The Chain Rule is a formula for the derivative of the composition of two functions. Suppose the real-valued function g(x) is defined on some open subset, of the real numbers, containing the number x; and h[g(x)] is defined on some open subset of the reals containing g(x). If g is differentiable at x and h is differentiable at g(x), then the composition h o g is differentiable at x and the derivative can be computed as
Table of contents |
2 Example II 3 Proof of Chain rule 4 The Fundamental Chain Rule 5 Tensors and the chain rule as cocycle |
By subsituting (2) into (1), we get
Therefore
The chain rule is a fundamental property of all definitions of derivative and is therefore valid in much more general contexts. For instance, if E, F and G are Banach spaces (which includes Euclidean space) and f : E -> F and g : F -> G are functions, and if x is an element of E such that f is differentiable at x and g is differentiable at f(x), then the derivative of the composition g o f at the point x is given by
A particularly nice formulation of the chain rule can be achieved in the most general setting: let M, N and P be C^{k} manifolds (or even Banach-manifolds) and let f : M -> N and g : N -> P be differentiable maps. The derivative of f, denoted by df, is then a map from the tangent bundle of M to the tangent bundle of N, and we may write
As an advanced explanation of the tensor concept, one can interpret the chain rule as applied to coordinate changes also as the requirement for self-consistent concepts of tensor giving rise to tensor fields. Abstractly, we can identify the chain rule as a cocycle. It gives the consistency required to define the tangent bundle in an intrinsic way. The other vector bundles of tensors have comparable cocycles, that come from applying functorial properties of tensor constructions to the chain rule itself; which is why they also are intrinsic (read, 'natural') concepts. What can be read as the 'classical' approach to tensors tries to read this backwards - and is therefore a heuristic approach rather than a foundational one.