vectorialcalculus: Numerical Tools for University Calculus
Independent reconstruction of numerical tools for teaching
university calculus. Provides finite difference derivative approximations
with numerical sensitivity diagnostics and local linear models. Optional
interactive graphics illustrate secants, tangents, differentials and
numerical Taylor polynomials of orders one and two. Includes finite limit
probes, critical-point and concavity analysis, and Newton iteration with
explicit numerical diagnostics. Provides signed Riemann sums, partition
refinement, fixed and adaptive quadrature with error diagnostics, and
optional interactive illustrations of numerical integration. Evaluates
accumulation functions and compares their numerical derivatives with
integrands while retaining quadrature and sensitivity diagnostics. Computes
geometric areas, graph lengths, revolution volumes and surface areas,
with separate interactive region and solid illustrations. Approximates
coordinate partials, gradients and line-restriction directional derivatives
without inferring total differentiability from finite coordinate probes.
Provides candidate affine models and total differentials with actual-change
comparisons and separate plane, section and tangent illustrations. Supports
rectangular Jacobians, direct Hessian stencils and matrix refinement views
with entry-level numerical sensitivity and roundoff diagnostics. Searches
for multivariable stationary candidates with explicit starts, retained
evaluation histories and conditional second derivative classifications.
Solves square nonlinear systems with Newton iteration, Jacobian
diagnostics, optional step damping and independent zero-contour playback.
Assesses equality-constrained candidates using Lagrange multipliers,
numerical rank and conditional tangent-space curvature diagnostics.
Integrates Cartesian type I and II regions with nested fixed quadrature,
retained inner diagnostics and independent sampled-domain graphics.
Supports polar regions with a radial Jacobian and Cartesian domain views.
Integrates nested Cartesian solids in six coordinate orders, retaining
slice diagnostics with independent fiber and base-projection displays.
Supports cylindrical-coordinate solids with a radial Jacobian and
separate Cartesian fiber and polar-base views. Spherical regions use
explicit azimuth, polar angle and radius with their volume Jacobian
and an independent Cartesian radial-fiber display. Assesses implicit
related rates with feasibility, differentiability hypotheses, denominator
sensitivity diagnostics and a separate local velocity display. Provides
product partitions, signed double Riemann sample rules, sampled cell
statistics, explicit domain indicators, refinement tables and prism views.
Samples standalone planar regions with reusable strip grids and separate
filled-domain or horizontal-plate displays. Samples six faces of nested
Cartesian solids with shared endpoints and independent face/wireframe
displays, without coupling geometric sampling to quadrature. Maps stored
cylindrical parameter patches into Cartesian walls, caps and angular cuts
with explicit seam and axis-degeneracy limitations. Samples spherical
radial shells and angular cuts with explicit pole/origin degeneracies
and separate stored-face displays. Shows stored stationary candidates
on sampled scalar graphs without rerunning search or classification.
Samples spatial parametric curves, displays stored samples, and integrates
speed using supplied velocities or numerical Jacobians. Computes spatial
Frenet directions, curvature and signed torsion with derivative diagnostics
and displays stored results. Samples osculating circles and ribbons
with explicit geometry gaps and separate interactive views. Samples
spatial vector fields, estimates divergence and curl, and integrates
streamlines or time-dependent pathlines with fixed-step Runge-Kutta.
Computes signed scalar line integrals and oriented vector work in two
and three dimensions, with separate quadrature and derivative diagnostics.
Samples parametric and ruled surfaces and integrates scalar densities,
area and oriented flux over rectangular parameter domains. Compares
the two integrals in Green's circulation theorem over a planar strip
and in Stokes' theorem over a rectangular parameter patch. Compares
outward flux and integrated divergence over a Cartesian box.
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