Question

If \(\delta_{j i}\) denotes the Kronecker delta symbol (16.115) and a is a vector with components \(a_{j}\) \((j=1,2,3),\) prove that \(\sum_{j} \delta_{j i} a_{j}=a_{i} .\) In the same way, show that \(\sum_{j} \delta_{j i} \partial_{j}=\partial_{i},\) a result we used in proving the important identity (16.116).

Short answer

The Kronecker delta selects terms, proving \( \sum_{j} \delta_{ji} a_j = a_i \) and \( \sum_{j} \delta_{ji} \partial_j = \partial_i \).

Step-by-step solution

Understand the Kronecker Delta

The Kronecker delta \( \delta_{ji} \) is a function of two variables, \( j \) and \( i \), which is equal to 1 if \( j = i \) and 0 otherwise. This is crucial in simplifying expressions involving sums over indices.

Apply the Kronecker Delta to a Vector

The expression \( \sum_{j} \delta_{ji} a_j \) involves a sum over the index \( j \). Using the property of the Kronecker delta, we observe that only the term where \( j = i \) survives, while all other terms are zero. This simplifies to:\[\sum_{j} \delta_{ji} a_j = \delta_{ii} a_i = a_i.\]

Apply the Kronecker Delta to a Partial Derivative

Similarly, consider the expression \( \sum_{j} \delta_{ji} \partial_j \). Again, by the property of the Kronecker delta, only the term where \( j = i \) survives. Hence, we have:\[\sum_{j} \delta_{ji} \partial_j = \delta_{ii} \partial_i = \partial_i.\]

Conclusion

Both expressions \( \sum_{j} \delta_{ji} a_j = a_i \) and \( \sum_{j} \delta_{ji} \partial_j = \partial_i \) are derived using the selective nature of the Kronecker delta, highlighting its use in picking specific terms from summations based on the indices.

Key concepts

Vector Components

Vectors are fundamental entities in mathematics and physics, characterized by magnitude and direction. A vector is often represented as an ordered sequence of numbers, each corresponding to a component of the vector. For a three-dimensional vector \( \mathbf{a} \), the components are typically denoted by \( a_1, a_2, \) and \( a_3 \). These number components define the vector in terms of a chosen coordinate system.

Understanding vector components is crucial because they allow you to perform arithmetic operations like addition and multiplication in a straightforward manner. When analyzing vectors with respect to indices such as 'i' or 'j', these components play an integral role.

In essence, the decomposition of a vector into its components helps simplify complex vector algebra by breaking down vectors into elementary parts.

Index Notation

Index notation is a powerful mathematical language used to express complicated equations succinctly. It relies on indices to denote the position of elements in mathematical objects such as vectors or matrices.

For instance, if you have a vector \( a \) with components \( a_j \), the symbol \( j \) serves as an index to specify individual components. This notation simplifies expressions involving summations and multiplications.

• Index notation uses indices as subscripts.
• It facilitates operations by emphasizing the role of each component or element.
• It offers a clean and efficient way to represent mathematical operations.

This notation becomes particularly advantageous when combined with concepts like the Kronecker delta, allowing us to perform quick evaluations without manually selecting components.

Partial Derivatives

Partial derivatives are a core concept in calculus, especially when dealing with functions of several variables. A partial derivative measures how a function changes as one of its input variables changes, keeping other variables constant.

Consider a function \( f(x, y) \). The partial derivatives \( \partial f / \partial x \) and \( \partial f / \partial y \) represent how \( f \) changes with respect to \( x \) and \( y \) individually. Such derivatives are essential in multivariable calculus, providing insights into the behavior and rate of change of functions.

When used together with index notation, partial derivatives become even more powerful. For example, \( \partial_i \) might imply differentiation with respect to the \( i^{th} \) variable, making it easy to generalize across different scenarios. This notation saves time and helps streamline calculations.