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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✍️Mathematical Proof
$\complement\cdots$Counselor
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Orthogonality
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.
Geometric Interpretation
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 $(0,0)$.
Scale Factors & Operators
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.
✍️Mathematical Proof
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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
- Dot Cross and Triple Products
- 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
- The Vanishing Curl Integral
- 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
- Verification of Vector Calculus Identities in Different Coordinate Systems
- Analysis of a Divergence-Free Vector Field
- The Uniqueness Theorem for Vector Fields
- Analysis of Electric Dipole Force Field
🧄Proof and Derivation-2
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