Understanding Taylor Expansion Using Tensor Notation

When dealing with vectors and tensors, the use of tensor notation can provide a more elegant and coordinate-free approach to mathematical expressions. This article explores how a Taylor expansion can be represented using tensor notation, providing a deeper understanding of the underlying concepts and simplifying the often-complex tensor formulas.

Linear Functions and Vector Spaces

In vector spaces (V) and (W), a function (f : D to W) can be considered as a (W)-valued function on a subset (D) of (V). The simplest type of function is a linear function, where for any (x, y in V) and scalar (r), we have:

When (V) and (W) are finite-dimensional, we can represent the function (f) in terms of a basis, but the coordinate-free approach is often more conceptually simpler. A linear function (A : V to W) can be denoted simply as (Ax) without explicitly mentioning the basis.

The differential of a function (f) at a point (p in V) is represented by the directional derivative (f'_{pv}). If (f) is continuously differentiable, then this directional derivative is also linear in (v in V). The second derivative of (f) at (p) is a bilinear map, which can be viewed as a tensor.

Taylor Expansion in Tensor Notation

The Taylor expansion of a function (f) at a point (p) in a vector space (V) can be expressed using tensors. For a (C^infty) function (smooth function), the Taylor expansion at (p) up to the (n)-th term is given by:

[ f(p x) f(p) sum_{k1}^{n} frac{1}{k!} f^{[k]}(p) cdot x^k R_n(x) ]

Here, (f^{[k]}) denotes the (k)-th derivative of (f), and (x^k) represents the (k)-fold tensor product of (x), i.e., (x otimes x otimes cdots otimes x) (k times).

The (k)-th derivative of (f) at (p) is a tensor, and evaluating it on a basis of (V) results in the familiar index notation for tensors. The symmetry of higher-order derivatives in the tensor notation can be expressed as:

[ f''(p)(x, y) f''(p)(y, x) ]

This symmetry simplifies the tensor formula, making the multi-linear nature of the derivatives more evident.

Tensor Notation Simplified

The transition from tensor notation to more familiar index notation involves the choice of a basis. Expanding all the multilinear operations into components in terms of the basis results in a formula that can be computed explicitly but is less interpretable in a general sense. The tensor notation, however, provides a powerful tool for understanding the underlying structure of the function and its derivatives.

Conclusion

In conclusion, the Taylor expansion of a function using tensor notation offers a coordinate-free and conceptually simpler approach to understanding the behavior of functions in vector spaces. The symmetry and multi-linear nature of the derivatives become clearer in this notation, making the analytical properties of the function more transparent.