Question

Use \(Y_{k}^{i} X_{j}^{k}=\delta_{j}^{i}\) to show that $$ \partial_{k}^{m} Y_{j}^{i} \equiv \frac{\partial}{\partial X_{m}^{k}} Y_{j}^{i}=-Y_{k}^{i} Y_{j}^{m} $$

Short answer

By differentiating the given tensor relation with respect to \(X_{m}^{k}\), and applying the chain rule for differentiation, then simplifying and substituting back, we proved \(\partial_{k}^{m} Y_{j}^{i} \equiv \frac{\partial}{\partial X_{m}^{k}}Y_{j}^{i}=-Y_{k}^{i} Y_{j}^{m}\)

Step-by-step solution

Define the Initial Tensor Relation

Initially we are given a tensor relation: \(Y_{k}^{i} X_{j}^{k}=\delta_{j}^{i}\). This relation exists between the coefficients of two different coordinate systems that we are considering.

Differentiate the Tensor relation

We differentiate this relation with respect to \(X_{m}^{k}\), where m and k are arbitrary indices. So, we have \(\frac{\partial}{\partial X_{m}^{k}}Y_{k}^{i} X_{j}^{k}=\frac{\partial}{\partial X_{m}^{k}}\delta_{j}^{i}\)

Apply Chain Rule

By applying chain rule on the left side of the above equation, we will get two terms : a derivative of \(Y_{k}^{i}\) times \(X_{j}^{k}\) plus \(Y_{k}^{i}\) times derivative of \(X_{j}^{k}\). As Kronecker delta \(\delta_{j}^{i}\) is a constant tensor, its derivative will be zero.

Simplify the Equation

Simplify the above expression to get \(\frac{\partial Y_{k}^{i}}{\partial X_{m}^{k}}X_{j}^{k}+ Y_{k}^{i}\frac{\partial X_{j}^{k}}{\partial X_{m}^{k}} = 0. \) On further simplification, the second term on left side becomes \(Y_{k}^{i}\delta_{j}^{m}\).

Final Step

Substituting this back in our equation, we get our desired result \(\partial_{k}^{m} Y_{j}^{i} \equiv \frac{\partial}{\partial X_{m}^{k}}Y_{j}^{i}=-Y_{k}^{i} Y_{j}^{m}\).

Key concepts

Kronecker Delta

The Kronecker delta, denoted as \( \delta_j^i \), is a fundamental entity in tensor calculus used to represent an identity operation in different dimensions. It is defined as being 1 if the indices are the same, \( i = j \), and 0 otherwise. This simple yet powerful tool is used in various mathematical and engineering fields to simplify operations involving tensors.

For example, let's consider a matrix product involving the Kronecker delta: \(Y_k^i X_j^k = \delta_j^i\). In this context, the Kronecker delta ensures that the resulting matrix is an identity matrix. As observed in mathematical operations, the derivative of the Kronecker delta with respect to any variable is zero since it does not depend on any variable—it's a constant with respect to its indices.

Understanding the Kronecker delta and its properties is crucial to grasping more complex tensor operations, as it often serves as the starting point for expressing transformations between coordinate systems or in simplifying tensor equations.

Chain Rule in Calculus

The chain rule is a fundamental rule in calculus for taking the derivative of composite functions. When dealing with tensors, the chain rule helps us calculate how a tensor changes with respect to a change in its components under a transformation. Specifically, if you have a function of a function, say \( h(g(f(x))) \), the chain rule allows you to differentiate \( h \) with respect to \( x \).

In the context of tensor calculus, applying the chain rule allows breaking down a derivative that may not be directly computable into parts that are. For instance, differentiating the product of two tensors will result in the sum of two terms: one for the derivative of the first tensor while holding the second constant, and another for the derivative of the second tensor while holding the first constant. This is an essential technique for solving various problems in physics and engineering where tensor equations describe dynamic systems.

Partial Derivative

Partial derivatives are used to explore the sensitivity of a function to changes in one of its variables while keeping the other variables constant. In the context of tensors, partial derivatives measure how a tensor component changes as you slightly vary one of its associated coordinates, without considering the others.

A partial derivative is represented by \( \frac{\partial}{\partial X_{m}^{k}} \) when taking the derivative of a tensor component \( Y_j^i \) with respect to the coordinate \( X_{m}^{k} \). This concept is essential when analyzing phenomena in which different variables can change independently, such as in fluid dynamics or electromagnetic fields. In tensor calculus, partial derivatives help us understand how a tensor field varies point by point, and can lead to differential equations that describe physical laws.

Coordinate Transformations

Coordinate transformations are mathematical operations used to convert the representation of a point or vector from one coordinate system to another. This is particularly important in tensor calculus as tensors are geometric objects that do not depend on the coordinate system used to describe them; they merely transform according to certain rules when moving from one coordinate system to another.

Coordinate transformations are necessary to ensure that physical laws remain consistent regardless of the observer's viewpoint or the measurement system. The transformation rules are encoded in the tensor equations that describe how each component of a tensor transforms under a change of coordinates. When you change coordinates, tensors will behave according to their specific transformation laws, which ultimately preserves the nature of the physical phenomena they describe. In practical applications, understanding coordinate transformations is crucial for tasks ranging from correcting satellite data to predicting weather patterns in climate models.