Closed-form expressions of vector gravity and magnetic field due to a rectangular disk.
Bibliographic record
Abstract
In the case of applying magnetic exploration to detect underground man-made objects precisely, it is important to calculate magnetic responses analytically due to various shapes, such as one-dimensional line segments, 2D disk types, and 3D prismatic bodies. As part of these contributions, in this study, I derive the closed-form expressions of the magnetic field of one of 2D disk types, a rectangular disk. First, the gravitational potential due to a rectangular disk parallel to the x-y plane is defined by the two-dimensional surface integral. The vector gravity can be derived by differentiating the gravitational potential in each axial direction. The surface integrals that include the multiple square roots of the distance between observation points to the rectangular disk are required. Differentiating the vector gravity once more in each axial direction yields the gravity gradient tensor. For a causal body with constant magnetization, Poisson's relation is applied to convert the gravity gradient tensor to the magnetic field. The derived expressions of magnetic response are validated by comparing them with a three-dimensional rectangular prism with thin thickness. For the inclined rectangular disk, the magnetic fields are computed by transforming the observing coordinate system to the coordinate system affixed to the rectangular disk, and then the magnetic fields can be obtained by the inverse coordinate transformation.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".