Question

If \(B_{r s}\) is an anti-symmetric tensor, show that \(B_{r r}=0\).

Short answer

Using anti-symmetric property, \(B_{rr} = -B_{rr}\), solving for \(B_{rr}\) gives \(B_{rr}=0\).

Step-by-step solution

Understanding an anti-symmetric tensor

An anti-symmetric tensor, also known as a skew-symmetric tensor, is one that changes sign upon swapping its indices. So for an anti-symmetric tensor, we have the property that for all index pairs (r, s), the relation holds that if you swap r and s, the tensor value flips sign, i.e., \(B_{rs} = -B_{sr}\).

Applying anti-symmetry to the diagonal elements

For the elements on the main diagonal of the tensor (where the indices are equal, r = s), apply the anti-symmetry property. For the diagonal element at the r-th position, we would have \(B_{rr} = -B_{rr}\).

Solving the equation for the diagonal element

In the equation, \(B_{rr} = -B_{rr}\), add \(B_{rr}\) to both sides of the equation. This results in \(2B_{rr} = 0\). Dividing both sides by 2, we get \(B_{rr} = 0\), demonstrating that any diagonal element of an anti-symmetric tensor is equal to zero.

Key concepts

Tensor Mathematics

Tensor mathematics plays a crucial role in various fields of physics and engineering, embodying concepts which extend beyond the simpler notions of scalars and vectors. A tensor can be thought of as a multidimensional array of numbers that transforms according to certain rules when changing the coordinate system.

Tensors have ranks that denote the number of indices required to describe them. For instance, a rank-0 tensor is simply a scalar, a rank-1 tensor is a vector, and higher ranks such as rank-2 or rank-3 tensors might refer to matrices or more complex geometrical entities, respectively. The power of tensor mathematics lies in its ability to succinctly capture the relationships between different directions and dimensions, which is particularly useful when dealing with physical phenomena like stress, strain, and electromagnetic fields.

One can perform various operations on tensors, much like with vectors and matrices, including addition, subtraction, contraction, and multiplication. The rules that govern these operations are strict to ensure that the outcome still behaves as a tensor. This aspect is fundamental for consistency across different coordinate frames, preserving the physical laws regardless of the observer's point of view.

Vector Calculus

Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space. It provides powerful tools like the gradient, divergence, curl, and Laplacian, used to analyze physical quantities having both magnitude and direction.

These tools help in understanding how vectors change in space, which is essential in fields such as fluid dynamics, electromagnetism, and even in computer graphics. For instance, the gradient of a scalar field represents the direction and rate of maximum increase, and the curl of a vector field indicates the rotation or twisting behavior of the field.

Real-world Applications

Practical applications of vector calculus include calculating work done by a force field, studying the flow of liquids or gases, and modeling the behavior of electromagnetic fields. Its concepts are fundamental for engineers and physicists who use it to derive key equations governing the natural world and to create simulations that predict how physical systems will behave.

Skew-Symmetric Tensor

A skew-symmetric tensor, also known as an anti-symmetric tensor, possesses a defining property where exchanging any two of its indices results in a change of sign for the corresponding component.

In more technical terms, for a skew-symmetric tensor of rank 2, denoted as \(B_{rs}\), we have the property that \(B_{rs} = -B_{sr}\) for all indices \(r\) and \(s\). This property means that flipping the order of the indices negates the value, which can lead to some interesting consequences. For example, if both indices are the same, as on the main diagonal of a matrix representation of the tensor, the element must be zero since it would be equal to its own negative.

Implications in Physics

Skew-symmetric tensors are important in physics, particularly in the formulation of electromagnetism and angular momentum, where they represent fields and forces acting in a rotational sense. This anti-symmetry is a reflection of fundamental conservation laws and often indicates the absence of certain types of scalar potential for the fields represented by these tensors.