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researcherdean.bsky.social
Multifaceted Researcher
@researcherdean.bsky.social
17 followers 1 following 72 posts
Cross-disciplinary perspective integrates knowledge and methods from multiple academic disciplines to examine a topic or solve a problem.

Notion: https://viadean.notion.site
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researcherdean.bsky.social
the covariant derivative is indispensable in non-Cartesian or curved systems that distinguishes it from the partial derivative

viadean.notion.site/the-covarian...
the covariant derivative is indispensable in non-Cartesian or curved systems that distinguishes it from the partial derivative | Animated Results
The animation vividly illustrates why the covariant derivative ( $\nabla V$ ) is indispensable in nonCartesian or curved systems, distinguishing it from the partial derivative ( $\partial V$ ). In the...
viadean.notion.site
October 1, 2025 at 7:51 AM
researcherdean.bsky.social
a 2D type (1,1) tensor under rotation under both linear transformation and non-linear transformation

viadean.notion.site/a-2D-type-1-...
a 2D type (1,1) tensor under rotation under both linear transformation and non-linear transformation | Animated Results
The defining characteristic of a true Type (1,1) Tensor Transformation is physical invariance: the resulting vector fields (the
viadean.notion.site
October 1, 2025 at 7:51 AM
researcherdean.bsky.social
display the instantaneous line element at a moving point and compare the standard Cartesian length calculation to the one using the non-orthogonal metric

viadean.notion.site/display-the-...
display the instantaneous line element at a moving point and compare the standard Cartesian length calculation to the one using the non-orthogonal metric | Animated Results
The metric tensor is the
viadean.notion.site
September 29, 2025 at 1:42 PM
researcherdean.bsky.social
visualize the underlying geometry and the tangent basis vectors defined by the metric demonstrating the non-orthogonality of the hyperbolic system and the varying scale factors
viadean.notion.site/visualize-th...
visualize the underlying geometry and the tangent basis vectors defined by the metric demonstrating the non-orthogonality of the hyperbolic system and the varying scale factors | Animated Results
The tangent vectors ( $E_u, E_v$ ) rapidly rotate and stretch as $u$ changes, even though the Christoffel symbols that depend only on $v\left(\Gamma_{u u}^v, \Gamma_{u v}^u\right)$ remain constant. Th...
viadean.notion.site
September 29, 2025 at 1:41 PM
researcherdean.bsky.social
Finding the Arc Length in Spherical Coordinates
viadean.notion.site/Finding-the-...
Finding the Arc Length in Spherical Coordinates | Animated Results
The powerful relationship between the abstract integral calculation and its concrete geometric result. The animated point visually confirms that when the radial distance ( $r= R_0$) and the polar angl...
viadean.notion.site
September 29, 2025 at 1:41 PM
researcherdean.bsky.social
Cylindrical has one scaled tangent direction while Spherical has two

viadean.notion.site/Cylindrical-...
Cylindrical has one scaled tangent direction while Spherical has two | Notion
When comparing these two dynamic bases is their scale factors (the magnitude of the tangent vectors) and how many vectors change length based on position. While both systems have basis vectors whose d...
viadean.notion.site
September 28, 2025 at 8:10 AM
researcherdean.bsky.social
the geometrical meaning of the Christoffel symbols in flat Euclidean space using the non-Cartesian spherical coordinate system

viadean.notion.site/the-geometri...
the geometrical meaning of the Christoffel symbols in flat Euclidean space using the non-Cartesian spherical coordinate system | Animated Results
Christoffel symbols are necessary to describe parallel transport in curvilinear coordinate systems, even in flat space. While a vector being parallel transported maintains a constant direction in Cart...
viadean.notion.site
September 27, 2025 at 8:17 AM
researcherdean.bsky.social
how the divergence field emerges from the antisymmetric tensor

viadean.notion.site/how-the-dive...
how the divergence field emerges from the antisymmetric tensor | Animated Results
For an antisymmetric rank-(2,0) tensor built from a scalar amplitude $t$ ($T^{12}=t, T^{21}=-t$), the divergence is the 90°-rotated gradient, $∇_a T^{ab} = (−∂_y t, ∂_x t)$. The demo shows that the an...
viadean.notion.site
September 27, 2025 at 2:05 AM
researcherdean.bsky.social
how to calculate divergence in a curved coordinate system

viadean.notion.site/how-to-calcu...
how to calculate divergence in a curved coordinate system | Notion
In a curved coordinate system, the naive divergence, which is a simple calculation of partial derivatives that becomes physically meaningless as the grid warps, must be precisely corrected by the Chri...
viadean.notion.site
September 25, 2025 at 3:20 PM
researcherdean.bsky.social
the fictitious forces that arise in non-inertial frames of reference

viadean.notion.site/the-fictitio...
the fictitious forces that arise in non-inertial frames of reference | Animated Results
The animation clearly shows that the path of the particle is entirely dependent on the observer's reference frame. An inertial observer sees a straight line, while a non-inertial (rotating) observer s...
viadean.notion.site
September 24, 2025 at 2:56 PM
researcherdean.bsky.social
the gradient of a scalar field is a covariant vector

viadean.notion.site/the-gradient...
the gradient of a scalar field is a covariant vector | Animated Results
The gradient of a scalar is a covector (1-form): its components change under coordinate transforms as $\alpha^{\prime}_a = \frac{\partial x^b}{\partial y^a}\alpha_b$. Despite the changing components, ...
viadean.notion.site
September 24, 2025 at 12:35 PM
researcherdean.bsky.social
how the magnetic field vector points in the azimuthal direction and how its magnitude changes with distance

viadean.notion.site/how-the-magn...
how the magnetic field vector points in the azimuthal direction and how its magnitude changes with distance | Animated Results
The strength of the magnetic field is inversely proportional to the distance from the current-carrying wire. This fundamental principle of electromagnetism is visualized by the length of the arrows in...
viadean.notion.site
September 24, 2025 at 6:15 AM
researcherdean.bsky.social
a tensor as a physical object that exists independently of the coordinate system used to describe it

viadean.notion.site/a-tensor-as-...
a tensor as a physical object that exists independently of the coordinate system used to describe it | Animated Results
The animation shows that while the numerical components of a tensor depend entirely on the chosen coordinate system, the physical object it represents (the ellipse) does not change. A tensor is the un...
viadean.notion.site
September 24, 2025 at 2:34 AM
researcherdean.bsky.social
visualize a general type (3,0) tensor as a 3D grid of spheres where each sphere corresponds to a component of the tensor

viadean.notion.site/visualize-a-...
visualize a general type (3,0) tensor as a 3D grid of spheres where each sphere corresponds to a component of the tensor | Animated Results
This animation that visually demonstrates the cubic growth of a general type (3,0) tensor. The program cycles through dimensions from $N=2$ to $N=7$, with each frame displaying a three-dimensional gri...
viadean.notion.site
September 23, 2025 at 2:50 PM
researcherdean.bsky.social
visualize a generic type (0, 3) tensor and then show how a zero tensor appears
viadean.notion.site/visualize-a-...
visualize a generic type (0, 3) tensor and then show how a zero tensor appears | Animated Results
Mapping Components to Space: The 3D scatter plot provides a tangible representation of the tensor's structure. Each point's position on the grid corresponds directly to a specific component based on i...
viadean.notion.site
September 23, 2025 at 12:30 PM
researcherdean.bsky.social
A tensor contraction only depends on its symmetric part if the contracted indices are symmetric in the vectors

viadean.notion.site/A-tensor-con...
A tensor contraction only depends on its symmetric part if the contracted indices are symmetric in the vectors | Animated Results
The contraction $T_{a b} v^a v^b$ computes a scalar from a tensor and a vector, and only the symmetric part of the tensor contributes. Without knowing the explicit values of $T_{a b}$ and $v^a$, the r...
viadean.notion.site
September 23, 2025 at 12:29 PM
researcherdean.bsky.social
Show symmetric and anti-symmetric components of 2D tensor represented by a stretching ellipsoid and a rotational arrow respectively

viadean.notion.site/Show-symmetr...
Show symmetric and anti-symmetric components of 2D tensor represented by a stretching ellipsoid and a rotational arrow respectively | Animated Results
Any tensor's transformation can be visually broken down into two distinct actions: a stretching or squishing component and a rotational component. The demo uses an ellipsoid glyph to represent the sym...
viadean.notion.site
September 23, 2025 at 12:28 PM
researcherdean.bsky.social
a type (2,0) tensor that is initially symmetric and the transformed tensor remains symmetric

viadean.notion.site/a-type-2-0-t...
a type (2,0) tensor that is initially symmetric and the transformed tensor remains symmetric | Animated Results
A tensor's symmetry is a fundamental property that is preserved under coordinate transformations. The demo visually confirms that even as the tensor's individual components change and the coordinate s...
viadean.notion.site
September 23, 2025 at 12:27 PM
researcherdean.bsky.social
how the components of the tensor and two vectors change as the coordinate system rotates while the final scalar value remains invariant

viadean.notion.site/how-the-comp...
how the components of the tensor and two vectors change as the coordinate system rotates while the final scalar value remains invariant | Animated Results
While the components of the tensor and vectors change as the coordinate system rotates, the final scalar value calculated from them remains constant. This demonstrates the defining property of a type ...
viadean.notion.site
September 23, 2025 at 12:26 PM
researcherdean.bsky.social
how a tensor's components change when you transform the coordinate system while the result of the contraction remains constant

Proof: viadean.notion.site/Proof-of-Coo...
September 22, 2025 at 8:34 AM
researcherdean.bsky.social
how tensors behave under a change of basis

Computational Analysis: viadean.notion.site/Verifying-Te...
September 22, 2025 at 2:31 AM
researcherdean.bsky.social
how the conductivity tensor affects the relationship between the electric field and current density in an anisotropic medium

Computational Analysis: viadean.notion.site/Analysis-of-...
September 20, 2025 at 7:23 AM
researcherdean.bsky.social
the polarisation is proportional to the electric field within an isotropic medium that is contrary to non-isotropic medium

Computational Analysis: viadean.notion.site/Tensor-Analy...
September 20, 2025 at 7:22 AM
researcherdean.bsky.social
how the difference in how an electric field propagates through a vacuum versus how it propagates through a dielectric material

Computational Analysis: viadean.notion.site/Verifying-th...
September 19, 2025 at 4:05 AM
researcherdean.bsky.social
The polar coordinate tangent vectors depend on Cartesian basis vectors

Computational Analysis: viadean.notion.site/The-Polar-Te...
September 19, 2025 at 4:04 AM