The Importance of Taylor's Theorem

In physics, Taylor's Theorem is crucial when we need to express the value of a function at one point based on its value at a nearby point. Often, the linear approximation suffices, as we can assume a scale where the contributions of higher powers of ε are negligible.

For instance, if we know the value of f(x) at a certain point and have the first derivative f′(x), we can estimate f(x + ε) by drawing a line through the point (x, f(x)) that has a slope of f′(x). If we also know higher derivatives at x, we can refine our estimate of f(x + ε) using this additional information. The concept naturally extends to multiple variables, where we simply replace the tangent line with a tangent plane.

The tangent plane to the graph of f at point P provides an approximation for nearby points on that graph.

Taylor's Theorem also comes into play when we possess qualitative insights into how physical processes relate at adjacent points. This qualitative information can be mathematically expressed by linking it to the derivatives found in a Taylor expansion of the function describing the phenomenon in question.

For example, in my recent discussion, I explored the relationship between the velocities of two test particles in an infinitesimal parcel of fluid. The relevant empirical insight was that the motion of a parcel of incompressible Newtonian fluid comprises translation, rotation, and deformation components. By employing a linear approximation to relate the motion of one particle to that of another, I derived expressions for translation, rotation, and deformation.

Such scenarios are commonplace in both theoretical and experimental physics. Introductory courses often overlook Taylor approximations in favor of more straightforward problems with exact solutions, as these serve better educational purposes at that stage. There's an argument to be made that physics fundamentally involves the study of linearization as it applies to the natural world. Regardless of one's stance on this interpretation, a solid understanding of Taylor's Theorem is indispensable for physicists, and lacking this knowledge can hinder progress.

Example: The Multipole Expansion

Consider the scenario where we wish to analyze charge distribution within a material sample. To achieve this, we need to infer the distribution from its interaction with an electric potential. We assume complete knowledge of the applied potential, as we control the experimental parameters. Our approach will involve representing the unknown distribution as a superposition of elementary charge distributions, each of which interacts only with the nth-order terms in the Taylor expansion of the potential. This method is known as the multipole expansion.

Let ϕ denote the known potential, while ρ signifies the unknown charge density function within the sample. Let V represent the region occupied by the sample, which we assume is small and centered around the origin. The charges responsible for ϕ are located far from the sample.

The energy associated with the interaction between the two sets of charges can be expressed as the volume integral of ρϕ over V. We can formulate ϕ as a Taylor expansion around the origin, with ϕ₀ representing the potential at that point. Therefore, the potential at (x, y, z) within V is:

The subscript 0 on each derivative indicates evaluation at the origin.

The leading term in the total energy becomes:

The elementary distribution interacting with the zero-order term is a point charge located at the origin, with its value equivalent to the total charge of the sample. This point charge is referred to as the monopole, thus we designate Q as the monopole moment of the sample.

For the subsequent term, we simplify as follows:

Here, E₀ represents the electric field at the origin due to ϕ. The interaction energy for this component is given by:

The integral of ρr across the entire charge distribution is known as the dipole moment, denoted as p. Hence, the elementary distribution interacting with the first-order terms in the Taylor expansion of ϕ is a pure dipole with moment p. A pure electric dipole consists of two equal and opposite charges separated by a fixed distance. A nonzero dipole moment indicates a net separation of positive and negative charges along the line through the origin in the direction of vector p.

The video titled "Oxford Calculus: Taylor's Theorem Explained with Examples and Derivation" provides an insightful explanation of the theorem, showcasing its applications through various examples.

Next, we simplify the remaining terms by separating mixed partials, for instance:

To streamline our calculations, we replace 1/2 with 3/6. The rationale for this adjustment relates to the mathematical principles underlying multipole theory, which we cannot delve into here.

Given that no charges producing ϕ exist within V, it follows that ϕ satisfies Laplace's equation at the origin. This allows us to add any multiple of (∇²ϕ)₀ to the total sum of second derivatives without altering its value. Hence, we can add -(r²/6)(∇²ϕ)₀, where r² = x² + y² + z².

Let i and j be elements from the set of variables {x, y, z}, leading to the assignment:

The symbol δᵢⱼ is known as the Kronecker delta. For instance, if i = x and j = y, then αᵢⱼ = 3xy; if i = j = z, then αᵢⱼ = 3z² - r². The sum of second partials can be expressed as:

Now we calculate the energy integral for the second-order terms:

The integral of ραᵢⱼ across the charge distribution corresponds to the quadrupole moment, represented as a tensor in a 3×3 array with components Qᵢⱼ. The elementary distribution engaging with the second-order terms in the Taylor expansion of ϕ is a pure quadrupole, with its moment tensor defined by Qᵢⱼ. A pure quadrupole consists of two equal dipoles oriented in opposite directions. The quadrupole moment provides insights into how the distribution deviates from spherical symmetry.

Continuing this process would yield interaction energies for the octopole moment, 16-pole moment, 32-pole moment, etc. In practice, the interaction energies associated with higher-order multipole moments diminish rapidly with increasing order, allowing us to achieve a good approximation by truncating at the quadrupole moment.

Consequently, we can express the total interaction energy as:

The moments' values can be determined experimentally by observing changes in energy as ϕ varies, offering a method for analyzing the internal electrical structure of a material sample. This highlights the significance of multipole expansion theory in both molecular and atomic physics. For instance, to assess whether a substance possesses polar or non-polar molecules, one can test a sample to check for a dipole moment.

The video titled "Proof of Taylor's Theorem from Real Analysis" offers a rigorous proof of the theorem, further solidifying its foundational role in advanced mathematics.