So, we have started looking at double integrals, integral of functions of two variables, so over a domain. So, we had a function f defined in a domain d in R2 to R. So, we looked at double integral over d of f xy dxy. So, that was the double integral. So, whenever it exists, we said there was a theorem called Fubini's theorem that this double integral under suitable conditions, namely the function exists, the double integral exists. For example, when d is closed bounded and f is a continuous function, this will exist and can be given as integral over, okay. So, let us write case 1 is equal to x is equal to a to x is equal to b integral y goes from some function eta x to phi x of f xy d by dx whenever the domain d can be written as all points x y such that x lies between a and b and y lies between some function eta x to phi of x. So, this was type 1. So, these are type 1 domains. So, how does one go ahead? Page done, more pages, okay. You have to add a page, okay, right. And similarly for type 2 domain, if d is of type 2, namely x y such that y is between c and d and x is between some function say eta y to phi of y, then the double integral can be computed as iterated integral. So, y is going from c to dx goes from eta y to phi of y f xy sorry dx. So, meaning that the double integral for a function of two variables if the double integral exists can be computed by computing one variable integral at a time. So, these were called the iterated integrals. So, that gave us lot of maneuverability of computing double integrals. Let us look at what is a situation in three variables. So, let us take a domain d in R 3. So, a domain d in R 3 can be quite complicated, right. So, we are going to look at one possibility which says d is projectable onto x y plane. So, we want to look at d is a domain which is projectable into x y plane. So, there is think of the domain. So, that is z this is x this is y. So, this is a domain which can be projected onto x y plane. So, what does it mean? That means I can look at the projection of this. So, that will be something. So, this is my domain d let us call it as region R. So, that means what? So, mathematically means that d can be written as x y and z such that x y belongs to a region R contained in R 2 and z. So, at any point here so that is x y in the projection my z starts somewhere and goes up to somewhere. Let us make this as a. So, top is some surface. So, let us call it as the top one is z is equal to a function of x and y. So, let us call it phi x y and the bottom this one is z is equal to something eta of x y. So, the domain looks like this it can be projected onto the x y plane projection is a region in R in x y plane. So, how does the points in the domain look like? They look like x and y belong to R that is a projection and for every point x and y the z starts at this surface that is eta of x y and goes up to the top surface that is phi of x y. So, it is a kind of a cylinder you can think of some kind of a cylinder you can think of in R 3. So, that can be projected onto the it has a projection onto x y plane. So, when you project it onto x y plane that is the region R. So, how are the points described for a projection that is projection meaning z coordinate becomes 0. So, for any point x and y in the projected region R how high you will go so that you are in the domain. So, when you want to go up you start at this surface and end up this surface. So, this is the value of z so that you are inside the region D. So, we start at R is x y belong to R. So, point is here and you go along z x is how how high you have to go you have to start at the surface eta x y and go up to the surface of phi x y. So, this is a domain which is projectable onto x y plane. So, what is the advantage of this advantage of this is that Fubini's theorem states for R 3 the computation says the following. So, Fubini's theorem for D projectable onto x y onto x y plane that means D is written as x y and z where x y belongs to the region R that is inside the plane R 2 and z goes from some function we said say eta of x y to phi of x y. So, with that description of D as projectable onto x y plane the triple integral of F D x y z. Can be computed as x y belong to R and z. So, double integral over R D x y and integral of F x y z with respect to a variable z z goes from eta of x y to phi of x y. So, that is what we are saying the Fubini's theorem says if the triple integral exists then and the domain is of this type projectable onto x y plane then I can split the triple integral into two parts integrate the variable z from the bottom surface to the top surface. And so that is this from the bottom surface in the domain is x y in R. So, you start here at this surface and end up at this surface. So, from this surface so integrate the variable z once you have integrated the variable z it is a function of two variables. So, it is a function of two variables. So, that says that this as a function of two variables can be integrated over R with respect to x and y and that will give you the triple integral clear. So, we are splitting the triple integral into two iterated integrals. So, one is a double integral other is the ordinary integral of one variable. Now, suppose we further this region R which is a projection into x y plane also is of either type 1 or type 2. So, let us write further. So, that will be using the further if this R the region R which is in x y plane x y belonging to R2 is of type 1. So, whatever is type 1 in the plane x lies between A and B y lies between some function. So, let us call that as xi of y theta of x sorry theta of x that was type 1. X lies between A and B and y goes from one lower curve to the upper curve. So, lower curve is ix upper curve is theta x. Then what does this triple integral look like? So, this triple integral over d f x y z dx y z can be computed. Now, this double integral when R is of this type I can write as integral x goes from A to B y goes from xi x to theta x. So, this was the double integral this was a function. So, what is that function? So, that function is z equal to eta x y to phi of x y f x y z dz right. Here is with respect to dy and then with respect to right. So, what we are saying is the triple integral if your domain is special. First it is projectable on to x y plane. Then I can split the triple integral into two parts. One is the double integral over the projected region R and the third variable that is z that is the ordinary integration of one variable. Further if the projected part looks like domain of type 1 in the plane then you can use the earlier Fibonius theorem. The double integral will look like x goes from A to B iterated integral x A to B y goes from some function psi x to phi x theta x and the double integral. So, this means that if the surface if the domain d is projectable on to x y plane and the projection itself is of type 1 then the triple integral becomes integration of one variable at a time right. So, it will be three integrations which will be coming with respect to x y and z iteratively. So, that is if the projection R is of type 1 supposing it was of type 2 that is also possible. Then what will happen then this one will split according to the type 2 right will be integrating y first and then with respect to x. So, this is when the domain is projectable on to x y plane. So, two possibilities right projection on to x y plane is possible and the projection is either type 1 or type 2.  Then again two possibilities will arise right and similarly on to say the y z plane. So, in all there are six possible iterated integrals possible for a function of three variables on a domain d right possible you can rise. So, let us try to do some examples on this. So, this is what it says. So, this is just recalling what we have said. So, this is a domain which is projectable on to x y plane. So, this is the region R which is the projection. So, every point you go from this lower surface right that is written here as h and g. So, from here to here right. So, can you see the pointer moving you cannot see the pointer moving. But I do not know whether you can write on top of it or not let us just see. So, if you take upon it good additional thing. So, if you go up you will start here and you will go up to here. So, that will be your z. So, for a point x y in the region R right you want to go it is projectable on to x y plane. So, you are moving along z axis. So, z goes from this region to this region. So, this is for the triple integral. So, it says if you can write it this way right projectable on to the x y plane. So, we will call it as type 1 elementary region. So, this is what x goes from A to B projection is again of type 1 right. So, x goes from A to B y goes from phi 1 to phi 2 and z goes from lower surface to the upper surface dx dy dz right. So, suitable conditions have to be put that namely this phi 1 phi 2 have to be continuous g is to be continuous and so on. So, we will assume all those nice conditions hold that it all the iterated integrals exist and R equal to the double integral. So, let us see. So, similarly if this projection is of type 2 that will be y going from C to D right x going from lower curve to the upper curve psi 1 to psi 2 and then z going from lower surface to the upper surface right. So, let us do one example. So, it is this example. So, we want to compute triple integral of the function 1 where D is the solid in R 3 bounded by the ellipsoid right. So, this is ellipsoid. So, ellipsoid is 4 x square plus 4 y square plus z square equal to 16. What does that ellipsoid look like? What is the ellipsoid? How do you visualize the ellipsoid? If you take z is equal to 0 what will that give me? z equal to 0 will give me the projection of this solid onto x y plane right and that looks like 4 x square plus 4 y square right. So, that is 4 x square plus 4 y square equal to 16 right okay. And similarly if you want projection on to other we have put y equal to 0 and projectible. So, this is a solid which is projectible on to all 3 coordinate planes x y, y z and z x. So, let us project it on to the x y plane that means z is equal to 0 you get 4 x square plus 4 y square equal to 16. So, what is that? So, that is a region R that is the projection okay. So, R is 4 x square plus 4 y square equal to 16. So, that gives me okay. So, I can write it as x y belonging to the region R the region R was the region R that is 4 x square plus 4 y square less than or equal to 16 right inside of that and that itself I can write it as there is a circle I can write it as a type 1 or type 2 right 4 x square plus 4 y square equal to 16. So, what will be that? So, x goes from so this is a region R okay and that itself I can write it. So, that will be the region R in terms of so that that is a circle. So, you can write type 1 or type 2 right 4 x square plus 4 y square equal to 16 that is x square plus y square equal to 4 right. So, type 1 where does x vary from that circle minus 2 to plus 2 lower part of the circle to the upper part of the circle or if you like you can write it also as type 2 where y goes from minus 2 to 2 x goes from the left side of the circle to the right side of the circle right. So, if you do that then you can write your so x goes from minus 2 to 2 that is a circle and where does y go from upper to the lower. So, what is the upper circle? So, upper circle is 4 minus x square positive square root upper part lower part is minus of that right. So, what does your domain look like? Domain looks like x goes from minus 2 to 2 right y goes from here to here and where does z vary from when you project you get the circle. So, what is the lower part of the circle that is a lower part of the ellipsoid to the upper part of the ellipsoid right. In the projection to be inside the domain D you will start with the lower part of the ellipsoid to the upper part of the ellipsoid. So, what will be the lower part of the ellipsoid? z in terms of y x and y. So, z square equal to so what are the equation of that ellipsoid? So, 16 right. So, you divide it by that. So, we will get z in terms of so I think it is quite clear. There is a lower part of the ellipsoid okay z square is it okay and that is upper part of the ellipsoid. So, this is a region ellipsoid when projected on to x y plane the points inside the ellipsoid can be described as x goes from minus 2 to plus 2. So, this is x goes from minus 2 to plus 2 and where does y vary from for every point x y varies from minus 4 minus x square to 4 minus x square plus square root. So, lower part of the circle to the upper part of the circle that is the region R. So, this is described in the region R the projection on to x y plane okay and then looking at to be z where in the region if you take a point in the region how high or how much high or how much low you should go to be inside. So, we will start at the lower part go to the upper part. So, this is a lower part and this is a upper part. So, what we have done is a function of three variables instead of integrating it three variable at a time right all together we have split integrate with respect to z integrate with respect to y and then integrate with respect to x okay. So, this is integrating one variable at a time. So, that is illustration of Fubini's theorem for okay. So, you can integrate all that and finally your answer comes out something. So, let us not bother about okay sign inverse and all that thing okay. Let us look at this. Let us find out what is what is integrate z dv say dx yz right that is also written as in short as d of v volume basically it comes from that. So, this dv is same as d of x y and z okay where d is the solid wedge in the first octant. So, your imagination also will come into picture in the first octant cut from the cylindrical solid by the planes y equal to x and x is equal to 0. So, how do we visualize that this is solid wedge in the first octant cut from the cylindrical solid. So, there is a cylinder y square plus z square less than or equal to 1 what is that cylinder if you want to just look at the cylinder y square plus z square less than or equal to 1 it is a three dimensional solid right cylinder three dimensional solid cylindrical solid. So, how do we visualize it is a cylinder with x is going through x axis it is symmetrical around x axis right x is independent. So, the x is goes minus infinity to plus infinity right and if you look at any point y square plus z square less than or equal to 1 at any point x how does y and z vary is a circle right. So, sections of the solid are circles y square plus z square less than or equal to 1 is that okay cylinder you are able to visualize okay in that cylinder we are looking at first octant that means we are looking at only part of the cylinder when x is bigger than 0 y is bigger than 0 and z is bigger than 0 on that part only right. So, you can see here. So, there is upper part only we are looking at in this picture where it is bounded it is bounded by the planes y equal to x that we do not bind it then it will be infinite right. So, y equal to x what is the plane y equal to x y equal to x is a plane right z is independent here that means what it is a plane which meets x y plane in the line y equal to x. So, in the in this line. So, this is a plane going up okay y equal to x how able to visualize y equal to x is a plane in R3 in right in coordinate plane x y it is a line but if you visualize it as a object as a set in R3 then it is a plane and what does it mean it is cutting the line it is cutting x y plane at the line y equal to x. So, it is going through that so it is a plane okay. So, that is one side of the plane bottom is x y plane anyway we are going up only right and then and x is equal to 0. So, what is x is equal to 0 that is a y z plane right x is equal to 0 the y z plane y and z vary. So, this is the wedge. So, this wedge is predictable on to x y plane right this wedge is predictable on to x y plane and what is the projection projection is this triangle okay bounded by y equal to x. So, this is the projection. So, that is visualized here in this side on the right hand side that is a projection okay. Is it clear the wedge when it is projected on to the x y plane that will give you a triangle. So, how do you write that now? So, x y z x and y z x and y in the projection z goes from bottom that is x y plane 0 how high it will go up to the part of the cylinder. So, that is z less than or equal to 1 minus x square okay. So, and the projection that are that is that triangle. So, let us write down the x between 0 and 1 y goes from the line y equal to x right that is y equal to x that line up to y equal to 1. So, it starts and that triangle it goes up to there or you can write other way around if you like the projection here is both type 1 and type 2 or you can write y equal to 0 to 1 x goes between 0 and x less than or equal to y either way it is okay. So, you can write down the integral by using Fubini's theorem it is a double integral over the region R which is a projection of z the function z integrated 0 to 1 minus y square right. So, that is the inside thing that is a projection is R which is type 1 or type 2. So, you can further write it as okay you can further compute 0 to 1 0 to y that integrated z integrated out gives you this and you integrate all. So, basically the idea is the region in R 3 you have to visualize whether you want to project it on to the x y plane or y z plane or z x plane which one it is easier to visualize okay the same way if you try to project it on to the y z plane that looks like very weird one and you cannot describe it as going from one point to another right. So, that is where your visualization power will come into picture.


Today we have a combination unboxing staff pick and writing sample for you This is the Tabalbi Bononia limited edition divine fountain pen and Deb and customer service fell in love with this Pin and she thought about it for a while and she made it hers So we're she's gonna let us ink it for you today And you notice the nib I already showed you the nib there This limited edition has an 18 karat gold nib with an engraving of Christ on the cross and a lot of detail It's really beautiful We're neat. We're gonna go ahead and ink it today with diamine writers blood ink Which is a very popular diamine ink and it comes out really dark and clear on paper. You can see here Deb picked this pin up in a medium nib But we have found that with Tabaldi pins when you get a Tabaldi pin with a gold nib It's gonna white it's gonna write. It's a really wet writer. So While this is a medium nib you might you might think gosh, that's a lot of ink for a medium nib But so with a Tabaldi gold nib you're gonna see that I really really liked the way this rose so smooth and the ink did flow really really well It just was there was like no friction against the paper and It's a beautiful pen. It's got a really warm white and brown resin that really Works well with that binonia silhouette and paired with the 18 karat gold nib with the very detailed engraving. It's a really beautiful piece Okay, and as you saw when I inked it, this is a piston filler and the filling mechanism is on the back of the barrel It is made in Italy and And it is a limited edition So if you want more information about this pin visit us pinchlay.com and see what nib sizes we have left Thanks for watching


Greetings, everyone. I'm speaking to you from the USS Gerald R. Ford, home of the Iron Nine Watch team, the first all-female listed-bridge watch section on a carrier. And earlier this year, I attended the USS Constitution Change of Command, where Commander Billy Farrell became the first woman to assume command in the ship's 225-year history. From the American Revolution to the 21st century, the history of our fleet and our nation has been strengthened by the contributions of women serving in our Navy and our Marine Corps. For many years, the contribution of women warriors remained hidden from view, yet their courage and their sacrifice often made the difference between victory and defeat throughout our nation's history. Recognized or unrecognized, they were trailblazers. We recognize women like Deborah Stamson, who many others, who as early as the American Revolutionary War circumvented the barriers to serve by disguising themselves as men. It's also the story of Loretta Walsh, a sailor who in 1920 became the first woman to legally enlist in the military and eventually became the Navy's first female chief pediocer. Trailblazers, like Olfa Mae Johnson, the first woman to enlist in the Marine Corps, and Laura Marmola, who in 2018 became the first African-American woman promoted to Brigadier General in the Marines. And it continues today with the stories of women like Captain Amy Baunschmidt, who commands USS Abraham Lincoln, currently deployed and operating in the South China Sea, and Lieutenant Commander Kyla Barron, who serves as a mission specialist on the NASA SpaceX Crew 3 on the International Space Station. As we mark Women's History Month, I ask you to remember all the women who served in the past and thank all who serve amongst us today. There are now over 85,000 women in our Navy and over 16,000 in our Marine Corps, 20,000 serve as civilians in our Department of the Navy. Today women operate every domain around the globe, but many of you can remember a time and that wasn't the case. Even while it was known that women were deployed to combat zones, women were restricted by law from serving in certain combat operational positions until 2013. And it wasn't until 2016 that all career restrictions were lifted for women in the service. This inequity did not just impact women in combat roles. In 1975, it took an order from President Ford, the namesake of the ship, to open our service academies to women. I was privileged to attend the U.S. Naval Academy with the first female graduates and served in the fleet with some of the first female surface warfare officers and naval aviators. And I'm very proud to have commanded one of the first Ollie Burke destroyers with a mixed-gender crew. For decades, these women were held back by others, arguing that changes would weaken our force and detract from the mission. And unfortunately, at times today, we still hear from some of those same negative voices. But time and time again, those voices are proven wrong. Throughout my career, and today as your secretary, I have seen firsthand what you all know. From the E-ring to the air wing, and from the deck plate to the battleground, with women in our ranks, our force is stronger today than it's ever been. But our work is unfinished. To protect our nation, the coming years we must recruit, retain, mentor, and promote the very best of all of our nation has to offer. Together, we must continue building a more inclusive force, defined by a worth fighting culture built on leadership, dignity, and respect that welcomes and encourages the service of everyone who is able and willing to serve. We must listen to our people and respond where we fall short. We will strengthen our promotion pipelines to recognize and value the service of all our enlisted and commissioned personnel, male and female. We will ensure our policies respect the needs of dual service and single parent families. And we will do everything in our power to ensure all who serve can do so in an environment that is safe and free from sexual harassment, or violence, or extremism of any kind. That is how we will truly honor our trailblazers, continues to support the legacy of empowered women, and build the strength of our Navy and our Marine Corps. So to all the extraordinary women who power our Navy and Marine Corps today, and those who honor them too, thank you for your service. I am truly honored to serve at your side as your Secretary of the Navy.


Okay, move out. They're commonplace in the war in Vietnam. Helicopters used as transports, carrying men, weapons, and materiel, soaring easily over terrain obstacles that would have immobilized ground vehicles. Helicopters bombarding the enemy with rockets, preying him with high-volume automatic weapons fire. Helicopters as aerial cavalry, swooping down from the skies to achieve the classic effects of speed and shock action, but with immensely more firepower than ground cavalry. Only a few short years ago, air-mobile operations was just an experimental doctrine, a concept never tested under actual battle conditions. It was only fitting and proper that air-mobile operations should have been first tried in combat by the First Cavalry Division of World War II in Korea fame. Writing helicopters instead of horses or mechanized vehicles, it was renamed the First Cavalry Division Air-Mobile. The air-mobile concept of using organic army aircraft to perform the combat functions of wheeled and tracked vehicles was carefully developed and tested in two-and-a-half years of intensive effort at the Army's Great Infantry Training Center, Fort Benning, Georgia. These revolutionary innovations were tried out by an experimental organization known as the Eleventh Air Assault Division. Its trained cadres and tested equipment provided a major component of the First Cavalry Division Air-Mobile when it was activated July 1st, 1965. Shortly afterward, it received orders to move from Fort Benning to South Vietnam. Its mission? To set up an air-mobile base in the central highlands of Vietnam. This base was called Anche. It set many precedents as an air-mobile combat base of operations. It was built around helipads. Compactness was a prime element in design. To make the base as impregnable as possible, it was encircled by a cleared strip 100 meters wide. Howitzer batteries were dug in ready to deliver pre-planned artillery fire on potential enemy mortar sites. The posts of riflemen and machine gunners dotted the perimeter of the base on watch day and night while combat patrols scoured the area on foot. Within the base, military police manned additional checkpoints and barriers. Another security force was posted on top of a mountain overlooking the base. Because of this strong web of defenses, Anche could be protected by a minimum number of men leaving the rest free to hunt down the enemy. So highly trained and motivated was the division that the Anche base serving every type of fixed wing and rotary aircraft was completed in record time. The army's first air-mobile division was ready to prove itself in battle only two months after being sent to Vietnam. The division was given a vast tactical area of responsibility ranging from the South China Sea to the Cambodian border and approximately 150 miles from north to south. Only certain parts of the coastal plain and nearby valleys were built up and populated. The interior of the operational zone was sparsely settled. Its highlands and mountains were thickly covered with jungle and forest. Into this wild, rugged environment, wholly favorable to a wily enemy skilled in guerrilla warfare the Sky Cavalry launched its first air assaults in September and October of 1965. Though limited in scope, these engagements with the enemy proved that the helicopter had brought remarkable new advantages to the strategy and tactics of ground warfare. Greater mobility for infantry forces and supporting artillery. Additional firepower, quick-firing howitzers on the ground. We're supplemented by aerial rocket artillery. As well as high-volume machine gun fire. Reconnaissance by fire gained ineffectiveness. When door gunners swept suspected areas of concealment with machine gun bursts the enemy was provoked into shooting back thus revealing his presence and location. Command posts could be made air-mobile so that unit commanders were able to direct air and ground attacks from a vantage point right over the battlefield. Equipment damaged in the field could be retrieved by air, swiftly repaired and returned to battle. Logistics operations were speeded up enormously. Vital cargo was delivered in record time right into the hands of troops in the battle zone. Helicopters as flying ambulances saved many lives. Carrying the wounded from firing line to field hospital, sometimes in a matter of minutes. All these advantages of air-mobile warfare were confirmed in the first days of combat. This early baptism by fire gave the Division invaluable experience for its first major engagement, the Plaku Campaign. This was to have historic significance. For many years, North Vietnamese and Viet Cong forces had been operating with impunity in the area of Plaku Province. Securing a remote sanctuary near the Cambodian border a North Vietnamese Army Division was getting ready for a major offensive. This was to be a powerful thrust eastward with the ultimate objective of cutting South Vietnam in two. The first step in this plan was carried out on October 13th when the North Vietnamese 33rd Regiment attacked the Special Forces Camp at Playa Naye. This attack was meant to lure the South Vietnamese and ascending a strong relief force to the aid of Playa Naye. Another North Vietnam Army regiment, the 32nd, was waiting in ambush along the line of March. The trap was sprung on the unwary arvan troops on October 24th. It almost succeeded. However, the South Vietnamese fought back strongly even though taken by surprise. The first cav was called to the rescue. Air assault infantry was rushed into the jungle to secure an artillery landing zone. They were sent from the Division's advanced command post at Plaku to an area close to the ambush site north of Playa Naye. Known as LZ Field Go, this area was immediately occupied by artillery flown in from Plaku. These guns supported the arvan troops in their effort to break out of the ambush and protected the relief column when it resumed its march to Playa Naye. At the same time, another battery harassed the North Vietnamese 33rd Regiment attacking the Special Forces Camp. This shelling was so accurate, the MVA broke off its siege and withdrew to the west. As a result, the Air Cav's limited role was changed. By order of General Westmoreland, it was sent on an all-out offensive to find and engage the enemy, thus blocking further attack plans. Air Cavalry Squadrons promptly searched large areas west of Playa Naye, seeking out communist forces concealed in the lush foliage below. Helicopters swooped down to skin the ground at low level, sweeping the landscape with reconnaissance by fire to provoke the enemy into revealing himself. Next, infantry battalions were air assaulted deep into North Vietnamese strongholds. The rifle companies were dispersed over several landing zones. Each company set out patrols to search aggressively for the enemy. On November 1st, 1965, the Skytroopers hit Maynard. They attacked and overran an enemy outpost on the Yate, the Teh River, a full-sized battle erupted when the North Vietnamese Army threw in a battalion of men against three Air Cav platoons. However, helicopter troop carriers took off night and day, pouring reinforcements onto the battlefield. It was a brilliant display of the air mobility to replace combat weary and wounded soldiers with fresh troops. Assisted by deadly accurate artillery support, air force tactical strikes, and aerial rocket artillery, the troopers chopped up the NVA 33rd Regiment that had besieged Playa Naye. The enemy was forced to withdraw to its base camp at Anta Village in the Yadrang Valley. The North Vietnamese were stopped cold for the moment. The 1st Cav continued to push westward in its relentless search for the enemy, licking its wounds in its hideouts in the Yadrang Valley. As part of this continuing search, a battalion of Skytroops flew into a tiny landing zone. Soon to be famous under the name of LZ X-Ray. LZ X-Ray was a clearing deep in the valley of the Yadrang and located at the foot of the mountains of the Choupon Massif in the very heart of enemy country. The NVA, reinforced by units of the newly infiltrated 66th Regiment, made an all-out effort to destroy the air-caved battalion. Their idea evidently was to make X-Ray the scene of a miniature Dien Bien Phu. They had not reckoned on the valor of the Skytroopers, who fought all the more fiercely when units were cut off and temporarily isolated from their comrades in arms. Nor did the enemy understand how the seemingly unprotected infantryman could call in the support of air-mobile artillery with such unbelievable speed. Shelled incessantly. Blasted by aerial rockets. Seared with fiery strikes by U.S. Air Force fighter bombers. The NVA was drenched in a rain of fires that decimated his attacking forces. The Skytroopers had to pay a price, but they had turned the Communist plan into a disastrous nightmare. The enemy losses in dead and wounded were tremendous. Once more, the enemy survivors withdrew to regroup their broken units, but they were hounded by demoralizing B-52 raids that pounded them in their mountain strongholds. The X-ray victors were ordered to get out of the way of the B-52 bombings by marching overland to two landing zones. One was an artillery landing zone dubbed LZ Columbus, and the other was for helicopters, its name LZ Aldenay. The Columbus-bound unit reached the LZ without incident. But the truth was that the LZ Aldenay ran headlong into a strong enemy column marching to attack the air-mobile artillery at LZ Columbus. A fierce firefight resulted. It was, in the end, another disaster for the enemy. By November 24th, the NVA was recoiling in defeat, abandoning its camps in the Yardrang and the Chupong Mountains. They retreated, taking their way to the LZ. Left behind were almost 2,000 enemy dead in prisoners, plus large stores of supplies and many weapons. The Plaku Campaign, better known to first-cav-men as the Battle of the Yardrang Valley, was over. Its effect was profound and long-lasting. The American victory proved that the LZ Aldenay was the only country in the United States The American victory proved that the best of the enemy forces, North Vietnam Army Regulars, could be stopped cold and destroyed in their own sanctuaries. No longer could the communists hope to win the war on the battlefield. Most important to the men of the 1st Cavalry Division, Yardrang had proved beyond question the soundness of the air-mobile approach to ground warfare. It was a well-deserved award when the air-cav received a presidential unit citation for the entire division and its supporting units. An award so rare, it had been made only five times in Army history. The division's main thrust in 1966 turned toward the Central Coastal Region of South Vietnam, particularly in the area of Bong Son, a town which gave the campaign its name. Strong air-mobile forces carried out a lightning series of forays against the Viet Cong, who held over 140,000 people in subjection. Each operation was carried out with polished efficiency, using tactics now thoroughly tested in battle. Devastating pre-strike bombardment by Air Force fighter bombers. Attached supporting artillery. Gunships darting in to strafe the landing zone perimeter with rocket fire. Automatic weapons. Infantry assault waves rushing from troop carriers to set up a defense line on the LZ perimeter, while succeeding waves of riflemen advanced through them. Beginning with an airlift into the An Lao Valley, the troopers pushed through all kinds of terrain in their pursuit of the enemy. They routed out the VC from well-camouflaged nests of bunkers and tunnels. The fighting was bitter, and casualties inevitable. By the spring of 1966, scores of villages had been flushed to the enemy, and hundreds of suspects identified for questioning. The immediate threat to Bong Son and other coastal cities was ended. In the operations that followed during the rest of the year, Thayer, Pershing, John Paul Jones, Paul Revere, and many more, the first Cav, often working with Arvin or Korean troops, continued to clear the huge rice-producing province of Bendin of VC control. The cavalrymen took time whenever they could to feed the liberated South Vietnamese who had been stripped of their crops by the Vietcong. Many Vietnamese received medical care for the first time for ailments that had afflicted them all their lives. The speedy air-mobile assaults of the first Cav brought impressive results, as many NVA soldiers surrendered, bringing large quantities of their weapons with them. With every operation, the first Cav continued to improve its air-mobile techniques. One lesson learned was to travel as light as possible. Troopers carried only weapons, ammo, and two canteens of water, depending on helicopters for further supply. The artillery learned how to be weight watchers, too. The M102 version of the 105-millimeter holizer was 1200 pounds lighter than the earlier model. Bokey radio equipment was scaled down in size and weight. This assembly of two radio sets and an intercom was reduced from 600 to 150 pounds. Collapsible fuel bladders, each with a 500-gallon capacity, replaced metal containers, and were air-transportable. This technical ingenuity, always air-mobile oriented, paid off as operations rolled on into 1967, with Pershing, Wheeler, Jeb Stewart, and others. The pressure on the enemy never relented as the choppers took to the air day and night in foul weather as well as fair. Nowhere in this wild, trackless country was the Viet Cong safe from air-mobile probes. They were pursued deep into the jungle, sawed out high on mountaintops. They were challenged in their most distant hideouts. It made no difference that the first Cav men who had come to Vietnam in 1965, the original first team, had all been rotated home. It was a completely fresh division now, but it was still very much the first team. They took over with smooth professionalism, planning each operation minutely, then lashing out at the Viet Cong with swift, stunning air assaults. The most battle-hardened enemy troops broke under the rain of fires that lashed at them incessantly from the air, and from the ground, as giant 8-inch howitzers reinforced the air-mobile batteries. Throughout 1967, the first Cav troopers constantly took the initiative, showing that they too could use the cutting arts of guerilla warfare, stalking the enemy across rice paddies, swamps through jungle and forest, frequently stopping to slug it out in one firefight after another, virtually toe-to-toe, breaking his will to resist, destroying him when he took refuge in tunnel or bunker. The toll among the Viet Cong and North Vietnamese rose steeply, with a record number killed in Operation Jeb Stewart. While scores of others surrendered. It was a grind, but only the troopers' feet tired, not their fighting spirit. They were heartened by the armor that joined the ground forces whenever terrain allowed. Armored personnel carriers and foot soldiers spraying enemy positions with machine gun fire, small arms, grenades.  When he chose to fight because of superior numbers, the first CAV poured in reinforcements with air-mobile swiftness. This quick reaction capability paid off early in 1968. When the Viet Cong launched its Tet offensive against South Vietnamese cities, diversionary tactics to pin down first CAV units were quickly thrown back by the fast-moving, hard-fighting Skytrucks. Not long afterward, the Air CAV men played a major role in the relief of Que Son, where a marine regiment and South Vietnamese troops were encircled by North Vietnam Army units. To relieve them, elements of the first CAV were committed to Operation Pegasus in April of 1968. Jumping off from an advanced base at LZ Stud, the Skytroopers set out to link up with the defenders of Que Son. Resistance was light. The air-cavalry men easily broke through and found Que Son to be a very dusty place. When the dust settled, it was business as usual for the Skytroopers. The aerial supply chain went into high gear. Attack plans were made with the usual first CAV deficiency. The troopers went about the job they knew and liked best. To engage the enemy, Harris and punish him with firepower. Men fearlessly closed with him, storming the bunkers and strong points with which he had surrounded the marine base. This cleanup operation was pursued vigorously into the summer of 1968, when Que Son, having outlived its strategic usefulness, was closed down. The northwest corner of South Vietnam, where Que Son is located, has a long irregular border in common with Laos. It is ideal country for infiltration. The North Vietnamese were quick to move troops and supplies into the remote Achao Valley, via the Ho Chi Minh Trail. High in the mountains, the valley is covered with a triple canopy jungle, 100 feet tall, and shrouded in fog and mist 10 months of the year. This near-perfect sanctuary was broken into by the air-cavalry men in the spring of 1968. The flying troopers coordinated their battle actions with paratroopers of the 101st Airborne Division and the South Vietnamese Army. Murderously heavy fire from high-altitude anti-aircraft guns attempted to turn back the Achao invasion. Several aircraft were lost, but the troop-carrying choppers made their touchdown. The cavalrymen found an extensive road net, but few signs of the enemy in person. There was little resistance. The North Vietnamese had literally run for the hills. The cavalrymen consoled themselves with the knowledge that they had captured, virtually intact, one of the greatest storehouses of weapons and war material ever assembled by the communists in South Vietnam. The air-cav met each blow with lightning-swift counterblows, sending its air-mobile power ranging far and wide to support other units of the Allied forces in combined operations. The primary confident in its brilliant battle record and the success of its air-mobile doctrine, now accepted throughout the United States Army, the 1st Cav continues to serve far from home as long as needed. The pioneering days of air-mobile operations are past. Changes and improvements impend. Bigger, more powerful army aircraft promising even more effective helicopter tactics. In these advances, the 1st Cavalry Division air-mobile will as always lead the way.