectors −→e_r and −→e_θ change direction as point M moves, while their magnitudes remain equal to 1.
The elementary displacement vector −−−→dOM is expressed as the differential of −−→OM:
- ⃗dOM = d(r−→e_r + z−→e_z) = −→e_r dr + r d−→e_r + −→e_z dz + z d−→e_z
We can simplify d−→e_z, which is invariant regardless of rotation, thus it is zero:
- Thus, we have:
- ⃗dOM = −→e_r dr + r d−→e_r + −→e_z dz**
Since d−→e_r/dθ = −sin θ ⃗e_x + cos θ ⃗e_y = −→e_θ**, it follows:
- d−→e_r = −→e_θ dθ
Therefore, we can express elementary displacement in cylindrical coordinates as:
- −−−→dOM = dr−→e_r + r dθ−→e_θ + dz−→e_z**

C. Elementary Surface and Volume

The elementary volume is defined by the differential elements of:
- Radial distance dr
- Angular interval dθ
- Vertical coordinate dz
Therefore:
- dV = r dr dθ dz
The term rdθ is the product of the radial distance and the angle dθ, reflecting the arc length that varies with r. Thus, the multiplier accounts for the variation in θ.
dV forms a cylindrical volume element when integrated:
- **V =

‬ dV =
rdr
dθ
dz**

When integrating the total volume V from dV, remember that the radius r must be considered during integration:
- Typically:
- θ will vary from 0 to 2π
- r will vary from 0 to R
- z will vary from 0 to h
The integral to calculate the volume in cylindrical coordinates:
- **V =

‬ dV =
rdr
dθ
dz**

The differential surface element dS in cylindrical coordinates depends on the surface orientation:
- Surface normal to the z-axis (parallel to the r-θ plane): −→dS = r dr dθ −→e_z
- Surface normal to the r-axis (radial, parallel to the θ-z plane): −→dS = r dz dθ −→e_r
- Surface normal to the θ-axis (parallel to the r-z plane): −→dS = dr dz −→e_θ

3. Spherical Coordinates

A. Definition

The spherical coordinate system describes the position of a point in space using three coordinates:
- r: radial distance OM.
- θ: polar angle (0 to π), limited to a half-turn.
- ϕ: azimuthal angle in the P(xy) plane (0 to 2π), allows for a full rotation.
The basis vectors in spherical coordinates are (−→e_r, −→e_θ, −→e_ϕ)**.
The relationship between Cartesian and spherical coordinates is given by:
- x = r cos ϕ sin θ
- y = r sin ϕ sin θ
- z = r cos θ

B. Elementary Surface and Volume

The elementary volume in spherical coordinates is defined by the product of the three infinitesimal dimensions:
- dV = r² sin θ dr dθ dϕ
- The factor r² comes from the spherical projection accounting for the surface area of a sphere of radius r; the sin θ is a geometric factor adjusting the contribution of the polar angle.
The differential surface element dS varies based on the orientation of the surface in space:
- Surface normal to the r direction (spherical surface): −→dS = r² sin θ dθ dϕ −→e_r
- Surface normal to the θ direction (cap-shaped): −→dS = r sin θ dr dϕ −→e_θ
- Surface normal to the ϕ direction (sector-shaped): −→dS = r dr dθ −→e_ϕ

C. Applications to Volume Calculations

1. Volume of a Sphere of Radius R

To calculate the volume of a sphere of radius R, we use:
- dV = r² sin θ dr dθ dϕ.
The volume V is obtained by integrating dV over the bounds of the spherical coordinates:
- **V =

‬ dV =
** from 0 to 2π in ϕ,

from 0 to π in θ, and from 0 to R in r:
V = R² dr
sin θ dθ
dϕ.
- Steps to compute the volume:
Integrate r² wrt r from 0 to R:
\int0^R r^2 dr = \left[\frac{r^3}{3}\right]0^R = \frac{R^3}{3}
Integrate sin θ wrt θ from 0 to π:
\int0^π \sin θ dθ = [−\cos θ]0^π = −\cos π + \cos 0 = 2
Integrate 1 wrt ϕ from 0 to 2π:
\int_0^{2π} 1 dϕ = 2π
- Finally, combining the three integrals yields:
V = \frac{R^3}{3} \cdot 2 \cdot 2π = \frac{4πR^3}{3}.
- Therefore, the volume of a sphere of radius R is given by:
V = \frac{4πR^3}{3}.

2. Volume of a Cylinder of Radius R and Height h

To calculate the volume of a cylinder of radius R and height h, we use:
- dV = r dr dθ dz.
The volume V is obtained by integrating dV over the bounds of the cylindrical coordinates:
- **V =
dV =
** from 0 to R in r,
- from 0 to 2π in θ, and from 0 to h in z:
- V = R dr
  dθ
  dz.
Steps to compute the volume:
1. Integrate r from 0 to R:
\int0^R r dr = \left[\frac{r^2}{2}\right]0^R = \frac{R^2}{2}
1. Integrate 1 wrt θ from 0 to 2π:
\int_0^{2π} 1 dθ = 2π
1. Integrate 1 wrt z from 0 to h:
\int_0^h 1 dz = h
Combining the three integrals, the volume of the cylinder is:
- V = \frac{R^2}{2} \cdot 2π \cdot h = πR²h.

3. Volume of a Parallelepiped in Cartesian Coordinates

In Cartesian coordinates (x, y, z), a parallelepiped with sides a, b, and c is defined by:
- x ∈ [0, a]: length in the x direction.
- y ∈ [0, b]: width in the y direction.
- z ∈ [0, c]: height in the z direction.
The differential volume element in Cartesian coordinates is:
- dV = dx dy dz.
The total volume is obtained by integrating dV over the bounds defined by the dimensions of the parallelepiped:
- **V =
dV =
** from 0 to a in x, 0 to b in y, 0 to c in z:
- V = a dx
  dy
  dz.
Steps to compute the volume:
1. Integrate 1 wrt x from 0 to a:
\int_0^a 1 dx = a
1. Integrate 1 wrt y from 0 to b:
\int_0^b 1 dy = b
1. Integrate 1 wrt z from 0 to c:
\int_0^c 1 dz = c
Combining these results yields:
- V = a · b · c.
Thus, the volume of a parallelepiped with sides a, b, and c is:
- V = abc.