Coordinate Systems

Chapter 1: Coordinate Systems

Introduction

• A coordinate system is defined as a mathematical system used to describe the position of a point in space.

• Space is represented in three dimensions (3D) using three axes that are mutually perpendicular (x, y, z).

• Basis vectors are utilized to describe the direction and scale of the axes, denoted as ⃗e_i where i indicates the direction of the unit vector (which is synonymous with basis vector).

• Three main coordinate systems for locating a point or a solid in space are:   - Cartesian Coordinates   - Cylindrical Coordinates   - Spherical Coordinates

1. Cartesian Coordinates

A. Definition

• In Cartesian coordinates, the position of a point M in space is represented by the coordinates (x, y, z).

• The vectors (⃗e_x, ⃗e_y, ⃗e_z) form an orthonormal basis. This means any vector in space can be expressed as a linear combination of the basis vectors.   - Unit Vectors:     - ⃗e_x = (1, 0, 0)     - ⃗e_y = (0, 1, 0)     - ⃗e_z = (0, 0, 1)

• A unit vector, such as ⃗e_x, is defined as a vector of magnitude 1, oriented in the direction of a given vector.

• If −→AB represents a vector that connects two points A and B, it can be decomposed as:   - $−→AB = | |−→AB| |−→e_x$

• The unit vector in the direction of −→AB can be found by dividing −→AB by its magnitude:   - $−→e_x = rac{−→AB}{∥−→AB∥}$

• A point M with coordinates (x, y, z) can be expressed as:   - $−−→OM = x \, ⃗e_x + y \, ⃗e_y + z \, ⃗e_z$

B. Elementary Displacement

• The concept of elementary displacement refers to an infinitesimal change in position from point M to point M'.

• It can be described by the vector −→dl = −−−→ MM', where each component of this vector denotes a small change (dx, dy, dz):   - $M' = M + dM$

• Therefore, the relationship for elementary displacement between M(x, y, z) and M'(x', y', z') is:   - $−→dl = dx \, ⃗e_x + dy \, ⃗e_y + dz \, ⃗e_z$

• Here, x' = x + dx, y' = y + dy, z' = z + dz.

C. Elementary Surface and Volume

• An elementary surface is denoted as dS, and an elementary volume is denoted as dV.

• An infinitely small cube of volume dV can be defined with edges of lengths dx, dy, and dz.

• The expression for elementary volume is:   - $dV = dx \, dy \, dz$

• The total volume V can be calculated as:   - $V = ext{∑} dV = ext{∫} dx \, dy \, dz$

• The elementary surface can also be summed based on each face of a cube:   - If z is constant, the surface is:   - $S = ext{∫} dS = ext{∫} dx \, dy$

2. Cylindrical Coordinates

A. Definition

• In cylindrical coordinates, the 3D position of a point M in space is represented 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 denoted as {⃗e_r, ⃗e_θ, ⃗e_z}. Unlike Cartesian coordinates, these are not fixed in space and are considered mobile.

• The position in the cylindrical system can be expressed as:   - $ext{OM⃗} = ext{OM⃗ ′} + ext{M⃗′M} = r⃗e_r + z⃗e_z$

• The basis vectors are projected on the two Cartesian axes:   - $ext{⃗e_r} = ext{cos} heta \, ext{⃗e_x} + ext{sin} heta \, ext{⃗e_y}$
    - $ext{⃗e_θ} = - ext{sin} heta \, ext{⃗e_x} + ext{cos} heta \, ext{⃗e_y}$

• The basis vectors ⃗e_r and ⃗e_θ are orthogonal, with ⃗e_θ being tangent to the angular displacement.

B. Elementary Displacement

• For a point M' (r + dr, θ + dθ, z + dz), the point displaced relative to M.

• As point M moves, the directions of the basis vectors -→e_r and -→e_θ change, but their magnitudes remain equal to 1.

• The elementary displacement vector −−−→dOM is expressed as:   - $ext{⃗dOM} = d(r−→e_r + z−→e_z)$

• Simplifying, we get:   - $ext{⃗dOM} = −→e_r \, dr + r \, d−→e_r + −→e_z \, dz$

• Since d−→e_z is invariant, we can remove it, leaving:   - $ext{⃗dOM} = −→e_r \, dr + r \, d−→e_r + −→e_z \, dz$   - $d−→e_r = −→e_θ \, dθ$ becomes a factor to consider. Thus, the relation simplifies to:   - $−−−→dOM = dr−→e_r + rdθ−→e_θ + dz−→e_z$

C. Elementary Surface and Volume

• The elementary volume in cylindrical coordinates is defined by:   - $dV = r \, dr \, dθ \, dz$

• The component rdθ accounts for variations in arc length with radial distance:

• The total volume integration considers the elementary volume:   - $V = ext{∫∫∫} dV = ext{∫}<em>{0}^{h} ext{∫}</em>{0}^{2 ext{π}} ext{∫}_{0}^{R} r \, dr \, dθ \, dz$

• Differential surface elements based on orientation include:   - Normal to the z-axis (r-θ plane): $−→dS = r \, dr \, dθ \, −→e_z$   - Normal to the r-axis (θ-z plane): $−→dS = r \, dz \, dθ \, −→e_r$   - Normal to the θ-axis (r-z plane): $−→dS = dr \, dz \, −→e_θ$

3. Spherical Coordinates

A. Definition

• In spherical coordinates, the position of a point is described using three coordinates:   - r: radial distance from the origin O   - θ: polar angle ranging from 0 to π (limited to half-turn)   - ϕ: azimuthal angle in the P(xy) plane (from 0 to 2π, indicating full rotation)

• The basis vectors are denoted as (−→e_r, −→e_θ, −→e_ϕ).

• The relationship between Cartesian coordinates and spherical coordinates is given by:   - $x = r \, ext{cos} ϕ \, ext{sin} θ$   - $y = r \, ext{sin} ϕ \, ext{sin} θ$   - $z = r \, ext{cos} θ$

B. Elementary Surface and Volume

• The elementary volume in spherical coordinates is defined by:   - $dV = r^2 \, ext{sin} θ \, dr \, dθ \, dϕ$

• The term r² results from the spherical projection and sin θ adjusts for the contribution of the angle.

• Differential surface elements depending on orientation include:   - Normal to r direction (spherical surface): $−→dS = r^2 \, ext{sin} θ \, dθ \, dϕ \, −→e_r$   - Normal to θ direction (cap-shaped surface): $−→dS = r \, ext{sin} θ \, dr \, dϕ \, −→e_θ$   - Normal to ϕ direction (sector-shaped surface): $−→dS = r \, dr \, dθ \, −→e_ϕ$

C. Applications to Volume Calculations

1. Volume of a Sphere    - Formula: $dV = r^2 \, ext{sin} θ \, dr \, dθ \, dϕ$    - Calculation can be completed by integrating this over bounds for spherical coordinates:    - $V = ext{∫∫∫} dV = ext{∫}<em>{0}^{2 ext{π}} ext{∫}</em>{0}^{ ext{π}} ext{∫}<em>{0}^{R} r^2 \, ext{sin} θ \, dr \, dθ \, dϕ$    - Steps:      - First integrate ${R^2}$ w.r.t r from 0 to R:        - $ext{∫}</em>{0}^{R} r^2 \, dr = rac{R^3}{3}$      - Next, integrate $ext{sin} θ$ from 0 to π:        - $ext{∫}<em>{0}^{ ext{π}} ext{sin} θ \, dθ = 2$      - Lastly, integrate 1 over ϕ from 0 to 2π:        - $ext{∫}</em>{0}^{2 ext{π}} 1 \, dϕ = 2 ext{π}$    - Final volume calculation results in:      - $V = rac{R^3}{3} imes 2 imes 2 ext{π} = rac{4 ext{π}R^3}{3}$

2. Volume of a Cylinder    - Formula: $dV = r \, dr \, dθ \, dz$    - Volume integration:    - $V = ext{∫∫∫} dV = ext{∫}<em>{0}^{h} ext{∫}</em>{0}^{2 ext{π}} ext{∫}<em>{0}^{R} r \, dr \, dθ \, dz$    - Steps:      - First integrate r from 0 to R:        - $ext{∫}</em>{0}^{R} r \, dr = rac{R^2}{2}$      - Then integrate over θ from 0 to 2π:        - $ext{∫}<em>{0}^{2 ext{π}} 1 \, dθ = 2 ext{π}$      - Finally, integrate over z from 0 to h:        - $ext{∫}</em>{0}^{h} 1 \, dz = h$    - Total volume of the cylinder results in:      - $V = rac{R^2}{2} imes 2 ext{π} imes h = ext{π}R^2h$

3. Volume of a Parallelepiped in Cartesian Coordinates    - Defined by dimensions a, b, and c for sides:      - x spans [0, a], y spans [0, b], z spans [0, c].    - Differential volume element is:      - $dV = dx \, dy \, dz$    - Total volume by integrating:      - $V = ext{∫∫∫} dV = ext{∫}<em>{0}^{a} ext{∫}</em>{0}^{b} ext{∫}_{0}^{c} dx \, dy \, dz$      - Each integral step results in dimensions:        - $V = a imes b imes c$

• The resulting volume of a parallelepiped defined by sides a, b, and c is:   - $V = abc$