Question

Write the divergence theorem in the form of suffix notation and hence obtain the analogue of the divergence theorem for a secondrank tensor \(T_{i j}\) : $$ \iiint_{V} \frac{\partial T_{i j}}{\partial x_{j}} d V=\oiint_{S} T_{i j} n_{j} d S. $$

Short answer

The divergence theorem for a second-rank tensor in suffix notation is: \[\iiint_{V} \frac{\partial T_{i j}}{\partial x_{j}} \, d V=\oiint_{S} T_{i j}n_{j} \, d S\], with the implied summation over the index j, according to the Einstein summation convention.

Step-by-step solution

Understanding Divergence Theorem for Vector Fields

Firstly, the divergence theorem for a vector field is given by: \(\iiint_{V} abla \cdot \mathbf{F} \, dV = \oiint_{S} \mathbf{F} \cdot \mathbf{n} \, dS\). In suffix notation, assuming \(F_i\) is the ith component of the vector field \(\mathbf{F}\), the divergence \(abla \cdot \mathbf{F}\) in suffix notation becomes \(\frac{\partial F_i}{\partial x_i}\), summing over i due to the Einstein summation convention. This gives us the integral of the divergence over volume V equal to the integral of the field dotted with the normal to surface S times the surface element, written in suffix notation as: \(\iiint_{V} \frac{\partial F_{i}}{\partial x_{i}} \, dV = \oiint_{S} F_{i} n_{i} \, dS\).

Writing the Divergence Theorem for Second-Rank Tensor

For a second-rank tensor \(T_{ij}\), we take the divergence with respect to the second index j since we deal with the vector components of the tensor. Following the logic of the divergence theorem for vector fields, we express it in suffix notation: \(\iiint_{V} \frac{\partial T_{ij}}{\partial x_{j}} \, dV\), summing over j. The surface integral, analogous to the vector field scenario, will be the integral over the surface S of the tensor component \(T_{ij}\) contracted with the jth component of the normal vector, summing over j, which gives us: \(\oiint_{S} T_{ij}n_{j} \, dS\).

Resulting Divergence Theorem for a Second-Rank Tensor

Thus, combining the volume and surface integrals and applying the divergence theorem to a second-rank tensor, we obtain: \[\iiint_{V} \frac{\partial T_{i j}}{\partial x_{j}} \, d V=\oiint_{S} T_{i j}n_{j} \, d S\]. This expresses the theorem in suffix notation, with the implied summation over the repeated index j.

Key concepts

Suffix Notation

In mathematical expressions, particularly when working with vectors and tensors, suffix notation serves as a compact and efficient way to represent components without resorting to longer vector notations. An element of a vector, for instance, is represented by a single subscript, such as the ith component written as F_i. This eliminates the need for vector arrows or boldfaced characters.

In the context of calculus, suffix notation becomes immensely useful for representing the partial derivatives with respect to each variable. For example, if we're considering the gradient of a scalar field \( \phi \), the component derivative with respect to x is presented as \( \frac{\partial \phi}{\partial x_i} \), where i indicates the specific coordinate direction. When handling higher dimensions or more complex calculations, such as the divergence of a vector field, suffix notation provides clarity while reducing visual clutter, making mathematical operations more streamlined and legible.

Applying suffix notation to the divergence theorem, it transforms the theorem's statement about vector fields into a succinct form, providing a clear pathway to extend the concept to tensors of higher rank, as we'll discuss further in the next section.

Second-Rank Tensor

A second-rank tensor is an extension of the concept of vectors into a two-dimensional array of numbers that transform under a change of coordinates in a specific way. These tensors can represent various physical quantities such as stresses, strains, or moments of inertia, where the interactions or relations are not merely along a line but span a plane or involve multiple directional components.

In a three-dimensional space, a second-rank tensor T can be expressed as a 3x3 matrix, with components T_{ij} indicating the effect in the i-th direction due to a unit vector in the j-th direction. The importance of recognizing a second-rank tensor lies in its ability to encapsulate complex phenomena into a structured framework, thereby simplifying computations and conceptual understanding. For instance, in the exercise, T_{ij} stands for the components of a second-rank tensor, and these components can vary according to the position within a medium if the tensor describes a field.

The versatility of the second-rank tensor allows it to be applied to a wide array of disciplines, including continuum mechanics, materials science, and electromagnetism, demonstrating the fundamental role these tensors play in the description of physical laws.

Einstein Summation Convention

The Einstein summation convention is a notation technique that makes tensor equations more compact and easier to interpret. It is widely used in physics and engineering because of its capacity to significantly simplify the expressions involving summations of products of tensor components.

According to this convention, whenever the same index appears twice in a single term, once as an upper (contravariant) and once as a lower (covariant) index, it implies a sum over all possible values of that index. For example, the expression \( a^{i}b_{i} \) automatically implies a sum over the index i: \( a^{1}b_{1} + a^{2}b_{2} + a^{3}b_{3} \), without explicitly writing out the summation symbol \( \Sigma \).

This convention is crucial when dealing with tensor equations, as seen in the divergence theorem for second-rank tensors. With Einstein's summation, the volume integral over the partial derivative of the tensor's component \( T_{ij} \) with respect to \( x_j \) (where j is the repeated index) and the accompanying surface integral are cleanly presented without cluttering the equation with summation symbols. This simplified but powerful notation is key in theoretical and applied sciences, aiding in the elegant formulation and manipulation of complex mathematical relations.