# Where Did I Go Wrong in Deriving Tensor Component Derivatives?

• RicardoMP
In summary, derivatives on tensor components involve taking the derivative of a tensor with respect to one or more of its components. The method of calculation depends on the type of tensor and the type of derivative. These derivatives are significant in various fields, allowing us to understand how a tensor changes in response to changes in its components. Some common applications include calculating curvature, analyzing stress and strain, and feature extraction in machine learning. However, there can be challenges and limitations, such as the complexity of the chain rule for higher-order tensors and computational intensity for large tensors. Care must also be taken when dealing with non-coordinate basis transformations.
RicardoMP
Homework Statement
I want to prove that $$-\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=\frac{1}{\not{p}}\gamma^\mu \frac{1}{\not{p}}$$
Relevant Equations
$$\not{p}=\gamma^\mu p_\mu$$
This was my attempt at a solution and was wondering where did I go wrong: $$-\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=-\frac{\partial}{\partial p_\mu}[\gamma^\nu p_\nu]^{-1}=\gamma^\nu\frac{\partial p_\nu}{\partial p_\mu}[\gamma^\sigma p_\sigma]^{-2}=\gamma^\nu\delta^\nu_\mu\frac{1}{\not{p}^2}=\gamma^\mu\frac{1}{\not{p}^2}$$
Any hint would be great! Thank you!

Apply the Leibniz rule to $$\left( \frac{\partial}{\partial p_{\mu}} \frac{1}{\not\! p} \right) \not \! p \ .$$

## 1. What are derivatives on tensor components?

Derivatives on tensor components refer to the process of taking the derivative of a tensor with respect to one or more of its components. This involves finding the rate of change of the tensor's value as one or more of its components change.

## 2. Why are derivatives on tensor components important?

Derivatives on tensor components are important because they allow us to understand how a tensor changes in response to changes in its components. This is crucial in many scientific fields, such as physics and engineering, where tensors are used to represent physical quantities.

## 3. How are derivatives on tensor components calculated?

The calculation of derivatives on tensor components depends on the type of tensor and the number of components being varied. In general, the process involves using the chain rule and product rule of calculus to find the derivative of each component and then combining them to find the overall derivative.

## 4. Can derivatives on tensor components be applied to higher-order tensors?

Yes, derivatives on tensor components can be applied to tensors of any order. However, the calculations become more complex as the order of the tensor increases, as there are more components to consider.

## 5. What are some practical applications of derivatives on tensor components?

Derivatives on tensor components have many practical applications in fields such as physics, engineering, and machine learning. They are used to understand the behavior of physical systems, optimize designs, and train machine learning models. They are also crucial in the development of advanced technologies such as artificial intelligence and quantum computing.

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