Orthogonal Coordinate Systems (Spherical Coordinates)

Flashcards covering key concepts, definitions, and relationships within spherical coordinate systems and vector operations.

Spherical Coordinate System

A three-dimensional orthogonal coordinate system that describes the position of a point using three parameters: the radial distance r, the polar angle θ, and the azimuthal angle φ.

Spherical Base Unit Vectors

The unit vectors âᵣ, âθ, and âφ which are directed towards increasing values of r, θ, and φ respectively, and are dependent on the coordinates θ and φ.

Orthogonality of Spherical Base Vectors

A property of spherical base unit vectors (âᵣ, âθ, âφ) where the dot product of any two distinct vectors is zero, and the dot product of a vector with itself is one.

Cross-Product Relationships of Spherical Base Vectors

The anti-commutative cyclic relationships between spherical base unit vectors, such as âᵣ × âθ = âφ, which aid in vector cross product calculations.

Vector Representation in Spherical Coordinates

Expressing a vector A as a sum of its components along the spherical base unit vectors, A = Aᵣâᵣ + Aθâθ + Aφâφ.

Spherical Vector Component

The scalar projection of a vector onto a specific spherical base unit vector, found by the dot product Aᵢ = A • âᵢ.

Magnitude of a Vector in Spherical Coordinates

The length of a vector A, calculated as the square root of the sum of the squares of its radial, polar, and azimuthal components: |A| = √(Aᵣ² + Aθ² + Aφ²).

Cross Product of Two Vectors (Spherical)

A vector operation between two vectors A and B in spherical coordinates, resulting in a new vector A x B whose components can be found using a determinant involving the base vectors and components.

Spherical Differential Volume (dv)

The infinitesimal volume element in the spherical coordinate system, given by the expression dv = r² sin θ dr dθ dφ.

Spherical Differential Surface Element (ds)

An infinitesimal area element in the spherical coordinate system, such as dsᵣ = r² sin θ dθ dφ for a surface perpendicular to the radial direction.

Spherical Differential Displacement Vector (dℓ)

An infinitesimal vector representing a directed distance along a line contour in spherical coordinates, expressed as dℓ = drâᵣ + r dθâθ + r sin θ dφâφ.

Coordinate System Conversion (Vectors)

The two-step process of changing a vector's representation from one coordinate system to another, involving converting coordinates and then individually calculating new components via dot products with new base vectors.

Position Vector

A vector r that extends from the origin to a specific point, uniquely identifying its location in space.

Position Vector Representations

Different forms of the position vector r in various coordinate systems, such as xâₓ + yâᵧ + zâ₂ (Cartesian), ρâᵨ + zâ₂ (cylindrical), and râᵣ (spherical).