Parabolic coordinates form an orthogonal system, which simplifies the representation of geometric shapes like parabolas and enables straightforward calculations for vector operators due to their identical scale factors.
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$\gg$Mathematical Structures Underlying Physical Laws
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Unlike the hyperbolic coordinate system, parabolic coordinates are orthogonal. The inner product of the tangent basis vectors, $E_t \cdot E_s$, is zero. This means the coordinate lines for constant $t$ and constant $s$ always intersect at a right angle.
The coordinate lines for constant $t$ are a family of parabolas opening along the negative $x^2$ axis, while the lines for constant $s$ are a family of parabolas opening along the positive $x^2$ axis. Both sets of parabolas share the same focus at the origin $(0,0)$.
The scale factors for the system are identical: $h_t=h_s=\sqrt{t^2+s^2}$. This equality simplifies the expressions for vector operators like the gradient, divergence, curl, and Laplacian in three dimensions. The expressions for these operators are derived from general formulas for orthogonal curvilinear coordinates.
The demo provides a visual proof that the two families of parabolas always intersect at a right angle. It also shows that the position vector is a direct line from the origin to any point in space, with its components defined by the coordinate system's conversion formulas.
the orthogonal grid formed by the intersecting parabolas
the orthogonal grid formed by the intersecting parabolas
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Proving the Cross Product Rules with the Levi-Civita Symbol
Proving the Epsilon-Delta Relation and the Bac-Cab Rule
Simplifying Levi-Civita and Kronecker Delta Identities
Why a Cube's Diagonal Angle Never Changes
How the Cross Product Relates to the Sine of an Angle
Finding the Shortest Distance and Proving Orthogonality for Skew Lines
A Study of Helical Trajectories and Vector Dynamics
The Power of Cross Products: A Visual Guide to Precessing Vectors
Divergence and Curl Analysis of Vector Fields
Unpacking Vector Identities: How to Apply Divergence and Curl Rules
Commutativity and Anti-symmetry in Vector Calculus Identities
Double Curl Identity Proof using the epsilon-delta Relation
The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
Surface Parametrisation and the Verification of the Gradient-Normal Relationship
Proof and Implications of a Vector Operator Identity
Conditions for a Scalar Field Identity
Solution and Proof for a Vector Identity and Divergence Problem
Kinematics and Vector Calculus of a Rotating Rigid Body
Work Done by a Non-Conservative Force and Conservative Force
The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
Calculating the Area of a Half-Sphere Using Cylindrical Coordinates
Divergence Theorem Analysis of a Vector Field with Power-Law Components
Total Mass in a Cube vs. a Sphere
Momentum of a Divergence-Free Fluid in a Cubic Domain
Total Mass Flux Through Cylindrical Surfaces
Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field
Computing the Integral of a Static Electromagnetic Field
Surface Integral to Volume Integral Conversion Using the Divergence Theorem
Circulation Integral vs. Surface Integral
Using Stokes' Theorem with a Constant Scalar Field
Verification of the Divergence Theorem for a Rotating Fluid Flow
Integral of a Curl-Free Vector Field
Boundary-Driven Cancellation in Vector Field Integrals
Proving the Generalized Curl Theorem
Computing the Magnetic Field and its Curl from a Dipole Vector Potential
Proving Contravariant Vector Components Using the Dual Basis
Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates
Vector Field Analysis in Cylindrical Coordinates
Vector Field Singularities and Stokes' Theorem
Compute Parabolic coordinates-related properties
Analyze Flux and Laplacian of The Yukawa Potential
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