Question

Simplify the suffix notation expression \(\delta_{i j} a_{j} b_{l} c_{k} \delta_{l i}\) and write the result in vector form.

Short answer

After applying Kronecker delta functions, the expression simplifies to \(a_i b_i c_k\), which in vector form is \( (\textbf{a} \cdot \textbf{b}) \textbf{c} \) where \(\textbf{a}\), \(\textbf{b}\), and \(\textbf{c}\) are vectors.

Step-by-step solution

Understanding the Kronecker Delta

The Kronecker delta \(\delta_{ij}\) is a function on two variables, such that \(\delta_{ij} = 1\) if \(i = j\) and \(\delta_{ij} = 0\) otherwise. This means that when the subscripts are equal, any term multiplied by \(\delta_{ij}\) remains the same, but if they are different, the term is zeroed out. In the given expression \(\delta_{ij} a_j b_l c_k \delta_{li}\), the deltas will cause the indices \(i\) and \(j\), and \(l\) and \(i\), to collapse if they are equal.

Applying the Kronecker Delta

Applying \(\delta_{ij}\) first, we replace \(a_j\) with \(a_i\) as the index \(j\) will take on the value of \(i\) due to the delta function. Similarly, \(\delta_{li}\) forces \(l\) to take on the value of \(i\), which collapses \(b_l\) to \(b_i\). Now the expression simplifies to \(a_i b_i c_k\).

Vector Form Representation

The contracted expression \(a_i b_i c_k\) can be viewed as a dot product between vectors \(\textbf{a}\) and \(\textbf{b}\) followed by an element-wise product with the vector \(\textbf{c}\). So, \(a_i b_i c_k\) represents \(\textbf{a} \cdot \textbf{b}\) times the \(k\)-th component of vector \(\textbf{c}\), which may be represented as the vector \((\textbf{a} \cdot \textbf{b})\textbf{c}\).

Key concepts

Suffix Notation

Suffix notation, also referred to as 'index notation', is an efficient way to represent components of vectors and tensors in mathematical expressions. This form of notation is particularly useful in fields such as physics and engineering.

In suffix notation, each component of a vector or tensor is represented by a symbol with one or more indices (suffixes), which are typically lower-case letters from the middle of the alphabet, like 'i', 'j', and 'k'. For example, in the vector \textbf{a}, its i-th component is represented as 'a_i'.

A powerful aspect of using suffix notation is the ability to concisely express complex operations like summation over indices using the Einstein summation convention. When the same index appears twice in a single term, it implies a summation over that index. For instance, 'a_i b_i' signifies summing over the product of corresponding components of the vectors \textbf{a} and \textbf{b}, which is the dot product operation.

In our exercise, the expression \(\delta_{ij} a_{j} b_{l} c_{k} \delta_{li}\) utilizes suffix notation to indicate relationships between different vector components. Understanding these relationships is key to simplifying mathematical expressions efficiently.

Vector Calculus

Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields, typically in three-dimensional Euclidean space. It plays a crucial role in physics, engineering, and computer science, among other fields.

Essential operations of vector calculus include gradient, divergence, curl, and importantly for our context, the dot product. These operations take vectors as inputs and produce either scalar or vector quantities. They are useful in describing various physical phenomena such as electromagnetic fields, fluid dynamics, and forces.

In the context of our exercise, vector calculus helps us move from a component-wise expression of vectors in suffix notation to a more general vector form. This allows us to express complex operations in a compact and intuitive format, making it easier to visualize and solve problems involving vectors.

Dot Product

The dot product, also known as scalar product, is a fundamental operation in vector calculus which takes two vectors and returns a scalar. It is calculated as the sum of the products of the corresponding components of the two vectors.

Mathematically, the dot product of two vectors \textbf{a} and \textbf{b} is denoted by \textbf{a} \(\cdot\) \textbf{b} and defined as \(\textbf{a} \(\cdot\) \textbf{b} = a_i b_i\), where 'i' runs over all components of the vectors. The result gives us a measure of the magnitude of one vector in the direction of the other, and it is a key concept in understanding the geometric properties of vector spaces.

In our exercise, we see that the Kronecker delta functions simplify the expression by effectively performing a dot product between vectors \(\textbf{a}\) and \(\textbf{b}\), which is then further combined in an element-wise fashion with another vector \(\textbf{c}\). This offers a practical example of how the dot product can be used in conjunction with other mathematical operations to simplify vector expressions.