Here is a security session and we'll talk apparently about cyber security and all things related to that. So the plan for today, this session is a general session of interest related to security privacy and other topics. And we'll talk about why security is important or not. What should we do and about very, very common misconceptions in security area. And the second session is more technical deep dive, which will happen in a couple of hours from now. And we will go to a different room and talk more specifically about the HS2 hardening and about answer your questions, some feedback. And we'll look in more specific details on implementations and how security the HS2 work together and how we can make our systems more compliant, more safe and robust depending on your use case. A couple of words about myself. I'm working with cyber security for more than 20 years and did a lot of stuff, including security auditing, penetration testing, compliance, privacy, various application security, engineering, teaching students and other things. I've been working with the HS2 for two years and based out of possible way. And to make this session a bit more interactive. I have a backlog of questions. It's a literally a Google document that you can access it by this link either pure code as a link security HS2.org slash ask. And then you can write your questions there, or feedback or comments as the whole overall as I speak. And either we will pick up the interesting questions and discuss them in the orders here either I can provide the answers after after the meeting in document itself, and it can be used for future reference for other things, and we can do some discussion about all the second session as well. If you need me to share your comments, you can write your name or email if you'd like to, or we can stay anonymous so it's totally up to you but it's important to get feedback for us and maybe it's more handy than to ask a question during the session. So, to begin with the main topic. So we're talking of cybersecurity here came from two different situations. The first one is once people come to us and say, Oh, we would like to do security properly, and we don't know how to start with, or we bought a very, very expensive security device a firewall or something. What should we do. This is one station. And another one is when we meet our implementers and they say, Oh, we just had a security incident. And the system is not working. People can't use the HS2. And we don't know what's happening. We're always somewhere in between two situations when we would like to do something really cool and do everything properly. And sometimes it's too early because we are, it's hard to navigate and to understand what's actually we need to do. And the second is when it's a bit late. Oh, we are in the crisis now and we need to help our implementers to resolve the issues. So, we'll talk about what's happening in between and how to approach security properly because in the world of consulting in the world of what we hear and listen, there are a lot of different ideas and trends and security is not the easiest domain to understand to get to. It will try to untie all the stores, the important stores here and talk about it a bit more in detail. And very practical question. How would you define a risk to have any ready definitions or anything to share. Okay. Typically, it's, once I asked this question, and someone has a ready answer, there are like 10, 15 different definitions of risk. And it's, and I think in the previous presentation, Bernard talked something about risks. And he definitely had some meaning. And when I talk about risk, there's also a kind of definition of that. And there is a general one that was quite recently adopted by the ISO organization. And it is as simple as that risk is an effect of uncertainty on objectives. It comes from the standard. And in certain cases, it is important to understand that there can be different interpretations or different means of that. But in general, it's an effect of uncertainty on objectives. So, important takeaways from this very, very short definition, which may be not very familiar or common to everyone is that it can be either negative or positive. So the risk, including security risk, of course, can be either a disaster, or it can also an opportunity, something that can help us. So it's not always negative risk is always connected to the context to the goals or objectives of the organization. And it's also related to the uncertainty or likelihood of the event that may or may not happen. And in order to talk about security risks and to plan and to implement our security response properly, we need to start with a very, very high level risk management exercise to understand what is really important for organization and how to deal with that. And the first exercise that we'll do is kind of looking into what are our objectives. If we had a workshop, we would do it in groups or other things. But now you can, as a personal exercise, you can try writing down, as I talk, you can try writing down the main goals of your organization and you can try to see if you have enough protection against the risks. And you don't need to share it with anyone, but it's a good kind of a mental exercise for yourself to see, oh, I'm working for the public health institution, or I'm working for NGO, or I'm working for the private company. And these are the organization's objectives, the goals. Is there anything related to cybersecurity that can happen and can damage the impact delivery or impact reaching these goals? So typically there are really a couple of things that are the main, the principal objectives of the organization and understanding the potential risk effect of them is the second thing. So once we write what's the most important, the values, the processes, it may be the kind of the core activities. Then we try to think, okay, this is what we do and here is the potential risk. The third one is we try to assess the likelihood of what bad can happen and then create an action plan. So it can be a mental exercise and we'll try to go through different steps. And once you come back home, I suggest doing this exercise with your teams. And if you do it now, we can discuss it during the next meeting or outside of the sessions because in many of the cases people start implementing security measures or without making this kind of exercise or vice versa, they haven't done that and they face the real risks before recapping what the objectives are. Let's take to make this a bit more easy. Let's look at the point of view of the defender or the organization that is being attacked. So what bad actually can happen? So the most immediate is it's a failure to provide public service. It can be a violation of the law, for example, in the case of all privacy data leaks where the citizens or employee information becomes publicly exposed. In some cases it might be a direct financial loss. It might be a reputational damage and maybe something else. Do you have any examples of what is important or what bad can happen to your organizations? Is there anything missing on this list that is important for your teams? Yes, please. I guess what's important is that data security is more of a factor. You have a lot of information to ensure that. And you see that your data is secured. I guess VHS2 do provide end to end encryption. But I mean there are other security threats as well. So you need to look around that as well. Thank you. So you mentioned encryption. And encryption is treated as one of the solid protection measures. But how do we make sure that encryption works in this case? So it is a very common idea. Every product should have a lot of security features. But using them properly is also kind of an art and a kind of knowledge and skill. That's why we typically should ensure that we select and implement these measures properly based on the goals of the organization. So yeah, it's true. So from the practical experience, if we look at the history of security incidents in the last two, three years globally, for the private sector, clearly the biggest threat is a financial loss. Because once the organization loses money, it breaks a business model. And I think the most of the organizations that suffer its banks, insurance companies, all kind of the services that retail services where you have some material assets. But this is not our case. What are the biggest threats that impact the public sector or NGOs or the organizations that don't have any immediate financial assets that are attractive to attackers? It's often mentioned and I think it's the kind of a last resort threat that it is a reputational damage. So it's often believed that reputation is something that is very important. And if you have a security incident, it can be recovered and there is a significant loss. The practice shows that in reality it is not always true. So reputation is extremely important, but it's about handling the PR response. So it means that as we proceed in time and if we handle the response properly, it's not a big problem and the organization can recover from that. And one important thing to consider is, okay, we need to have someone to respond to the incident and to manage our public communication in crisis to do things and I think we are generally covered. There were, if you heard about major security breaches of the last two, three, five, ten years, all the organizations that suffered from these breaches, they survived, they continued operating, nothing happened to them. They had a lot of problems, but they were not existential problems. They did not stop, they didn't prevent them from working further. So it looks like it's a problem, but not the major one. For the public sector, I think I've been talking to different people and I've heard I think there are two bigger problems. One is a legal non-compliance and the violation of the law for institutions that should support and demonstrate a good example of following the law. And I think all kinds of non-privacy non-compliance, technical non-compliance, data governance non-compliance is a real effect which can lead to the court claims, can lead to the different issues related to the general national data security and other things. So I think that if we look at the security from the compliance perspective, following the compliance requirements is extremely important and if something is not in place, it means that we should pay quite a lot of attention to that because it undermines any security issue, undermines the whole legal structure, the whole legal approach for the organization. In the next session, we'll talk a bit what kind of compliance requirements can be implemented in DHS-2, what do we support, what is outside of DHS-2 as a product and as compliance is used as an extra burden to any organization, I think that our mutual goal is to make it as efficient as possible. So the best-winning organization is the one that can introduce and implement compliance in the most lightweight and the most easy way and this is what we're aiming for. And another thing is, like we're going from bottom to the top, is failure to provide public service. So if the government or the public function is not working properly and it typically impacts a lot of internal KPIs or a lot of inorganizational policies. So typically, if something stops working, it's a problem because citizens can't get the service, data is not available and all kinds of operational issues. So these are the most typical things that are at risk and it means that the investment or the money that was spent on the organization to make it work, it is kind of wasted. And the instrument, the tool, is not working properly. So I think these two are the biggest one and if you have any feedback and comments on that, let's discuss it later. Now let's look on the attackers' point of view. So how attackers see the same story, what's important for the attackers because once we understand what they're actually looking for, why do they attack our systems? It makes our task for protection much easier. All that I'm talking about is based on the experience that was reported by the HS2 users and these were real-life cases of what happened and they're supported by any other product in the industry. So from the attackers' perspective, the main interest is typically a kind of a financial interest, no surprise. So people are interested in getting something for free in order to be able to either resell, either to use for any kind of malicious purpose and get some monetary gain on that. The most typical thing is, okay, we have installed HS2 on a brand new server and in a couple of months nobody was checking what's going on on the server infrastructure there and we noticed that the server is overloaded, there is abnormal high system utilization and we don't know what's happening. We're running the latest HS2 version but we don't know what's in the server environment. Then we help the team to understand what's going on and see that there is a crypto mining tool consuming all resources and trying to mine cryptocurrencies in the server in the background. And from our perspective, it's a very strange use case. So people got into the system, not for stealing our data, but just for using our computing power. But it's a security breach, it's a violation. Our users can't work and there is someone who is not authorized who is sitting in the system and doing some bad stuff. For the motivation for people who do this kind of security attacks is just to get some extra CPU power or create a botnet or just proxy traffic to some other services like the platform for short attacking. This is a very, very typical example and we typically ask why and we say it just because it's a free resource that can be hacked. Second thing is selling user accounts and personal data. It's mostly related to, again, to banking institutions or kind of high profile accounts or high value accounts. But on the black market, if you get personal data of someone, they can be sold roughly from 10 cents in US dollars or like 10, 50 cents to 5, 10 dollars per account. So depending on how many user accounts you are able to mine and hack, including personal data, emails, phone numbers, social security numbers, whatever, depending on the country where the speaker originates, you can rate and create and sell a database of this account and get some extra money. So this is kind of a profitable business and then these databases of user data are used for spamming, they're used for different social engineering attacks and they like this aggregated and curated arrays of data they have some certain value in the criminal market in dark web. And if you by any chance have access to dark web, you can just see the marketplaces where different databases are sold and they have very, very specific and clear monetary value. But if you have roughly a thousand of users or thousand of records, it can be worth from like 50 dollars to hundreds or maybe thousands of dollars. So this is a clear financial motivation. The third case is, on one side I would say very exotic. On the other side it is increasingly common nowadays. It is related to the targeted interest in your systems and this is probably an attack that you will not even notice. It's related to APT actors or advanced persistent threats. So these are on-purpose hackers that are looking for data or for information in different systems. We know that all kinds of the state hacker groups, state control traffic groups, they combat in cyberspace and by some case your systems can be also the target if they consider that your data is valuable. To make this assessment, you need to be quite or justification for accessing this data. You need to have very specific interest and very specific purpose and we know the examples of data being collected kind of randomly but then analyzed and processed, not in the case of HS2 but in the case of other popular data leaks that happened in the joint public sector in the last 5-10 years. Typically it can be the case getting the voting database of the whole country if there were some cases when the whole population registered one Asian country leaked like two years ago and the same was in South America as I recall. So there are quite a lot of well-known examples of that and I think this data was some kind of processed and released to the public for a specific purpose. And the fourth case is script kiddies. It's all kind of random stuff happening on internet, young geniuses, students or other people who would like to try new brand security tools or regular internet noise, automated scanners or whatever. It can impact your systems even if you don't think about this. So it's a regular internet noise that happens all the time. To give a kind of understanding of how polluted internet is, for example, if you take a brand new computer connected to the internet on the public address, someone tries to connect to it within 30 to 15 minutes with some kind of security attack. So the internet space is scanned and assessed all the time by multiple parties and it's a part of the regular noise that happens there. And these kinds of attack may be quite harmful and they can be quite harmless. It's really hard to predict. What could go wrong? So what are the typical problems that happen and that everyone should be aware of on a very, very high level? In the last 20 years or all the time that I've been doing with security, weak passwords and all kinds of lack of security hygiene is the biggest problem ever. We talk about passwords all the time. We have implemented a lot of improvements to that but at the end of the day, both new users, experienced users, everyone, they got hit by password attacks and with the kind of development of technology they become more and more sophisticated. So it's not only that you have a password from 1 to 6 or a password which is called password or the password that is incorrect because it's an old meme on security. So it's different combinations of that but in the kind of a nutshell it's about lack of security hygiene and using the basic rules that you can either read online, either get from the security teams in your organization so it's about very, very simple human factor related security hygiene things. Second, we've just installed our system, we configured it in some way and in three days it got hacked. There can be different reasons for that but once we use the up-to-date software and we have designed it properly the most common case is lack of configuration or improper configuration or just not doing the full deployment to the end. So it means that with the complexity of the modern systems the more complex environment you have the bigger chance of doing something wrong or not fulfilling all the security requirements is kind of a bigger problem so we try to make the HS2 as easy to install as possible and to ensure that safe security defaults are always in place so the only solution here is to study a lot about how to deploy systems and to not overcomplicate things once you discuss your deployments and plan your deployments. The third most common thing is outdated software lack of updates, there were some discussions about it but running legacy software is always almost a bad idea and in the last two years this approach got extra flavor. There is a term which is called supply chain and it means that not only your software that you run the HS2 on may be vulnerable but all the dependencies all the components that were used to build this software they can be also vulnerable and if you update the package itself there is no guarantee that the dependencies that were included in this software package application they were not vulnerable so in our case we have an automated process of developing and checking all the dependencies for the HS2 but if you use a operating system or database with some other or dependencies that were not checked you have potential at risk so the attackers are always trying to find the weakest part in this chain and the weakest element in this chain and it means that if they are not able to attack your system they will be looking for the vulnerable parts of the supply chain to attack them and the last case is poorly designed to develop software including the same supply chain so if you are using some technologies to deploy systems on that are poorly designed or have lack of security controls it means that even if you use a very secure software the underlying layers at risk I've been talking to one of the NGOs in Norway last week and they told me that they had invested a lot of time and resources in building security of their systems but as they were running a lot of different applications and people in different countries where they operate had a lot of different technologies tax the effort was not worth that because they simply did not cover all the underlying layers and the design decisions that were made by people in different countries were not always good from security so whatever they built on the top of that was not secure just because of the vulnerabilities in the underlying platform so holistic approach to security always wins so it's not an easy thing to do at once so after talking about motivation after talking about impact and failing to reach the organizational goals the remaining component is likelihood and for example if we try to do some kind of the risk management for banking or insurance they've been doing it for hundreds of years and there are really good mathematical statistical models and you can predict the likelihood of such event or you can predict the likelihood of the credit score or the reputation of the loaner or many other things related to the case of foreign insurance based on the very proven mathematical models this is not the case for security yet if we look at the history of the whole cyber stack I think the first quantitative efforts the quantitative model for security appeared I think in 1992 or 1993 and we are still not there yet although we have a lot of data since then so most of the approaches they are very very qualitative so we talk about the exposure of the systems to the internet so if you have anything connected to the internet there are high chances that it is hacked if you run outdated software there are high chances that it is hacked if you don't have a really good understanding of security within your organization it is also kind of a signal that something bad can happen so there are a lot of criteria that we can discuss in more details during the next session or outside of this session rooms but in general we try to select the most important criteria based on the qualitative assessment this is what shows the real attitude to security so when everything breaks there are some who says okay there is no problem to that and these are common explanations to that I am sure that you've heard them from someone in your organizations we don't have budget we don't have time we don't have skills and it looks like security is not a priority for many although the amount of security incidents and the severity and the impact of the incident it grows from year to year so for the statistics that we have from 2022 this year is not available yet but I think that every third organization in the world had a security incident so it means that in this room there is a likelihood that one third of people, for example this role had faced a security issue or will face a security issue through this year as an organization so it's quite a lot and it's growing and the big problem is that attackers see more and more value of the user data or the cyber attacks and it becomes much more profitable for them to attack so it means that the stakes are growing and we have more and more problems and we are still not there yet because we really don't have enough awareness and sufficient knowledge because it's kind of a not very trivial domain to understand and, for example, general physical safety I think like when driving a car many people use seat belts or safety belts or if you go to the airport and go through security checks it's very different to what was in the car industry 50 years ago so I assume that in some years we'll come to the same state where we go through the scans we use some belts or similar things in the security domain but in general we don't understand the impact and the severity of risk that happened and once we don't understand them we don't make a proper risk assessment and our risk appetite is too optimistic and we are more on the incident mitigation side rather than on incident prevention side and what happens is the perception changes so people think more about security but typically it is based on the incident follow-up rather than on prevention measures so this is exactly what we're going to change together we're trying to encourage you to think more about cybersecurity and to educate your peers educate your colleagues and to grow the security awareness and knowledge within their organization so what can be improved? First, everything starts from the top and it is important to raise security awareness on the exact level and the biggest justification for that is we would like not to report our managers about the ongoing crisis but we would like to report that we have prevented that or we know how to prevent the potential crisis and it means that before doing anything we need some management commitment we need some management support and management understanding that it is important and otherwise we will be just mitigating one incident after another and it is a kind of a painful, low effective way of doing things the second is making proper risk assessment and risk choices and it's about agreeing what is the impact on what are priorities the third thing is having clear ownership of data and systems within their organizations in the same manner as someone owns all the physical assets and in the same manner as their people responsible for teams for team management for all kinds of physical assets and the processes and other things and it's about assigning roles and responsibilities based on the ownership this list can be extended and this is a very high level thing it is something that we start with but there is a huge layer of technical and practical security happening once we get to the management support or once we get to the technical department and this is what we are going to talk about in the next session about the HS2 security practices about hardening the HS2 about compliance making compliance easy and answering your questions that's all from me if you have any follow-up questions now, please ask thank you for your presentations I'm Intan from Indonesia I have two questions for you has the HIS to receive ISO certifications 27,000 because they are so very important for me in the next years we have certifications of 27 ISO just like standardizations from our government to connect the system of populations data so this is one question and the second is what is the differences between ISO 27,000 and 30,000 that you said before how to risk management approach you use 30,000 for the ISO what is the difference between two types ISO because I know about the ISO 27,000 there is an approach of the risk management so what is the differences between two types thank you thank you, great questions let's start with the first one so DHS2 does not have ISO 27,000 certification because it does not apply to the software so ISO 27,000 is information security management systems compliance and it tells about how organizations are or should be compliant or should be managing their security in accordance with the best practice with the standards so it means that once you come to the software there should be some different standard that applies if we look at the ISO 27,000 there is a chapter that is related to the software systems development where you describe how your organization acquires or develops software this is the only part where you can find some kind of connection or some compliance requirements that may apply to us if we talk about general requirements like logging like access control and password policy DHS2 supports them but it is always up to the implementation so it's how you configure the system rather than what software can do in short the standard itself doesn't apply we would say that for the software applications there are some better standards for that like all the standards to describe security in the products but if you would like to find some mapping between DHS2 and ISO requirements there is applications, software development and acquisition there is access control, logging and typical things that should be implemented in the organizations for all the systems including DHS2 and if you have some specific requirements on how to address DHS2 we can take it for the next session discussion to go into details and see these mappings in addition to that there is a really good website I'll share a link in the document a bit later which makes a compliance mapping between different standards for example ISO, NIST all kinds of French NS, AN, SSI and other standards so if you are compliant with one or would like to have some requirements in alignment with one standards you can see for example how it matches all the others and it's really helpful to mitigate for this kind of the set of the different requirements we use internally also when we check if there is anything that matches and that applies to our environment so that's it but I agree that there can be specific cases where you should consider how the system is implemented to ensure compliance with ISO 27000 internally for our organizations we didn't consider that because we are developing software and in order to establish some secure software development process we need a bit more detailed standard than ISO and we are aiming for compliance with these standards as well as all the standards second question the difference between ISO 27000 and ISO 31000 so as I mentioned 27000 is information security management system standards and it talks about information security 31000 is a risk management standard which tells how an organization can manage all kinds of risks including security risks and for some historical reason the definition of risk sits in ISO 31000 standard rather than in ISO 27000 but in ISO 21000 it refers to ISO 31000 so they kind of harmonize with that so if you are required to do some risk management as a general organizational practice you can have a registry of risks for your organization, ministry and so on you can look into the ISO 31000 because it has a really good guidance on that and you can apply these risks to the general organization risks to the policy risks to the compliance risks and security risks as well and the last addition it's a relatively new standard and it's actually developed good for like use because it's quite new and considering a lot of insecure area I see someone from the back row willing to ask a question Thanks Michael I'm actually just relaying a message from the document that you shared but I think would be a really great thing for us to talk about here today the question I'm just going to read it directly because I think it's pretty good is there any security document available for DHS-2? Do any PED testing is there any other testing that's done and are there any security standards that you choose to comply with which maybe was talked about a little bit already I think that would be great I have it open here Thank you for reading so many questions in one and really good that they were asked so security document available for DHS-2 so everything that we do as an open source is public so we have two web pages I'll write them in the document later it's DHS-2.org slash security it explains security features of the product and the second one is DHS-2.org slash trust and it explains how we do security and the high level explanation of our security approach is on the second link DHS-2.org slash trust for pen testing it's a bit more complicated because DHS-2 is open source and everyone can download and install software so it means that if we test something we can get a radically different result from what you test in your environment and for example when someone sends us pen testing reports from the field we see that roughly 80% of the vulnerability is found they are not related to DHS-2 but they are related to the specific implementations and configuration issues or software security issues in that installation so it means that what we can do is to have a reference setup of DHS-2 and make a pen testing report so we did it internally for the current release and we are going to make it public in the next months to have something available but it will say some estimates about our security level for our own setup so for your implementations we always recommend to have your own penetration test because they will consider your features installed how it is configured in your environment and for us we always love to receive penetration testing reports and if we see something important there that is relevant to DHS-2 we definitely consider proof and put into the bug fixes as well as you can see they have some security components and changes there so for other testing done we do security scanning of the code source code with the public tools and we have public dashboards I will share the link for your reference for the next session and we do security audit of every new big release as for now starting from the last version we released and for the senders I think I answered that but we are aiming to comply with the OWASP web and mobile application security standards and software assurance maturity model which is most relevant for security teams and software teams going to touch something around data disclosure agreements about what? We have one as a template that was created roughly three or four weeks ago and we can talk about this in the next session because it is a specific use case but we have a suggested form of data sharing agreements that DHS-2 implementers and contractors that support the system with some security component included Thank you everyone so we will end the session there just to remind you that Michael has a dedicated session on security and he is of course here for the next couple of days please feel free to reach out to him and speak to him but of course he has a dedicated session where we will get into more details so T is already outside but just before you break for T so the first thing is I again urge all of you please join the WhatsApp group because we always communicate everything in that WhatsApp group because we have one more city tour plan for today so if you are not in the WhatsApp group then you may miss important messages so please do that and secondly about the Google form that we have shared in the WhatsApp group only 80 people have submitted so far so we have around 25 missing unfortunately if you don't submit it by this by today noon we might not be able to share with you the certificates and other documents by the end of the week and finally about the people for the people who have paid us on site the registration fee you will be receiving a soft copy of the receipt during the day in case if you don't receive it by tomorrow morning we will meet our team at the registration desk and let us know that's it thank you we are breaking for T please if you will come back at 11am and we will continue with our country presentations we do have quite a few today so we just want to be on time so we will start back with our colleagues from Libya so please enjoy your break thank you and soon to be seen through some very different eyes those of his exotic wildlife from the crystal blue waves of the Indian Ocean where giants gather together along the sandy beaches of Gaul home to newly hatched turtles through the temples of Polonarua to meet cheeky locals and to the highlands seen for the first time from an eagle's eye Sri Lankan wildlife presents so majestic so playful so vibrant so Sri Lanka Sri Lanka lonely planet destination of the year 2019 beautiful vibrant and soon to be seen through some very different eyes those of his exotic wildlife from the crystal blue waves of the Indian Ocean where giants gather together along the sandy beaches of Gaul home to newly hatched turtles through the temples of Polonarua to meet cheeky locals and to the highlands seen for the first time from an eagle's eye Sri Lankan wildlife presents so majestic so playful so vibrant so Sri Lanka


Hello, my name is Matt Clark, and I'm a lieutenant in the Denver Police Department's Major Crimes Division. This critical incident debriefing is intended to provide you with information regarding an officer-involved shooting that occurred at 901 South Irving Street in Southwest Denver on January 1, 2020. You are about to see relevant video footage and learn about other evidence and police procedures related to the case, so you have an understanding of the details of this incident. The use of deadly force by a police officer demands a thorough investigation be completed. The Denver Police Department is committed to ensuring a full and timely investigation of these serious incidents. This allows for a comprehensive examination of the officer's actions to determine compliance with state statutes and department policies. In accordance with legislation passed in 2015, the investigation of police shootings in Denver are conducted by a multi-agency investigative team, made up of members of the Denver and Aurora Police Department homicide units, as well as the Denver District Attorney's Office. All critical incident investigations are actively monitored by the Office of the Independent Monitor. A word of caution, the images and information you are about to see may be disturbing. When a police officer uses force to arrest a suspect or defend against an attack, it can be graphic and difficult to watch. In addition, there may be strong language used by those shown in the video. Viewer discretion is advised. On Wednesday, January 1, 2020, around 12.50 a.m., Denver Police officers responded to a call of an assault with weapons at 901 South Irving Street. The victim was able to leave the residence after the assault and met officers at Mississippi Avenue and South Irving Street. The victim reported that his roommate, who was later identified as Gerardo Antonio Conchas Bustos, had assaulted him with a metal pipe inside the residence. He further advised that Mr. Conchas Bustos threatened him with a knife.  The victim also advised officers he believed there were two other females inside the residence. After completing their initial investigation and obtaining details from the victim, the officers escorted the victim back to the residence to ensure it was safe for him to re-enter and also to check the welfare of the other female residents. Three officers approached the front door and announced their presence in both English and Spanish. After receiving no response, the officers entered the residence and began conducting a protective sweep. A protective sweep is a quick and limited search of a premise that is conducted to protect the safety of police officers and others. Based upon the reports that Mr. Conchas Bustos was in possession of a knife, the officers had their weapons drawn while inside the residence. As the officers searched the house, they continually announced their presence in both English and Spanish. Officer Juan Gamboa and Officer Diego Archuleta completed a search of the kitchen and made their way towards a small mud room that can be accessed from the kitchen. Officer Gamboa was the first to enter the mud room. He stepped down two steps into the room and eliminated the confined room with a light that was mounted on his firearm. This room had three doors in it. One door accessed the kitchen, one open to the exterior north side of the residence, and the other provided access to the attached garage. Unbeknownst to Officer Gamboa, the door that led to the garage also provided access to a basement stairwell. As Officer Gamboa approached the threshold to the door leading to the garage, Mr. Conchas Bustos quickly came up the stairwell and rounded the corner of the doorway into the mud room. Mr. Conchas Bustos had a large knife in his right hand and was charging towards Officer Gamboa in a threatening manner. Based upon the layout of the room and the position of Officer Archuleta, Officer Gamboa was not able to back out of the room. Mr. Conchas Bustos was within five feet of Officer Gamboa when Officer Gamboa fired six rounds from his handgun at him. Officer Gamboa stopped firing his weapon when he recognized Mr. Conchas Bustos no longer posed a deadly threat to him. Officer Archuleta was on the two stairs between the kitchen and mud room floor when he observed Mr. Conchas Bustos advancing towards Officer Gamboa. Fearing for Officer Gamboa's safety, Officer Archuleta discharged one round from his handgun at Mr. Conchas Bustos. Officer Archuleta stopped firing his handgun when he believed Mr. Conchas Bustos was no longer a threat. Officers immediately called for an ambulance and secured Mr. Conchas Bustos. The knife Mr. Conchas Bustos possessed was secured and placed on a vehicle in the garage. Resuscitative efforts were attempted, but Mr. Conchas Bustos was pronounced deceased at the scene. Both officers involved in this incident were uniform patrol officers assigned to District 4, Southwest Denver. A female resident was located in a locked bedroom on the main floor of the residence. She was found to be unharmed. The victim who reported the initial assault was later transported to a local hospital for treatment. None of the officers involved in this incident were injured. Following a police shooting, any officers involved in the incident are separated and assigned to a supervisor once a situation is stabilized. Involved officers remain in the company of the assigned supervisor throughout the initial investigation of the incident. Through this investigation, it was determined that a total of seven rounds were fired by the officers. Investigators recovered a large butcher-style knife with a white handle on the hood of a vehicle in the garage. A toxicology report revealed the presence of alcohol in Mr. Conchas Bustos' system. All officers assigned to uniform patrol duties have been issued body-worn cameras. These devices are generally worn at chest level by patrol officers and are capable of recording both audio and video. Prior to being activated by the officer, the body-worn camera maintains a 30-second video buffer. Once the officer activates the camera, the 30-second buffer video is captured and the camera starts recording both audio and video. It is important to note that a body-worn camera captures a general perspective of what is in the camera's view. However, this footage may or may not be what the officer actually saw or perceived. At times, an officer's movement or hand positioning may inadvertently block the camera's view. Additionally, the camera may not capture light in the same way as an officer's eye. Here are the complete body-camera videos from the two officers involved in this incident. Again, viewer discretion is advised. We've got officers down there now, sir. That's locked. No, it's locked. Make cars, coats, and finish clear of this house. What's up? Make sure your camera's on. I got you. We're gonna be at 901 South Irving. Thought he was shooting at you, man. Hit or anything? I shot one. Okay. All right, buddy. Give us a second. All right. We're gonna help you. No, officers are injured. We just need EMS for one suspect. Clear. No, officers injured. Can you confirm the address for me one more time? The last one sounds like something different. We're gonna be at 901 South Irving Street. Officer Kuril will be right up front. Okay, clear. Make a firm, sir. Myself and 52 Ida. Hey, Kuril. I just want to confirm it. Finish getting that house cleared. Correct. We'll be for the suspect and confirm the address. 901 South Irving, 901 South Irving. Clear. Thank you. Clear. Thank you. Give cars over here. Hey. Clear this basement for us? We're gonna hold it here. Spare with us, man.  Give us a minute. CPR, bro. I'm gonna serve, sir. We haven't cleared that. I'm careful of clearing it. Stay with us, man. Stay with us. So, guys, see you soon. The Denver District Attorney will review the details of this incident and determine if the officer's actions were in compliance with Colorado law. After the district attorney renders the decision, the Denver Police Department's Internal Affairs Bureau and Conduct Review Bureau will complete an administrative review of this case. The details of the case will be presented to a use-of-force review board made up of community members and police command officers who will determine whether the officer's actions were in compliance with the high standards expected of every Denver Police Officer related to policies, training, and tactics. For additional information regarding the investigation of critical incidents or the Denver Police Department's use-of-force policies, you may visit denvergov.org. Thank you for taking the time to view this critical incident briefing.


Hi, I'm Zor. Welcome to a new Zor education. We continue talking about vector fields. We have already covered gradient of scalar field, which produces the vector field of gradients. We have also covered divergence. Today we will talk about the characteristic, which is called circulation of vector field. Well, obviously you understand that circulation is somehow related to circular character of the vector field. Like a tornado, for example, if our vector field is three-dimensional velocities of each molecule of the air, just as an example. Anyway, this lecture is part of the course called physics for teens. That's the wave part of this course, and in particular part of the topic field waves, because we are gradually moving to the waves of the fields. But we have to know the characteristics of fields, including the circulation, for example. Now it's presented on the website called Unisor.com. The site contains also a prerequisite course, which is called mass routines. I do recommend you to go through material presented in the mass, because we definitely use a lot of math in the physics course. Definitely calculus, definitely vector algebra. Mathesis sometimes. Now the the course contains basically video-recorded lectures like this one, which you are looking at. And every lecture has a textual complement, supplement. It's basically like a textbook. So every lecture has like a chapter of the textbook, which is related to that particular material. There are some problems solved. There are exams. The site is completely free. There are no advertisement, no strings attached. You don't even have to sign in if you don't want to. Exams, you can take exams as many times as you want, until your score is 100 percent, basically. Alright, so let's return to circulation. First of all, I would like to talk about circulation in two-dimensional case, because the subsequent lectures will cover two-dimensional curl of the vector field, which is basically a subsequent topic after the circulation. And then three-dimensional, which basically is reduced to three two-dimensional cases. So one three-dimensional is three two-dimensional cases. Basically, that's how I approach this. So today we'll be talking about two-dimensional vector fields, which means that every point on the plane has a vector coming from it into some direction of some magnitude. That's what basically vector field is. So obviously this vector at point xy, this is a vector. So at every combination of xy, which are coordinates on the Cartesian plane, we have a vector which has certain direction and certain magnitude. And obviously this vector has projections on x-axis and on y-axis. So the vector has two components, obviously. So this is x-component, projection turned to x, and this is y-component, projection on the y. Okay. Now we are assuming that the vector field is not changing with the time right now. So it's a stationary vector field. I mean, all the vectors can be either the same, maybe or different or whatever, but they are not changing with the time. Now, now we would like to introduce the concept of time, but not to the changing of the vector field, but to the movement within this vector field. So let's assume that there is a steady wind, which is blowing into some directions, not necessarily in one direction. Maybe there is some kind of small whirlpool or whatever, doesn't really matter, but it's there. And let's assume that we have a surface of the lake with paper ships on it, or some particles, whatever, and the wind is blowing, and we would like actually to move our paper ship from one place to another, along certain paths, and I'm interested in work, which I'm supposed to do. Now, if I'm working against the wind, so I'm having certain, I have to exert certain effort to go against the wind. Now, the wind is known, there is F, this is basically the force of the wind, which I have to overcome, and I know my trajectory, so in theory I can calculate the work, which I'm supposed to do. If, on the other hand, I'm moving with the wind, then there is also work, but it's kind of negative. So the work in this particular case becomes positive or negative, depending on whether I'm going against the wind, or the wind helps me. How can I calculate this amount of work to basically make some kind of a judgment about, well, which way to, which paths to take from A to B? We will consider three different cases of what kind of a wind actually we have, what kind of vector field we have, and the first and very simple situation is when the wind goes straight along one line, so all these vectors are equal in both direction and magnitude, and my path, my trajectory, is a straight line. So I'm moving from A to B and the wind goes this way. Doesn't matter actually. Where, as long as we are considering the case when all these vectors are exactly the same, same direction, same magnitude, and I'm moving along the straight line. Well, let me just start from a simpler case. What if I'm moving from A to B exactly opposite to the wind, exactly opposite to the force? Well, then I have to exhort efforts, my force supposed to be basically compensated. It's supposed to compensate the force against which I'm going, so I have to exhort force the same F as the wind. These are all F and I have to cover the distance from A to B, and the work is basically force times the distance. We all know that, right? Now, if my path is not parallel to direction, there is a slight modification to this, cosine of alpha, cosine of angle. Why? Because, well, if in this particular case, my force goes this way, the force can also, can be represented perpendicular to my path and parallel to my path, right? So these are two components. Now, the components which is perpendicular to my path doesn't really resist my movement, but this component, which is exactly opposite, it does. So I have to really take only this component into consideration, and this component is equal to, if I would take the vector F, and I would take the vector AB from A to B, the way how I'm moving actually, then their scalar product would be exactly the multiplication of this force, the projection, by lengths of this vector, and the proper angle will be, because the scalar product is magnitude of this times magnitude of this times cosine of angle between them, so that would be basically my work, which is the same as this one. So this is the simple case, and it's just elementary problem in mechanics. Whenever we were talking about dynamics in the part of mechanics of this course, even I, we can actually take a look at the lectures of this particular course. That's where it's all explained. Okay, so the first case is simple. That's uniform vector field and straight line path. Now, just as an example, what if my force is perpendicular to my path? Well, then obviously this is equal to zero, because I don't have to really, my, my, my, wind does not resist my movement against this particular path. Now, if my force is directed this way, it helps me. So, then basically I will have the negative negative scalar product. Well, it obviously depends on how we count the the angle. So the cosine of zero is equal to one, and the cosine of 180 is minus one, so it all depends. So the sign is kind of not as important right now, but the magnitude is definitely correct in this particular case. Plus or minus depends on whether I'm going from A to B or from B to A and how I calculate the angle from from the vector to from one vector to another or vice versa. So sign is basically not that important. In any case, this is basically the correct representation of the work, which is supposed to be done either by me, who is moving these paper ships, or by wind itself. So it depends on who I am actually, which side I am representing. Okay, fine. So that's a simple case. Now, let's consider we have a different case. We still have a straight line movement, but our forces, our our wind, it's not, so this is again from A to B, but our forces are different, and they still have to basically calculate work, which either I'm supposed to do or the wind is doing, whatever, moving the paper ship from one location on the surface of the lake to another. Okay. Well, how do we approach it? Well, first of all, for simplicity, I will use the system of coordinates. This is y, and this is x. Doesn't really matter. I mean, it's all up to me. I know my paths from A to B, so I established this particular type of coordinates. Now, in this coordinates, every vector has some kind of magnitude and direction. Okay. Now, again, obviously every vector can be represented as this type of thing. Now, as I'm moving from A to B, my y-coordinate is always zero, right? So I don't really have to talk about any f of x, y. I'm basically talking about f of x, zero. So vectors which are at this particular point, they are moving in some direction. So I'm interested to analyze these, because all these are not important. I don't go through these points. I'm going only through these points, and the wind in these points is important. Okay. So now, this thing, this vector, each vector, again has two components. It has f of x component, and it has a y component. Obviously, depending on the coordinate x, zero, x, zero. Okay. So this is my x component. From this vector, for example, this is my x component, and this is my y component. This is fx, this is fy. And again, since fy is perpendicular to my paths, I don't care about it, and I'm only involved with fx, because only fx goes either helping me or resisting me, right? Okay. Now, how can I basically calculate the work which is supposed to be done here? Well, I'm doing exactly the same as I've done with whenever we were approaching integration. If you remember, if you would like to have some kind of an area under the curve, we break the argument into small pieces, x0, x1, etc., xn. Then we consider this to be almost like a rectangle with the height being equal to f of xi, right? This is xi.  And that will be, this is f of xi. This is delta xi in between these. And then we will summarize them, and that would be the total area, and then we will go to a limit when the number of these points is increasing to infinity with each one shrinking to zero. That's exactly the same approach which we will take here. So, there is nothing new here. What I will do here is I will break from a to b, this would be my x0, it would be my xn, and I will break them into equal parts. Each one is equal to delta x. Let's say it's equal. It doesn't really matter, but let's, for simplicity, consider it's equal. All right? So, this is my xi, this is my xi plus 1, this is my xi minus 1, right? Now, at each interval from xa minus 1 to xi, I consider that the strength of the horizontal component of my wind is not really changing. Now, it's basically something which mathematicians are much more sensitive than physicists. Physicists always assume that something like a smoothness of anything which they're dealing with exists. Mathematicians are more sensitive, they're basically asking, is it a smooth function or something like this? Smooth in terms of existing some kind of limits, etc. So, we do assume, as physicists do, that we can consider the wind is not changing on this small infinitesimal, ultimately it will be infinitesimal segment. So, during this particular piece of the pass, the wind is exactly the same. And it's equal to f of fx of xi comma zero. That's the horizontal component, the projection of the x-axis of the vector x at point xi. So, what we have to do is, now we have this projection of the vector, we can multiply it by delta x. So, we have the distance which is delta x, it's between xi minus 1 as xi. We have the strength of the force which is this one. And then we will summarize it, i from 0 from 1 actually, from 1 to n, to get all the work done on each particular segment of our pass. And then we will take the limit when delta x goes to zero. That's the approach. And if I know this particular function, if my vector field is given as a function, and by the way, this is the function of one argument right now, right? It's a function of one argument of x. So, basically the result would be integral from a to b, from zero to b, whatever, of function fx of x zero dx. So, if my function fx is given, and it is given because my field, vector field is given, which means I know this at every point, x and y. So, that's why I know projection onto x-axis. And I can take this integral. So, the straight pass is fine. Now, let's go to a curve pass, my third option. So, I'm trying to do it like in a sequentially more difficult cases. Now, speaking mathematically, we are talking right now about introduction into integral along a curve, basically, which is very much similar to regular integral. But in any case, we just have to really spend some time. So, we're talking about a non-uniform vector field f of x, x, y, and we are talking about direction which is not really straight. This is a, this is b. Now, how can I calculate this particular work? Okay. Now, I think it would be easier if my way from a to b is measured in time. Now, my field remains the same within the time frame. But my movement, so my position is changing. So, somehow, I actually have to have my position, my position is x, y. And I have to have my position specified as some kind of function of time. That what defines the curve. Now, define the straight line, I define basically by a and b. And this is a straight line. So, b has certain lengths and basically that completely defines my pass. If pass is not straight, something like this, I have to define it somehow. So, this is how I define it. So, my position, let's call it r of t vector has two components, my abscissa coordinate as the function of time and my position, my ordinate position. So, that's what defines my pass. Okay. Now, I will do exactly the same thing as before, but instead of dividing a straight line into pieces, I define time into pieces. Let's say I'm at a at t is equal to zero and I'm at b at t equals to some kind of a t. So, I divide time segment from zero to t into small segments from t zero, which is zero to t one, then from t one to t two, etc., from t n minus one to t n, which is my end of time. And let's assume that I divide in such a way that delta t is constant. So, difference in time between t two and t one, same as from t three to t three, t three, etc., so all the different intervals are the same. Now, what am I buying by doing this? Well, now I have positions. Now, this is position at t one, this is position at t two, this is position at t i. This is position of t i minus one. Now, I'm making my delta t is so small that this particular segment of my curve is practically almost approximately straight. Now, if I'm talking about work, I have to have the force at this point and again assume that the force is the same during this period of time. So, the force which is acting at this point is exactly the same as this and this and this. Again, mathematician would probably ask, okay, is that true that the function is smooth enough that whenever we are making small enough distance in time between these two positions, we can assume approximately that the wind will be basically the same along the same straight line. So, I'm basically reducing my micro problem into a certain number of micro problem. Now, every micro problem is exactly like the first one which I was talking about when I have a uniform vector field and straight line pass. So, this is a straight line pass approximately and this is a uniform vector field approximately. So, if I will multiply my lengths by projection, now, if this is, let's say, my force at this particular point. So, I'm representing it as perpendicular to this and parallel to this segment. Obviously, perpendicular is not involved at all. So, I have to find projection on the segment itself. Okay, how can I characterize this particular segment? I need its direction and its length, right? Okay, fortunately enough, physics has its own approach. There is a concept of a speed. Now, speed, velocity, velocity is a speed and direction. So, basically, we're talking about velocity. So, the velocity of my movement at this point, and at this point, we are assuming that delta t is small enough, is always a vector which is tangential to my segment itself. No matter how curvy my way is, velocity is always tangential to my curve. Again, if you remember, if you are moving along the circle, velocity is always tangential to a circle. So, it doesn't really matter how you move. It's always, because velocity is actually calculated based on position at this point, position at this point, and then you have a difference, and you reduce the difference between this point and this point, and basically divide it by the length, divide by time, and that will give you the tangential length. And velocity is, as a vector, is a derivative of position, this is of xy, and this is of xy. So, at any point, velocity is a vector which is x of t, y of t. These x and y are two components of my r vector, the position vector, and their derivative, which is the prime, derivative is basically the vector of force. So, that's what my velocity is, okay? So, I have a vector which is definitely related to this segment since it's tangential, and the delta t is very small, so the points are very close to each other, and obviously the closer points are to each other, the closer the tangential line is to the segment between these two points. When this point is moving towards this one, this segment is directed more and more as a tangential line. That's the definition of the tangential line. Okay, so as you see, we do have some mathematics in this, so we need to understand these simple concepts. Okay, so what's the length of this segment?  Well, the length of this, that's my length of the segment, and direction is actually the direction of the vector. So, I know all the components I need. I need the projection of my vector f onto the direction which is defined by v, so I need f of xy. I have to multiply it as a scalar product by v of xy. This is a scalar product, and that scalar product will give me magnitude of this as projected onto this direction times magnitude of this, which is actually what I want, magnitude of projection of f onto v times the length of this particular vector. All I need is multiplied by delta t, right? So, delta t times v will give me the lengths, this times this will give me this and the projection, so I have my work. This is i's, right? So, I assume this is i, this is i, this is i, and this is i. This is the piece of work which I am actually performing during this. What should I do then? Well, obviously I should summarize it, which is actually and go to a limit when delta t goes to zero, right? When these points are closer and closer to each other, which basically gives me integral from zero to t f of xy a scalar product v of xy dt. Or alternatively it can be expressed differently from zero to t f of xy dr of xy. Well, they are obviously xy, x of t y of t. That's position. Or simply dr times f. That's how in the simplest form it actually expressed. So, this is basically not a simple integral, it's integral along a curve. But it's approached exactly the same as integral of any regular functions. Well, why did they do all this? Why is it called circulation? Okay, basically this particular integral, the last one, is called circulation from point a to point b. It's the definition basically. What's more interesting is that there are certain vector fields when their circulation from a to b would be equal to circulation from a to b by any other pass. So, no matter how we move from a to b, the result, work which we have to really spend, will be the same. Now, these fields are called conservative. For instance, gravitation field and electrostatic fields are possessing this particular quality and they're called conservative. So, what's interesting is, and I'm not going to prove it right now, but maybe I will do it as a problem later on, later on that any conservative field f of x, y is actually a gradient of some scalar field. You remember what gradient is? That was a couple of lectures before. That's the operator df of x, y to dx, df of x, y to dy vector. So, if I have a scalar field, which is function of two arguments basically, there the partial derivative by first argument and partial derivative by second argument is a gradient. Physical sense, physical meaning of the gradient is which direction it changes the fastest, these values. So, it's a vector of fastest change. So, any conservative field is actually a gradient of some function which is called potential. Remember, gravitational potential, electric potential for electrostatic field. From these fields, we can always get the force. And again, maybe we'll just have a couple of problems later on solved in this particular case. Okay, now, if this is true, then what's interesting is that if I'm moving on any closed circuit, let's say, from A to A, in the conservative field, my work, my circulation should be equal to zero. Right? Why? Because I'm just choosing any other point from here to here supposed to be the same from here to here. But integral from here to here is opposite with a sign to move in an opposite direction because I'm moving in my line, my old little pieces are in a different direction. So, from here to here, and then from here to here, which is the same as this one was an opposite sign, would give you zero. So, that's an interesting fact, just by itself. So, if the field is conservative, then the circulation is equal to zero. Well, actually, because of this type of property, the name circulation comes because we're always going around the circle. But in some cases, for some fields, it's not zero. And let's analyze these cases. So, we're talking about something new. Now, imagine you have some kind of a field, and you have to move around some kind of a closed path. Okay, that's some work which is probably not equal to zero if the field is not conservative. But what is the relationship of the work which I'm spending here and the area of this inside this circle? We're talking about two-dimensional. Everything is in two-dimensional. So, this is area. Okay, what's interesting is that if I would like to know the circulation property in one particular point of the field, what I can do is I can surround this field with some paths, have the circulation calculated, and have the area calculated. Now, obviously, if I'm squeezing, if my path is shrinking to the point, both my work is changing down, reducing probably to zero, and my area is reducing to zero. And what physicists have introduced, well, I'm not sure actually who introduced it. They have introduced an interesting characteristic which is called curl. Curl is actually of the field f of x, y. This is a characteristic of the field at the point x, y. And it's a limit of circulation along a closed path divided by area when I am shrinking, shrink my circle around it. Now, when I'm saying circle, it doesn't necessarily mean that it is circular. If the field is smooth enough, and again, this is, mathematician is talking to me right now, you still have to prove that this limit exists, because maybe it depends on how exactly the path is formed. Is it a circle? Is it a rectangle? Is it something else? Will this limit exist? And again, if it does exist, maybe it would be different depending on what kind of a shape of the circular path, or any path, or rectangular path, or any closed path would be. So, right now, I'm completely leaving these issues out. These are relatively complex mathematical issues. But we assume that the field is smooth enough, so that this particular limit indeed exists. And now, I will just go through the calculation of this limit, how it looks. And again, all we need for this, from the mathematical standpoint, a relative smoothness, well, differentiability, if you wish, of the field. So, the field is supposed to be differential, by x, by y, by whatever. But let's try to basically calculate this thing, assuming it exists. All right, what we do is, we will choose a particular path, a rectangular path around my point, this is my point, p of x, y. And I'm choosing the delta x here, delta y here, and I will squeeze these sides of this rectangle. So, I will calculate my circulation around this particular path. And then I will squeeze down to zero delta x and delta y, and see what happens. Well, let's see what happens. If the middle point of this rectangle is x, y, then this point is x minus delta x, y minus delta y divided by two divided by two, right? Half. But this is half of delta x. This point is x plus delta x divided by two y minus delta y divided by two, right? This is x plus delta x divided by two y plus y delta y divided by two, and this one x minus delta x by two, and y plus delta y by two. Now, I assume that this is a very small rectangle. So, the force within each side of this rectangle, my force f of x, y, is the same. And again, that because my force field is really smooth enough, so whenever these points are close enough, there is no change of the vector of force on this segment. All right, so let's just calculate the work which is required in this particular case. My f of x is, has two components, f x and f y. f x would be parallel. f y would be parallel to this, right? So, whenever I'm moving along this side, I have to take into consideration only f x because f y would be perpendicular. And whenever I'm moving along this side and this side, I'm taking into consideration only f y component. All right, so if I move from here to here, my component is equal to f x. Let's take this one, for example. I'll take in the middle. That would be easier, okay? So, the component of this, f x component of this force at this point, this point has what corrosion as a what? x would be the same and y would be minus delta y divided by 2x, y minus delta y divided by 2 times. And this length is delta x, right? Okay, go this way. The component would be f y of, this point has x plus delta x and y, x plus delta x of 2y times delta y. Now, this way. Now, mind you, here we move from here to here. And here we move in the opposite direction. So, that would be with a minus sign. If I'm using the same component, it would be minus sign. So, minus sign would be what? So, plus, plus, minus f x at this point. It's x, y plus delta y over 2 times delta x and again minus here. And again, I'm moving against delta y's that way. So, I'm moving against, so it's minus sign. f y of x minus delta x over 2y delta y. Okay, we've got our formula. Now, it's simple. Remember this thing from calculus. Okay, so this is my a, this is my b. Now, the difference between them is this one, right? So, if I take the line which is parallel at some point c, then the difference between this and this would be equal basically to b minus a times tangent of this angle. And tangent of this angle is the derivative at one middle point. So, whenever my points a and b are very close to each other, I can say that this is approximately times b minus a. But since points are very close to each other, I can put a b or a, it doesn't really matter what it is. When a and b are very, very close. So, basically that's the definition of the derivative. So, from this simple calculus case, we can combine this and this. What is this? Well, both of them are delta x and both of them are f x. So, it would be f x. Now, the first argument is the same x. Second argument is y plus delta y divided by 2. No, the first is minus. Minus f x x y plus delta y divided by 2. Now, and both are multiple by the x. Now, let me just talk about the y component. The x component is fixed because right now where x is here and x is there. This is the difference between them. What's the difference is? Well, that's exactly the same as when I was talking about f of b minus f of a. b is this, a is this, and it's equal to the difference between them, which is delta y with a minus sign actually times derivative. In this case, partial derivative by y in some middle point. Middle point between these two is y. So, basically I can say that this is equal to f x of x y d by dy times the difference between arguments b minus y, b minus a, which is delta y and delta x. But I think it's with a minus sign, right? Because this minus this would be equal to delta y minus delta y. All right, fine. Now, very similarly, if I will take this and this, I will have df y of x y by dx times delta x delta y. Exactly the same consideration because y and y is the same and the difference between arguments is delta x. That's why I multiply by delta x and delta y itself. So, this difference between these two is my circulation. And they have to divide it by the area, right? Divide by the area. Circulation divided by area. That's my curl definition. Area is delta x times delta y. So, from this, I came to a formula that circulation divided by area when area is shrinking to a point x y is equal to df y x y dx minus df x dy. Because this is area which is cancelling when I divide the circulation by area. So, this is basically a final formula which we will return in the next lectures related to curl of the field. Curl is very important. For electromagnetic field, the Maxwell's equations related to electromagnetic field, they contain the curl. And that's basically why I have introduced the whole thing. This nubbler symbol with gradient and divergence. And now we have the curl. And what's interesting is that the curl of this f of x y of vector is equal to nubbler vector product with vector f of x y. Now, this will be explained later on. And that's what actually makes nubbler very important symbol in physics. So, you have gradient, you have divergence, and you have curl of the field expressed in nubbler. So, now it's vector product, the cross product. Okay, that's it. I would suggest you to read the notes for this lecture. And as I said, that you go to unizor.com. The part is called waves. And a particular topic is field waves, which contains this lecture and others about gradient, about divergence, and about Maxwell's equations. That's it. Thank you very much and good luck.


It seems like it was just only last week when I decided I wanted to make videos for YouTube. Hang out with my friends and skate. I didn't have much of an idea on how to make videos or even how to edit, and all I had to film videos with was my iPhone. As time passed, I continued to study YouTube and I kept making videos and skating. YouTube has definitely helped me get outside my comfort zone and experience a whole lot of new things. I've shared massive parts of my life online for years now and I've had a whole bunch of fun doing so. I still remember when I hit my first 100 subscribers on YouTube and how accomplished I felt and how excited and happy and grateful I was to have that amount of people be interested enough in what I was doing to subscribe. Since then, I've just continued to skate and make more and more videos, learning new things with each upload. I've spent the best part of the last two years skating and filming, trying my best to push this YouTube skate thing here in Australia. I've learned new things about myself and also learned that if I really put my mind to something in focus, I can progress forward on and off the skateboard. Growing up, I never wanted to do anything else apart from skate, even just subconsciously. I've always been drawn back to just jumping on my board and rolling around when life got a little uncertain and strange at times. Learning how to be confident in front of the camera didn't come naturally to me and I had to allow myself to be vulnerable in order for myself to leave my comfort zone and continue to grow. I became self-motivated to skate as much as I could and make good use out of each session to try to film at more parks and have new videos for you guys to watch. I've come to learn, I'm very stubborn and driven in what I want and what I want to accomplish for myself. There was never anyone else that was doing the same thing as me to bounce off and draw inspiration from directly and I felt alone and alienated because of that. I had a vision and I knew I was willing to put in the work and give everything towards making it happen. A lot of people didn't understand what I was doing and a lot of the people I was hanging around and skating with didn't want to support it or even accept it one bit. They made fun of me and told me I wouldn't be anything. I was alone and at times I was unsure if what I was doing would even work out. There had been a lot of times I've wanted to quit and delete my channel and just stop. Times were tough for me and at that point it was either quit and burn everything down I had built or keep pushing on forward. Knowing what I wanted for myself in life I knew I needed to get out of the environment I was in and experience life elsewhere. September 20th 2018 I packed my bags and I boarded a plane to California.  I got to skate so many red parks out there and I got to spend time with family. That trip to California changed everything for me. I experienced a whole new side to life and I really felt like I had found my sense of belonging there. I was too busy having fun and enjoying my time over there to worry about all the negativity and uncertainty I had gone through previously. I made friends with a whole bunch of like-minded people and got to experience skateboarding in its original birthplace. I met some really special people in the whole process and I got to experience the whole vibe and culture of LA. All the spots I grew up watching people skate in videos and online I got to see them in real life and skate them too. I never knew skateboarding could be so amazing and its presence be so strong. Parks I got to skate were unbelievable and when I would visit them I would feel like I was in a daydream. This changed my outlook on what I was doing completely and I knew from then on out I would continue to work hard in sacrifice to get towards where I wanted to be. The side of life I got to experience over there really helped me put things into perspective and put me back on track towards my goals and aspirations. I learned to trust my intuition and more importantly trust the process. Upon returning to Australia I continued to keep that nostalgic feeling of skateboarding alive within me and I've been able to use it as a driving force for what I'm pushing towards. As time has gone on the right like-minded people have gravitated towards me with full support and positivity towards my goals. And also the general understanding of skating in YouTube here in Australia has boosted which has allowed me to feel more accepted and proud of how far I've come. As I celebrate hitting 10,000 subscribers with you all I just want to say thank you to each and every one of you that have subscribed, clicked on a video of mine, left a comment or even just told me you dig what I'm doing and want to see me go far. Thank you from the bottom of my heart. I love you all.


I'm really pleased to speak to you today as Executive Director of Corporate Resources and Services here at West Sussex County Council. I'm going to tell you a little bit about the directorate itself, what motivates me, and very headline what we're trying to do as a Directorate. So I'm very lucky to be leading a team of very motivated, excellent individuals, about 450 people make up the corporate resources and services directorate, gyda gwych, rydych yn eich dữ yr ysgol, at urgynneud, ond i ni'n siwr bod i'r urgynneud colleagues yw sy'n dechrau hyn sydd wedi bodi gweithio'r dweud. Mae'r ddweud hwn ymwneud yn cael ein slogos, ond yna yn ei gweld byddoedd ychydig. Mae'r slogos yn ei ddwy i yn enw ydyn nhw mewn gweithio' i ni, ac os yna'r dweud yn cysyllt y pryd yna yw'r ddweud ymklwyd yma sy'n ei gweld peirion dweud yr ymlaen nhw. Mae'r cyfnod o'r gweithio? Bydd y gallwn awr gennymol i'r unig wych yn ei wneud i'r hyn ymddangos wedi'i gweld i Weithsaswydd a'r unig bydd y gallwn awr gennwysau i'r hynny o'r reufeidliadau i'r cyfrifysgau i'r cyfrifysgau, a'r cyfrifysgau i'n gweithio'r gwahanol i'r effeithiwn i'r uchwyn I'm also very pleased that we have a clear set of values as an organisation, that we are focused on our customer, that we are honest and realistic about what we can achieve, that we listen to people and we act upon what we hear, that we trust and we support each other and that all our members and staff are genuinely valued in what they do. As a directorate we have lots of things that we are seeking to achieve during 2015-16 in support of our future West Sussex plan, too many to mention in this short piece, but we will be doing our very best to support those values and to support our organisation as it moves forward on its future West Sussex journey. I'm very pleased to be part of an organisation that is focused, energetic and proactive.