Tensor analysis

Calculus for fields whose values are vectors or tensors, in coordinates that are allowed to bend. The algebra of tensors is taken as known; this subject asks what happens to their components when you change coordinates, and how to differentiate a field when the basis vectors themselves change from point to point. The corpus holds one account, chapter 9 and parts of chapters 10 to 13 of bowen-wang-vectors-tensors-vol-2, on a euclidean-manifold.

Coordinate systems and their transformations

A chart gives N coordinate functions; two charts give a coordinate transformation and its inverse. At each point the chart's natural basis is the set of vectors tangent to the coordinate curves. A change of chart changes the natural basis by the Jacobian, and the components of a tensor field change in the opposite, contragredient way, which is what "transformation rule" means in §45 bowen-wang-vectors-tensors-vol-2. The ring F^∞(U) of smooth scalar functions on an open set is set up first, because a tensor field is then something that behaves correctly over that ring.

Components that are not coordinates

§46 separates holonomic components (relative to a chart's natural basis) from anholonomic ones (relative to any set of N independent vector fields, not necessarily coordinate-induced). Physical components are the special case measured in unit-length directions, which is what an engineer reads off in cylindrical or spherical coordinates bowen-wang-vectors-tensors-vol-2.

Covariant differentiation

The natural basis rotates and stretches from point to point, so differentiating components alone does not give a tensor. The fix is the Christoffel symbols, which record how the natural basis changes, and the covariant derivative built from them: it gives the gradient of any tensor field in any chart, and a derivative along a curve (§47–48). In Cartesian coordinates the Christoffel symbols vanish and the covariant derivative reduces to the partial derivative bowen-wang-vectors-tensors-vol-2.

Where it leads

The Lie derivative (§49) differentiates one field along the flow of another and recovers the Lie bracket (grad v)u − (grad u)v as the vector-field case.
Antisymmetric covariant fields lead to differential-forms and the exterior derivative, which generalizes the classical curl without needing the metric.
Restricted to a hypersurface the covariant derivative becomes the surface covariant derivative, and its failure to commute gives the Riemann-Christoffel tensor (§56–59).

How the corpus's other calculus treats the same ground

vector-calculus (Corral) meets curvilinear coordinates in R³ as a change-of-variables device, without Christoffel symbols. matrix-calculus defines the derivative as a linear operator, which is the coordinate-free move this subject makes too. gallier-quaintance-math-deep teaches tensor and exterior algebra and differential calculus, but not covariant differentiation. This is the only place the spoke goes past the coordinate-patch calculus to the machinery that makes it coordinate-independent.

euclidean-manifold · differential-forms · linear-algebra · topology · mathematics