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Space Trip 5. Space Shoot 5. Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by .
It has many applications in mathematics, physics , and engineering. The space and product form an algebra over a field , which is neither commutative nor associative , but is a Lie algebra with the cross product being the Lie bracket.
But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.
This expands as follows: . A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral.
To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S , by considering a system of curvilinear coordinates on S , like the latitude and longitude on a sphere.
Let such a parameterization be x s , t , where s , t varies in some region T in the plane. Then, the surface integral is given by. Given a vector field v on S , that is a function that assigns to each x in S a vector v x , the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
A volume integral refers to an integral over a 3- dimensional domain. The fundamental theorem of line integrals , says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
If F is a continuously differentiable vector field defined on a neighborhood of V , then the divergence theorem says: .
The left side is a volume integral over the volume V , the right side is the surface integral over the boundary of the volume V.
Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers.
For example, at least three dimensions are required to tie a knot in a piece of string. Many ideas of dimension can be tested with finite geometry.
The simplest instance is PG 3,2 , which has Fano planes as its 2-dimensional subspaces. It is an instance of Galois geometry , a study of projective geometry using finite fields.
For example, any three skew lines in PG 3, q are contained in exactly one regulus. From Wikipedia, the free encyclopedia.
Geometric model of the physical space. For a broader, less mathematical treatment related to this topic, see Space.
For other uses, see 3D disambiguation. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations.
Please help to improve this article by introducing more precise citations. April Learn how and when to remove this template message.
Projecting a sphere to a plane. Outline History. Concepts Features. Line segment ray Length. Volume Cube cuboid Cylinder Pyramid Sphere.
Tesseract Hypersphere. Main article: Coordinate system. Main article: Sphere. Main article: Polyhedron. Main article: Surface of revolution. Main article: Quadric surface.
Main article: Dot product. Main article: Cross product. Jet manages to rescue Mitch, using a miniature rocket unit.
Frank Rogers, of 2, tells Lemmy he's not getting on at all well with Whitaker, his shipmate. Later, Control requests information about Whitaker; and Whitaker starts behaving very strangely while Jet is questioning him.
While Whitaker is in a strange deep sleep, both Frank and Lemmy have nightmares. Radar suddenly reveals a huge meteor swarm ahead of the fleet, blocking its path.
Just as Lemmy escorts Whitaker to his new ship, 7, a large meteor destroys the ship. Control discovers very odd information about Whitaker - the only man answering to his description was apparently born in , and has been missing since The fleet changes course twice to avoid the meteor swarm, but both times it moves to block their path again.
The crew realise they have only one option: to head straight through the swarm. The fleet enters the swarm, and for nearly 7 hours all electronic equipment is badly disrupted.
Peterson reports an emergency on board 6 with Whitaker. On leaving the swarm, 6 is nowhere to be seen, and the swarm is now blocking radio contact with Control.
Two weeks later, Discovery apparently re-establishes contact with home, but as they prepare to take an important message, the crew are puzzled by the short delay between replies.
A radio message apparently from Control orders the fleet to return home. However, a bearing on the radio signal reveals the transmitter to be somewhere in front of the fleet, thus proving the message to be a fake.
Jet decides they must enter the ship to investigate. Jet and Doc transfer to 6, and Whitaker is found badly injured in the ship's hold, but Peterson has vanished.
Jet has a strange dream, and Whitaker dies from his injuries, suddenly transforming into an old man. Recording tapes found in the cabin confirm the crew's suspicions that Whitaker faked the radio message.
Discovery and 6 both rejoin the fleet, and Jet and Lemmy are shocked to hear Peterson's voice on the radio. Peterson's suicide recording is found, and radio contact with Control is restored.
The fleet finally enters orbit round Mars, and Discovery lands at the South Pole, but a strange soporific noise troubles the crew and causes them to crash-land.
Jet reveals that the strange noise caused him to black out, but the crew are all safe. Whilst conducting experiments, they soon get lost in a dense fog.
They see a mysterious orange light, mistaking it for the landing light of the Discovery, and the same soporific noise heard during the landing causes them to lose consciousness.
Doc and Frank rescue Jet and Mitch using a land transport truck, and Jet says he dreamed of a ruined city in a valley.
Discovery' s crew begin the exploration of Mars in the land trucks, whilst Freighter 2 begins to ferry supplies down from the fleet in orbit.
However, on its second trip down, the freighter pilot hears the soporific noise and the ship apparently crashes. Racing to reach the crashed freighter before sunset, the Discovery's crew reaches the Mare Australis, the plain north of the ice cap.
After nightfall, they see a mysterious white light moving across the plain. In the morning, they spot the crashed freighter, but upon reaching it find its crew missing.
However, there are strange marks in the ground nearby which Jet deduces to belong to "the light", realising that a ship has visited and kidnapped the freighter crew.
Jet resolves to change the exploration route and go after the ship. The Discovery crew travel west across the Mare Australis in pursuit of the mysterious ship.
Meanwhile, Frank and Grimshaw, currently crew members of Freighter 1, reach the wrecked 2 and take up residence for the night.
However, during the night, a 'sphere' spherical UFO arrives and its crew attempt to enter the wreck. Whilst observing the crew, Frank realises they are human.
A few days later, Discovery's crew reach a valley in the Argyre Desert, and Jet spots a city which is identical to the one in his earlier dream.
Discovery' s crew explore the valley's ruined city, but Lemmy vanishes, and experiences a hypnotic dream.
He's later found with McLean pilot of 2 , who is now mysteriously able to breathe the Martian atmosphere without a helmet.
Jet and Doc return Lemmy to the caravans, and Doc deduces that Whitaker and McLean had been 'conditioned' hypnotised , allowing them to be controlled, as well as breathe in the Martian atmosphere.
Mitch becomes separated from the others, and meets a man who claims to be a dingo hunter in Australia. Meanwhile McLean, disguised in Mitch's spacesuit, enters the caravan and confronts Lemmy.
McLean reveals that only two of his three crew mates survived the crash, and declares that Mitch is alive but will never return to the fleet or Earth.
He then tries to overpower but Lemmy wins the fight. The sphere is spotted on top of the city, but takes off with Jet, Lemmy and Doc in pursuit again.
Meanwhile, Mitch, lost in the desert and now able to breathe the Martian atmosphere, comes across a house inhabited by a man and woman, who claim to be sheep farmers living in Australia in Driven to a frenzy, Mitch demands explanations, only for the farmer to turn his rifle on him.
Jet, Doc and Lemmy discover the farm house, and find Mitch inside it. But Mitch is now 'conditioned' to believe he's in Australia, and doesn't recognise the others.
He refuses to rejoin them, and attacks Jet, puncturing his helmet. As the others work to save him, the sphere lands outside and the crew are trapped.
A 'flying doctor' has arrived in the sphere, with Dobson and Harding of 2 'conditioned' and working as his assistants. Although Jet successfully escapes back to the land trucks, the doctor manages to capture Doc and Lemmy.
He tries to 'condition' them using the soporific noise, but they manage to resist. They find themselves being taken to a much larger Martian city.
Jet sets out after the sphere again, having ordered McLean to follow him in the second caravan. Meanwhile, the sphere arrives at Lacus Solis the Martian capital , and the flying doctor reveals the Martian plan to invade the Earth in Lemmy overpowers the doctor in a struggle, and he and Doc contact Frank, who is searching for them in 1.
Through him, they pass their news to Jet, but then Frank hears the soporific noise and his freighter crashes. Doc and Lemmy escape the city in the sphere with Mitch, who starts to recover from his 'conditioning', but can only remember being two weeks away from reaching Mars.
Meanwhile, Jet is contacted by Webster, a man trying to return to Earth, and learns more about the Martians.
Webster offers to help Jet find his lost crew mates, but while travelling across the desert, they spot a sphere hovering above them.
The sphere turns out to be Doc and Lemmy's, and Discovery crew are fully reunited at last Thanks to advice from Webster, Mitch completes his recovery.
Jet, Lemmy and Webster try to rescue Frank and Grimshaw from an underground factory, but they cannot be reached, and during a fight with the factory supervisor, Webster is injured and dies.
The crew return to Discovery at the North Pole, and take off for home, escaping a pursuing fleet of spheres.
Their news of the upcoming Martian invasion is passed to Earth, causing great commotion. They land back on the Moon six months later, but what happens next is another story The Discovery and two freighters return to the Moon from the disastrous Mars expedition.
The crew are subjected to intense questioning about their experiences on Mars, and Lemmy discovers that the Discovery is being readied for take off. Back on Earth, they visit an astronomical observer who has spotted suspicious objects in the heavens.
Jet and Lemmy go to investigate the objects in an orbital rocket, but are sent to sleep for 24 hours by a strange noise. Film footage suggests the asteroids are actually spacecraft carriers.
Jet discovers that a spaceship has crashed in the Lake District. Jet and Doc enter the ship. The crew and the Lunar Controller try to locate the crew of the crashed ship.
They interview Moore, a commercial traveller who may have had contact with the crew of the ship, who recounts his experience. The crew are ordered back to Mars and are driven to an airfield, but the car crashes when the crew tackle the 'Conditioned Type' driver.
The crew fly to Australia, then on a ship to the Moon. While preparing for the voyage to Mars, they spot unidentified objects on the lunar horizon.
The take off is complicated by strange communication with control, suggesting the Moon base has been infiltrated. The Discovery gets away safely; however, they cannot establish contact with Moon control, so Lemmy contacts Earth.
After ships from Earth to the Moon are lost, the crew concludes that the Moon has fallen under Martian control.
While refuelling from freighter 1, Jet and Lemmy hear a strange wailing voice. The crew slow the ship down, and discuss where on Mars to land and the objectives of the trip.
They regain contact with the strange voice, discovering it is a very confused Frank Rogers, ex-crew man of Freighter 2. Thinking Frank may be there, the crew decide to land near the wreck of freighter 2, but Jet and Mitch discover it to be empty.
From the Discovery, Doc and Lemmy spot a fleet of Martian spheres approaching. Jet and Mitch fail to make it back to Discovery , and the crew all pass out as the ships approach.
They wake up in a dark room and figure that they must be on one of the moons of Mars. They discover an unconscious man in the room, that Jet is missing, and a door in the floor opens.
Jet is found under his bed and the unconscious man is found to be Frank Rogers. Doc tries to bring Frank out of his conditioned state, and he reveals that the invasion is under way.
Jet and Lemmy go through the floor, and are directed by the voice of Paddy Flynn. Frank regains some of his memory, revealing that the crew are aboard an asteroid ship.
Jet and Lemmy meet Paddy face to face, and he reveals that he is the rebel captain of the asteroid, working with Jack Evans.
Paddy receives orders to have the asteroid join the invasion fleet. Paddy leaves the crew, who learn how to manipulate the 'visionphone'.
As Mitch and Frank follow them along a circular corridor using the cameras, Jet, Lemmy and Doc hear a voice they assume to be a Martian, ordering them back.
Jet, Lemmy and Doc ignore the Martian's warnings to turn back, and Jet is knocked out after touching a door at the end of the corridor.
The crew persuades Paddy to set the ship on course for Mars, so they can warn Earth of the invasion. However, he loses control, and is injured.
Mitch and Lemmy go to fetch a bed for Paddy, but become lost and enter the Martian's lair. They hear a familiar voice but can't place it.
Paddy dies, and Mitch and Lemmy discover that Jack Evans is really the Lunar Controller, who has been in league with the Martians from the start.
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