Spatial transport dossier: one vessel, three descriptions
Build and revise one dossier connecting a synthetic vessel’s geometry, mass distribution and instantaneous boundary transport. Reconcile independently derived volume and surface calculations, audit a coordinate change, retain numerical evidence and defend a changed-boundary case. This is a supported public investigation; saved text, downloaded calculations and matching an answer do not certify independent proficiency or a physical vessel design.
Planning estimate: 18 hours, including analysis, writing and revision. Use the investigation and working agreement below. Keep a single coherent report across milestones rather than unrelated short answers.
Preparation and working agreement
Physical x,y,z are in metres. Positive lengths a,b,H define E by cH≤z≤H[1−(x/a)²−(y/b)²], with dimensionless 0≤c<1. Its complete boundary is a curved top plus a horizontal floor. Density ρ=ρ₀(1+βz/H) uses ρ₀ in kg/m³ and dimensionless β≥0. Prescribed velocity u=(κx,κy,w) uses κ in s⁻¹ and w in m/s. Mass flux is J=ρu in kg/(m²·s). Volume rate and mass rate are different quantities.
Default: a=2 m,b=1 m,H=3 m,c=0,ρ₀=2 kg/m³,β=1,κ=0.1 s⁻¹,w=0.2 m/s. This is an instantaneous synthetic control volume, not a measured tank or a fluid-dynamics solution. It permits boundary transport and supplies no pressure, viscosity, wall law or momentum solution. A steady density requires source s=div J; with no source, its instantaneous time derivative must be −div J. Do not assert unchanged mass, no source and nonzero net outward mass rate together.
Freeze at least these comparisons before using public results: default versus β=0; default versus c=1/2; and unchanged default geometry under ε=0 and ε=1/2. The angular map is θ=ψ+εsinψ, with |ε|≤1/2 and 0≤ψ≤2π. Radius parameter t∈[0,1] maps to r=t√(1−c). A coordinate change does not change the physical vessel. Predict mass, centroid, signed floor/top/closed rates and which numerical disagreements should disappear with refinement.
The CPU workbench uses Simpson product quadrature at 16,32,64,128 even intervals per parameter, standard and distorted coordinates. One complete run uses exactly 44,488 parameter nodes, evaluating area and mass-flux integrands together. Measured elapsed milliseconds are separate from node counts. These deterministic nodes are not independent measurements. Refinement differences are observations, not certified error bounds. Analytic mass/flux references and circular special-case area references are separate checks; a general elliptic-area oracle is not supplied.
Download each JSON record before changing settings/reset. The workbench state is temporary, with no saved run history. Retain raw downloads, figures, exact derivations, code if used and the final dossier separately. A project draft holds up to 8,000 characters of evidence summary and artifact references, not uploaded or executed files. Self-reported guidance exposure in a download is a disclosure, not a verified claim of independence. Record public solution/assistance exposure truthfully.
A supported noncoding route requires complete derivations, labeled sketches, node checks and calculation tables. Implementation evidence additionally requires retained runnable code and an independently written checker. A decision brief cites the same mathematical evidence. Preserve initial predictions, errors and revisions. Ask a real reviewer to challenge the argument; if none does, mark independent review pending. Eighteen hours is an unvalidated planning estimate, with no time-based pass or automatic mastery.
Use each argument's planning fields, immutable revisions and revision-specific feedback. Save derivations, data provenance, numerical checks and a reproducible method in the report. External code or files should be identified by filename and instructions in your response; this page stores text and does not run code or upload artifacts.
Audit geometry, mass and boundary transport
Start with a question: does the outward rate through the curved top equal the rate through the whole boundary? Predict the floor’s contribution before calculating. The synthetic vessel has top z = H[1 − (x/a)² − (y/b)²] and floor z = cH. Its density is ρ = ρ₀(1 + βz/H), and its prescribed velocity is u = (κx, κy, w).
Here x, y and z are physical coordinates in metres. The dimensionless numbers c and β control floor height and density variation. The mass-flux vector is J = ρu. Integrating J through a surface gives kg/s; integrating velocity alone gives m³/s. These are different quantities.
This is an instantaneous mathematical model, with transport allowed across the boundary. It does not solve fluid dynamics or establish a manufacturing design. A steady density requires a source s = div J; with no source, its instantaneous change is ∂ρ/∂t = −div J. Settings and numerical records are temporary and excluded from completion.
Changing a model setting clears run evidence. Compare two settings by exporting each run before changing it. Reset restores the default geometry, but keeps your temporary guidance-exposure declaration.
Which boundary and parameter measure are you using?
The x–z section below is at y = 0. Its fixed scales are x from −3 to 3 m and z from 0 to 3 m. The horizontal floor closes the curved top; the meeting edge has zero surface area.
The short arrows show outward normal directions at x = 0: upward on the top and downward on the floor. Their lengths do not represent speed or flux magnitude.
Compare the same footprint in two coordinate systems. Both plan views use x,y from −3 to 3 m at equal scale. Dots mark sixteen angular samples at normalized radius t = 0.8, not particles or a flowing fluid. The physical boundary is unchanged.
The outlined ring on the distorted view is the inspected node below. Because |ε| ≤ 0.5, dθ/dψ = 1 + ε cos ψ is at least 0.5. The distorted angle is one-to-one and preserves orientation. Its derivative belongs in the integration measure; a similar-looking drawing does not justify omitting it.
Position R = (0.4185469, 0.4540976, 2.25) m. Tangent Rₜ = (0.8370939, 0.9081952, -3) and Rψ = (-1.229291, 0.2832628, 0) are derivatives with respect to dimensionless parameters. The selected oriented area vector is (0.8497884, 3.687872, 1.353553) m² per dt dψ. Its scalar magnitude is 4.019284 m² per dt dψ; J · area vector is 1.658103 kg/s per dt dψ. Angular derivative: 1.353553.
For the increasing-angle traversal, the top uses Rₜ × Rψ to point outward/upward. Check that order by hand. A positive vertical velocity enters through the floor and leaves through the top.
At t = 0 the polar area vector vanishes because this parameterization collapses all angles to one point. The surface itself remains a smooth Cartesian graph. Reversed traversal uses θ = 2π − ψ − ε sin ψ, so the same ψ can select a different physical point. In this symmetric model its scalar area density agrees, while signed transport reverses. Over the whole surface reversal changes flux sign and leaves area unchanged. The run table always uses the outward orientation.
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0 of 7 milestones have recorded responses. This counts submissions, not quality, independent review or project mastery.
Freeze the question, geometry and physical interpretation
Start one dossier with the default vessel’s plan view and vertical cross-section, axes/units, equations for every boundary and a manifest of the three required comparisons. Predict the mass/centroid and flux-sign changes before opening results. Define what you mean by an instantaneous balance versus a steady state and list what the prescribed velocity does not model. Record prior guidance exposure. Identify every surface needed to close the boundary, including the outward floor normal.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Derive volume, moments and the mass comparison
Derive volume I₀, vertical moments I₁=∫z dV and I₂=∫z² dV by horizontal slices. Independently set up the mass and vertical moment using x=arcosθ,y=brsinθ with z as the third coordinate; derive the Jacobian and bounds. Compute default mass/centroid and compare with β=0. Prove horizontal symmetry and density positivity; do not infer a centroid by visually averaging the drawing.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Close the boundary and audit the product rule
Derive div J without treating density as a constant. Find closed, floor and top mass rates analytically for the default and β=0 cases. Independently parameterize the top and evaluate its flux directly from tangent cross products. Diagnose two faulty reports: one applies the closed-volume flux to the top alone; another replaces div(ρu) by ρ div u. Quantify both errors for the default and explain their physical meaning.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Keep physical area while changing the parameter grid
Prove θ=ψ+εsinψ is a valid orientation-preserving coordinate map on the full angular interval. Derive the normalized-t and ψ tangent vectors, their vector area and the scalar area integrand. Hand-check a displayed noncentral node. Explain orientation reversal and diagnose the supplied omitted-angular-factor calculation. Why can its flux happen to agree while its area fails on the default ellipse?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Retain a bounded numerical audit with honest errors
Run the frozen comparisons, download records before each settings change, and retain a manifest of complete calculations. Build a table of n, node counts, area, top mass flux, coordinate disagreement, exact-reference errors where available and refinement differences. Audit one node and one contribution independently. Explain any apparent agreement of the faulty integrals, any nonmonotone refinement change, and why the elliptic area lacks a supplied exact error. Implementation evidence must include a separate checker; noncoding evidence must include independently checked arithmetic and node tables.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compare two proposals and revise the physical claim
Write a decision section comparing default c=0 with c=1/2 at the same other physical settings. Connect the lost volume, remaining mass, raised centroid, top/floor rates and computed top area. Check whether each initial prediction survived. State an explicit limited objective for the comparison, and explain why neither a lower top area nor a lower outward rate proves structural safety, efficient flow or a preferred real vessel. Preserve one repaired assumption or calculation rather than replacing the original draft.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Freeze the dossier and defend a challenged step
Freeze the manuscript, downloads, calculation sheets and code/checker if applicable. Ask a real reviewer to challenge one region bound, orientation, coordinate factor, conservation assumption or numerical inference. Record what was actually reviewed, feedback and your response, including a new calculation if needed. Prepare a reproduction guide and a concise list of unresolved limitations. If no review occurred, preserve self-review evidence and mark independent review pending; do not fabricate a reviewer or award yourself mastery.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review, defend and revise
Ask a reviewer to examine the complete reasoning, reproduce selected results and pose an unfamiliar case. Record their feedback against the specific revision discussed. Reviewer labels are entered locally and do not verify identity or judgment. A complete set of text submissions does not certify an independently defended investigation.
Targeted lesson returns
Freeze the question, geometry and physical interpretation
