Tutorials

Every page in this section is generated from a runnable script in scripts/ by Literate.jl, so each one can be executed, edited and re-run locally:

julia scripts/03_walpole.jl

The generated pages come with a pre-run Jupyter notebook and a cleaned standalone script under docs/generated_notebooks/ and docs/generated_scripts/.

Simplest first within each group.

Tensor algebra

Page	What it shows
Bases, variance and the metric	an oblique basis, dual basis and metric, covariant vs contravariant components, and where variance stops mattering
Tensor products, contractions and their identities	the ⊠/⊠ˢ algebra checked numerically, the identity that fails, isotropic projection, and why it does not commute with inversion
The Walpole basis end to end	the six as   matrices, the multiplication table, the correct decomposition of , and three parametrizations of one carbon/epoxy ply
The eight-dimensional axially invariant space	, the complex product rule, and what a five-parameter best fit silently discards
Projection and symmetry detection	fixed axis vs optimized, determinism, the detection cascade, and the cost of each path

Differential calculus

Page	What it shows
Differential operators, symbolically	the five operators in polar, cylindrical and spherical coordinates; the hoop strain that comes from the connection alone; the index-placement convention
Christoffel symbols	the definition checked term by term, every predefined system tabulated, symbolic and numerical routes compared, spheroidal harmonics
Differential operators, numerically	pointwise evaluation by AD, the Lamé hollow-sphere problem with its equilibrium residual, and sensitivities through the operators
Surfaces: fundamental forms and curvature	sphere, cylinder, plane and paraboloid; Gauss's Theorema Egregium; the array-comparison trap
Classical identities of the differential operators	every identity checked, not quoted, on four charts — including a deliberately non-orthogonal one, the only case that tests the covariant/contravariant bookkeeping

Applications in mechanics

Page	What it shows
Elastic Green's functions	the Kelvin solution in 2-D and 3-D,    matched against its closed form
A two-point Green tensor between offset frames	combining tensors that live in two different spherical charts
The elastic sphere, solved symbolically	hydrostatic and deviatoric problems, the Lamé exponents, and how they assemble into an -layer sphere
Acoustic and Hill tensors	and    , written once and run on an isotropic and a transversely isotropic stiffness

Types, differentiation and performance

Page	What it shows
Differentiation and interoperability	ForwardDiff through the algebra and through a projection, symbolic vs automatic derivatives, Tensors.jl interop
Performance of the structured types	measured speed-ups of the closed forms over the dense route, allocations, and where the structure stops paying

Where to start

If you are new to the library, read Getting started first, then Bases, variance and the metric — variance is the one notion a Cartesian-only tensor library does not have, and everything else rests on it.