# The operator of a distribution

• CptXray
In summary: The Attempt at a SolutionI think it has something to do with the fact that derivative of a distribution is defined with some test function ##\theta##, such that: ##T \theta' = -T' \theta##. And in more general case: ##T^{(\alpha)}\theta = (-1)^{\alpha} T \theta^{(\alpha)}##. Here for ##A(x)##: ##A^{(\alpha)} = \frac{\partial{A}}{\partial{x}}##. But here i have two derivatives of different parameters and also ##r##. I've found literature about distributions (quite few to be honest) but can't find any examples
CptXray

## Homework Statement

Let ##T## be a distribution in ##\mathcal{D}(\mathbb{R}^2)## such that:
$$T(\phi) = \int_{0}^{1}dr \int_{0}^{\pi} \phi(r, \Phi)d\Phi$$
$$\phi \in \mathcal{D}(\mathbb{R}^2)$$
calculate ##r \frac{\partial{}}{\partial{r}} \frac{\partial{}}{\partial{\Phi}}T##.

## The Attempt at a Solution

I think it has something to do with the fact that derivative of a distribution is defined with some test function ##\theta##, such that: ##T \theta' = -T' \theta##. And in more general case: ##T^{(\alpha)}\theta = (-1)^{\alpha} T \theta^{(\alpha)}##. Here for ##A(x)##: ##A^{(\alpha)} = \frac{\partial{A}}{\partial{x}}##. But here i have two derivatives of different parameters and also ##r##. I've found literature about distributions (quite few to be honest) but can't find any examples that could help me do the calculations and get the feeling about this. I'd be grateful for help and tips.

You already have the basic ingredients. Insert the definition for how a distribution derivative acts on a test function and the definition of T. The ##r## is just regular multiplication of a distribution by a function.

I tried to do that, but I'm afraid it doesn't get me anywhere. From definition:
$$\big(r\frac{\partial^2{T}}{\partial{r}\partial{\Phi}}\big)\phi = (-1)^2T\big( r\frac{\partial^2{\phi}}{\partial{r}\partial{\Phi}} \big)$$.
Evaluating the right side of this equation:
$$(-1)^2T\big( r\frac{\partial^2{\phi}}{\partial{r}\partial{\Phi}} \big) = T\big( r\frac{\partial^2{\phi}}{\partial{r}\partial{\Phi}} \big) = \int_{0}^{1}dr \int_{0}^{\pi} \big( r\frac{\partial^2{\phi}}{\partial{r}\partial{\Phi}} \big)d\Phi =$$
$$= \int_{0}^{1}dr \big( r( \frac{\partial{\phi(r, \pi)}}{\partial{r}} - \frac{\partial{\phi(r, 0)}}{\partial{r}} )\big) =$$ integrating by parts:
$$= (r(\phi(r, \pi) - \phi(r, 0)))\bigg|_{0}^{1} - \int_{0}^{1}dr(\phi (r, \pi) - \phi(r, 0)) =$$
$$= \phi(1, \pi) - \phi(1, 0) - \int_{0}^{1}\big( \phi(r, \pi) -\phi(r, 0)\big)dr$$.
I really don't know if that reasoning is correct, because I don't know how to find what that ##r\frac{\partial^{2}{T}}{\partial{r}\partial{\Phi}}## could be. I suspect that I somehow need to recreate ##\phi(r,\Phi)## but that's not obvious and seems more like guessing the answer.

Last edited:

## 1. What is the role of the operator of a distribution?

The operator of a distribution is responsible for managing the distribution of goods or services from a central location to various destinations. They oversee the logistics, transportation, and storage of products to ensure timely and efficient delivery to customers.

## 2. What skills are necessary to be successful as an operator of a distribution?

An operator of a distribution needs to have strong organizational and problem-solving skills, as well as the ability to manage and coordinate multiple tasks and teams. They should also have a good understanding of inventory management, supply chain logistics, and transportation systems.

## 3. What are the main challenges faced by operators of distribution?

Some of the main challenges faced by operators of distribution include managing inventory levels, optimizing transportation routes, and ensuring timely delivery of goods. They also have to deal with unexpected events such as delays, shortages, and equipment malfunctions.

## 4. How does technology impact the role of an operator of a distribution?

Technology has greatly impacted the role of an operator of a distribution by providing tools and systems for more efficient and accurate inventory management, route planning, and tracking of goods. It has also enabled real-time communication and data sharing, improving overall operations and customer service.

## 5. What are the potential career paths for an operator of a distribution?

An operator of a distribution can advance to higher-level positions such as distribution manager, logistics manager, or supply chain manager. They can also specialize in a particular area such as inventory control, transportation, or warehouse management. Some may also choose to start their own distribution company or consultancy business.

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