🧪Vector Calculus in General and Orthogonal Coordinate Systems

Vector calculus in general orthogonal coordinate systems hinges on the use of mutually perpendicular coordinate surfaces and variable scale factors that modify vector operators to fit curved coordinate grids. This framework is essential for handling physical and geometrical problems with symmetries not aligned with Cartesian axes.

The "Non-Cartesian Coordinate Systems" section provides a comprehensive exploration of curvilinear coordinate systems beyond the familiar Cartesian, emphasizing their foundational principles (basis vectors, transformations, orthogonality), practical applications in visualizing fields and physical phenomena (e.g., flux, charge distributions), and analytical methods for verifying their properties and transformations, ultimately deepening the understanding of how vector fields behave in diverse spatial representations.

🎬Animated result and interactive web

how position affects the orientation and scale of basis vectors in a polar coordinate system

linear independence of tangent vectors ensures a curvilinear coordinate system can uniquely map points

how a vector can be expressed as a linear combination of tangent basis vectors

Emphasis on the coordinate system and the tangent vector basis and the dual basis

The tangent and dual basis vectors for the non-linear coordinate system highlight the inverse relationship between their magnitudes

Visualize the tangent vector basis and their orthogonality

how scale factors are derived and how the orthonormal basis vectors can change direction depending on the point in space

Both divergence-free and curl-free of the field of a point source

The flux of the point source field through a closed surface

The implication for divergence and the Dirac delta function for the point source field

The spherically symmetric field outside a point source

how a Cartesian coordinate system works within an affine space

The difference between orthogonal and non-orthogonal coordinate systems

Display Cartesian components and the corresponding polar coordinates and the polar basis vectors