Dimension Definition and 906 Threads

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.
In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.
The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

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  1. J

    Motion in One Dimension - Finding two separate changes in time

    Homework Statement A commuter train travels between two downtown stations. Because the stations are only 1.30 km apart, the train never reaches its maximum possible cruising speed. During rush hour the engineer minimizes the travel interval Δt between the two stations by accelerating for a...
  2. A

    Calculating Overtaking Time and Distance in One-Dimensional Motion

    Kathy Kool buys a sport car that can accelerate at the rate of 4.90 m/s2. She decides to test the car by racing with another speedster, Stan Speedy. Both start from rest, but experienced Stn leaves the starting line 1.00 s before Kathy. Stan moves with a constant acceleration of 3.50 m/s2 and...
  3. M

    Kinematics 1 Dimension, Person Running Solve for Time

    Homework Statement A runner hopes to compleate the 10,000m run in less than 30.0min. After running at a constant speed for exactly 27.0min there are still 1100m to go. The runner must then accelerate at 0.20 m/s^2 for how many seconds in order to achieve the desired time. *note* person does...
  4. D

    Is Gravity Considered a Dimension in Physics?

    Hey everyone. I was recently on youtube, looking at 4-d objects, when I came across a comment which caught my eye: "You're forgetting Gravity. Without gravity there is no xyz and without two large bodies there is no concept of time. Therefore, gravity must be included as a dimension. They are...
  5. S

    Why phase transition not possible in one dimension?

    can anybody tell me why in general phase transition is not possible in one dimension? and for a lattice gas at low temp. why it requires long range order to occur phase transition?
  6. E

    Runner's motion in one dimension

    Homework Statement A runner hopes to complete the 10,000-m run in less than 30.0 min. After exactly 27.0 min, there are still 1100 m to go. The runner must then accelerate at 0.20 m/s2 for how many seconds in order to achieve the desired time? Homework Equations vf = vo + at avg...
  7. B

    An equation related to the dimension of linear operator

    Homework Statement Let V be a finite vector space, and A, B be any two linear operator. Prove that, rank A = rank B + dim(Im A \cap Ker B) The Attempt at a Solution Since rank A = dim I am A dim(Im B)+ dim(Ker B)=dim V dim(Im A + Ker B)=dim(Im A)+dim(Ker B)-dim(Im A \cap Ker B) It...
  8. C

    Time - a Dimension of Physics or Mathematics?

    I have read a number of threads that argue the ontological status of time in physics. One of the most concrete "descriptions" of time in physics is offered by Huw Price. Huw Price suggests time is symmetrical, but the order of events in time are not. http://arxiv.org/abs/physics/0402040...
  9. I

    Trace(matrix) = 0 and the dimension of subspace

    Homework Statement Claim: Matrices of trace zero for a subspace of M_n (F) of dimension n^2 -1 where M_n (F) is the set of all nxn matrices over some field F. Homework Equations Tr(M_n) = sum of diagonal elements The Attempt at a Solution I view the trace Tr as a linear...
  10. M

    Proving Inequalities Involving Vector Space Dimensions

    Hi all! I´m trying to prove following two inequalities but I somehow got stuck: U, W are subspaces of V with dimV = n 1) dimV >= dim(U+W) 2) dim(U+W)>=dimU and dim(U+W)>=dimW Could you give me some hints? thanks in advance!
  11. E

    Compactness and space dimension question

    I am having trouble accepting two well known results of analysis as non contradictory. First, given a vector space equipped with norm ||.||, the unit ball is compact iff the space is finite dimensional. Second, the Arzela-Ascoli theorem asserts that given a compact set X, a set S contained...
  12. G

    Elastic collision 2 dimension unequal masses not at rest

    Please help! i know how to do a elastic collisions in 1 dimension, but the 2D is too confusing...here is my problem: You have a blue ball with a mass of 1.5 kg moving with a speed of 4.5 m/s in a direction below the positive x-axis. You have a red ball with a mass of 3.6 kg moving with a...
  13. M

    Basis for null space, row space, dimension

    Homework Statement What are the basis for the row space and null space for the following matrix? Find the dimension of RS, dim of NS. [1 -2 4 1] [3 1 -3 -1] [5 -3 5 1] Homework Equations dim RS + dim NS = # of columns The Attempt at a Solution I reduced the matrix into...
  14. B

    Is Time a Dimension? Exploring the Differences

    is time a "dimension" I have been trying to figure this one out for some time now. I have read much on the subject, but it seems to be dealt with in such a cavalier fashion. So, here goes... We speak of space-time as if it were something packaged. I have a problem with this. Let me list the...
  15. C

    Lebesgue measure vs. box counting dimension

    Short version: What is the difference between the Lebesgue measure and the box counting dimension of a set? Long version: I was reading up on the definition of the Lebesgue measure, and the description of how to take the Lebesgue measure of a set (which I understood basically as "cover the...
  16. P

    Simple physics kinematic one dimension problem?

    Homework Statement Driving along a crowded freeway, you notice that it takes a time t to go from one mile marker to the next. When you increase your speed by 5.0 mi/h, the time to go one mile decreases by 13 s. What was your original speed in mi/hr? Homework Equations X (which in...
  17. P

    How Do You Calculate Initial Speed in an Inelastic Collision?

    1. Kevin has a mass of 87kg and is skating with in-line skates. He sees his 22-kg younger brother up ahead standing on the sidewalk, with his back turned. Coming up from behind, he grabs his brother and rolls off at a speed of 3.4 m/s. Ignoring friction, find Kevin's speed just before he grabbed...
  18. M

    What Acceleration Is Needed for Ion Deflection in Cancer Treatment?

    Homework Statement You are asked to consult for the city's research hospital, where a group of doctors is investigating the bombardment of cancer tumors with high-energy ions. The ions are fired directly toward the center of the tumor at speeds of 4.5 . To cover the entire tumor area, the...
  19. G

    Tricky Kinematics: Projectile Motion in 2D

    hi, I am new to this forum and i have been having great difficulties with this question... Homework Statement An airplane with a speed of 81.6 m/s is climbing upward at an angle of 44.7 ° with respect to the horizontal. When the plane's altitude is 944 m, the pilot releases a package. 1)...
  20. M

    Diffusion equation in d- dimension

    i know that idea would seem a bit weird but, let us suppose we have a surface or volume in d- dimension, here d can be any real number (fractional dimension) the question is that we do not know what value 'd' is \frac{\partial \phi}{\partial t} = D\,\Delta \phi D is a diffusion...
  21. S

    Acceleration in 1 dimension (Conceptual)

    Homework Statement Two cars C and D travel in the same direction on a long, straight section of a highway. During a particular time interval \Deltat0, car D is ahead of car C and speeding up while car C is slowing down. During the interval \Deltat0, it is observed that C gains on car D...
  22. Z

    How Do You Calculate Initial Velocity and Angle in Two-Dimensional Kinematics?

    Homework Statement Useful background for this problem can be found in Multiple-Concept Example 2. On a spacecraft two engines fire for a time of 871 s. One gives the craft an acceleration in the x direction of ax = 6.00 m/s2, while the other produces an acceleration in the y direction of...
  23. V

    Dimension of Hilbert Space in Quantum Mechanics

    We know that the Hilbert space of wavefunctions can be spanned by the |x> basis which is a non-countable set of infinite basis kets. Now consider the case of a particle in a box. We say that the space can be spanned by the energy eigenkets of the hamiltonian (each eigenket corresponds to an...
  24. A

    Physics Kinematics in One Dimension

    Wow... I'm fried from doing this problem... Any help would be nice... A 1000kg weather rocket is launched straight up. The rocket motor provides a constant acceleration for 16s, then the motor stops. The rocket altitude 20s after launch is 5100m. You can ignore any effects of air...
  25. X

    Exploring the Concept of a Fourth Dimension: Time or Space?

    What is a fourth dimension?? Is this time or any other space coordinate??
  26. N

    Dimension of vector space problem

    Homework Statement Suppose V1 (dim. n1) and V29dim. n2) are two vector subspaces such that any element in V1 is orthogonal to any element in V2.Show that the dimensionality of V1+V2 is n1+n2 Homework Equations The Attempt at a Solution The subspace V1 is spanned by n1 linearly...
  27. I

    Conformal dimension of a scalar function?

    Actually, I'm not fully understand what the meaning of conformal dimension is. But I know how to read off the conformal dimension of a tensor, say, t^{++}{}_+, then the conformal dimension is -2 + 1= -1, where the lower index carries conformal dimension 1 and upper index carries conformal...
  28. H

    Vectors and Motion: Calculating Average Velocity and Speed

    Homework Statement Suppose the positon vector for a particle is given as a function of time by vector r(t) =x(t)i + y(t)j with x(t) =at + b and y(t) = ct2 + d, where a=1.00 m/s, b=1.00m, c=.125 m/s2, and d=1.00m. (a) Calculate the average velocity from t=2.00s to t=4.00s. (b) Determine...
  29. P

    So confused Kinematics in one dimension

    A hot air balloon is ascending straight up at a constant speed of 8.80 m/s. When the balloon is 12.0 m above the ground, a gun fires a pellet straight up from ground level with an initial speed of 31.0 m/s. Along the paths of the balloon and the pellet, there are two places where each of them...
  30. X

    Motion in one dimension- Find acceleration and original velocity

    Homework Statement A truck covers 40.0m in 8.50s while smoothly slowing down to a final velocity of 2.8 m/s. a) Find the truck's original speed b) Find its acceleration Homework Equations a= [v-vo]/t-to Constant acceleration: v=vo + at x= xo + vot + (1/2)at2 Constant velocity...
  31. X

    Help with kinematics in one dimension free fall

    Homework Statement A wrecking ball is hanging at rest from a crane when suddenly the cable breaks. The time it taktes for the ball to fall halfway to the ground is 1.2s. Find the time it takes for the ball to fall from rest all the way to the ground Homework Equations v=vo + at The...
  32. P

    Usefulness of a 4th spatial dimension

    Are the any thoughts on this paper (which I copied the first part of as a reference)? They seem to be saying the existence of a 4th spatial dimension would help resolve questions. But, I think I've read other places that speculation of this type is meaningless. Hierarchy Problem in the...
  33. S

    Calculating Minimum Deceleration and Meeting Heights in Kinematics

    Homework Statement (1) A speed trap is set up with two pressure activated strips placed across a highway, 116 meters apart. A car is speeding along at 29.6 meters per second, while the speed limit is only 19.4 meters per second. At the instant the car activates the first strip, the driver...
  34. J

    Motion in One Dimension (Using Calculus)

    Homework Statement The speed of a bullet as it travels down the barrel of a rifle toward the opening is given by v = (-5.95 multiplied by 107) t 2 + (2.45 multiplied by 105) t, where v is in meters per second and t is in seconds. The acceleration of the bullet just as it leaves the barrel is...
  35. P

    Analyzing the Velocity and Direction of Fuel Tank Drop in Two-Plane Scenario

    1. Two planes are each about to drop an empty fuel tank. At the moment of release each plane has the same speed of 135 m/s, and each tank is at the same height of 2.00 km above the ground. Although the speeds are the same, the velocities are different at the instant of release, because one plane...
  36. P

    Kinematics in One Dimension Problem

    1. In getting ready to slamdunk the ball, a basketball player starts from rest and sprints to a speed of 6.0m/s in 1.5seconds. Assuming that the player accelerates uniformly, determine the distance he runs 2. Vf=Vo+at, VF^2=Vo^2+2a(Delta x) 3. Vf= 0m/s+(6.0)(1.5)= 9 Delta X=...
  37. L

    How to Calculate Height of a Cliff Based on Falling Object Time?

    i have a quetion for you Dr., I know this is a simple problem but my physics book is not very clear on how to figure out this answer to the question. "A stone dropped from the top of a cliff. it hits the ground below after 3.25s. how high is the Cliff?" *to figure this out do you not use this...
  38. redtree

    Why is the time dimension multiplied by c^2?

    Why is the time dimension multiplied by c^2?
  39. K

    Determining whether three points lie on a straight line in three dimension

    Homework Statement Determine whether the points lie on straight line A(2, 4, 2) B(3, 7, -2) C(1, 3, 3) Homework Equations The Attempt at a Solution I've looked up at the equation for lines in three dimension, and it appears to be x=x_0+at y=y_0+bt z=z_0+ct i tried to take the x y z for A and...
  40. K

    Dimension, Image, Kernels etc - Linear Algebra stuff

    Does linear algebra go in this thread?? Anyway, Homework Statement Let n be an integer at least 1, and x1,...,xn be distinct points in R. For any integer m>=1, let P denote the vector space of polynomials of degree at most m. Define a linear transformation T:P->R^n by [f(x1)]...
  41. Y

    Order parameter of charge density wave in one dimension

    In one dimensional electron gas in charge density wave phase, as I know , the density of electrons will be periodic. The order parameter of charge density wave is written as O_{CDW}(x)=\sum_s\psi_{L,s}^{\dagger}(x)\psi_{R,s}(x) For Luttinger model, the \psi is the Fermion annihilation field...
  42. F

    Dimensional Travel and the Human Body

    I am not a physicists, engineer or even a student of such, however, I am writing a book and so this leads me to research. Say traveling to another dimension is possible. Would the traveler lose his/her memory there, only to regain it upon return to the dimension they started from? What...
  43. T

    What is the TOPOLOGICAL DIMENSION of?

    Let {\mathbb I} = {\mathbb R} \setminus {\mathbb Q} the set of the irrational numbers of the real line. What is the topological dimension of {\mathbb R}^2 \setminus {\mathbb I} \times {\mathbb I} ?
  44. J

    Dimension of eigenspace, multiplicity of zero of char.pol.

    Does the multiplicity of zero of characteristic polynomial restrict from above the possible dimension of the corresponding eigenspace? For example if we have a 3x3 matrix A, and a characteristic polynomial \textrm{det}(\lambda - A)=\lambda^2(\lambda - 1) I can see that the eigenspace...
  45. J

    Subalgebra of one less dimension

    Let g be a complex Lie algebra, of dimension greater than one. Does there always exist a subalgebra of dimension dim(g)-1? If the claim is not true, is it true for some particular dimension? I can see that the claim is trivially true for dim(g)=2, but what about for example dim(g)=3? The...
  46. D

    Does the Expectation of Momentum Vanish for a Real Wavefunction?

    Problem Consider a free particle moving in one dimension. The state functions for this particle are all elements of L^2. Show that the expectation of the momentum \langle p_x \rangle vanishes in any state that is purely real. Does this property hold for \langle H \rangle? Does it hold for...
  47. S

    Dimension of momentum per metre

    Considering momentum to be in units of \rm kg^1 m^1 s^{-1}, is there any particular attribute associated with \rm kg^1 s^{-1} (other than "momentum per metre")? Or perhaps the inverse \rm s^{1} kg^{-1}? Just curious.
  48. S

    Report on dimension on the governing equation help

    Report on "dimension on the governing equation.." help Ciao all, I am sorry for my stupid but I really need some help of someone who knows about structure mechanics. I need to write an report on the topic that I myself don't understand it in normal English " DIMENSION ON THE GOVERNING...
  49. I

    Help with hausdorff dimension and countableness.

    My question, because I keep seeing this on the internet, is that if S is a subset of R and Hausdorff dimension greater than 0, it is uncountable... is this true. It seems not to be. If one were to modify the Cantor third set and remove some length of 1/n from the middle of the sets at each...
  50. K

    How Fast Must a Salmon Jump to Clear a 1.9m Waterfall?

    In its final trip upstream to its spawning territory, a salmon jumps to the top of a waterfall 1.9m high. What is the minimum vertical velocity neeeded by the salmon to reach the top of the waterfall? This uestion has no velocity so, do you think that final velocity is Zero... solution...
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