Surfaces Definition and 459 Threads

In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world.

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  1. S

    Calc 3-normal lines to surfaces

    Homework Statement Find the equations of the normal lines to the surfaces at the given points. z=(3/4)x^2+3y^2 @ pt. (2,1) 2. The attempt at a solution I have already found the equation of the tangent plane and know it is correct. Tangent plane => (z-6)=3(x-2)+6(y-1) Now, I am confused...
  2. X

    Angle between two surfaces and gradient

    In Marion & Thorton problem 1.29 asks to find the angle between two surfaces (x^2 +y^2 + z^2)^2 = 9 and x + y + z^2 = 1 at a point. The solution takes the gradient of (x^2 +y^2 + z^2)^2 - 9 and x + y + z^2 - 1, and using the dot product between the two vectors at that point gets the angle...
  3. M

    Vector parameterization of intersection of 2 surfaces

    Homework Statement Find a vector parameterization of the intersection of the surfaces x2+y4+2z3=6 and x=y2 in R3. The Attempt at a Solution I let x=t. Then y3=t I solved the first equation for z in terms of x z = cube root ((t2+(t(cube rt(t)) - 6)/-2) I know this is wrong...
  4. H_man

    So confused about reflection from metal surfaces

    :bugeye:Hi All... I am really confused about the mechanism of loss of energy when an EM wave hits a metal surface. I always thought the reflection was due to the motion of the electrons in the metal (due to the electric field of the wave). Which suggests that resistive losses would come...
  5. L

    Is This Mod 2 Cohomology Class on Surfaces Geometrically Significant?

    I am wondering if the following mod 2 cohomology class which can be defined on any compact surface, has any geometric meaning or is important in any way. triangulate the surface then take the first barycentric subdivision. This is a new triangulation. Define a 1 - cochain on this new...
  6. A

    Why polished surfaces are colder?

    Why of two floors made of the same material, the polished one is (at least feels) colder than the rough one. Just compare two concrete floors, one polished and one unpolished. Thanks
  7. jegues

    Identifying and Drawing Surfaces

    Homework Statement See figure Homework Equations The Attempt at a Solution Just to give the readers some background on my current situation, Recently I've been doing some independent study on some of the material that will be covered in upcoming math analysis course I'm...
  8. P

    Null Hyperplanes & Cauchy Surfaces in Spacetimes

    Is a null hyperplane a Cauchy surface in Minkowski spacetime? What in case of other spacetimes?
  9. P

    Equipotential surfaces - - Pictures

    Homework Statement Sketch the equipotential surfaces which result from the following charge configurations: (a) a point charge (b) a spherically symmetric charge distribution (c) a very large, plane, uniformly-charged sheet (d) a long, uniformly-charged cylinder (e) an electric dipole...
  10. N

    Calculating the Volume Between Three Surfaces: Is My Approach Correct?

    Dear all, I do really need your help. I'd like to find the volume contained between a sphere (x^2+y^2+z^2=r^2) , plane1 (ax+by+cz+d=0), and plane2 (z-h=0). Would you please check what I've done till now? From the sphere and plane1 equations I got: x1=sqrt*(r^2-y^2-z^2) x2=d/a-(b/a)y-(c/a)z...
  11. I

    Gaussian surfaces: Electric Field=zero

    Please explain to me in detail why a gaussian surface within a conductor has an electric field of zero? thanks.
  12. M

    2D surfaces in the third dimension?

    Hello, I just need to know whether or not surfaces with zero size in the third dimension, 6x8x0, is considered two-dimensional. The surface is there all the time. It has a location in the third dimension, so wouldn't it be a 3D object? I am not sure whether I should call a flat surface (as...
  13. R

    Understanding Surface Dimensions for Beginners

    How do we define the dimension of a surface? I know surfaces are 2-D but I don't really get where that comes from.
  14. T

    Learn Quadric Surfaces Basics for 3D Structures

    Hi! I need to know how to work with quadric surfaces to draw a 3D structures in a code. However I have no idea how to do this. I can't find any place in the internet where they explain quadric surfaces for newbies... Can someone point me in the right direction, please? I didn't post this...
  15. R

    Optics of spherical surfaces expressed as functions of y

    Homework Statement Problem is: A maksutov camera, which is made from a refelector with a spherical surface s and a transparent corrector with two spherical surfaces s1 and s2. The radii of s, s1 and s2 are 2f, r1 and r2, respecitvely. Z=0 is the centre of all these spherical surfaces...
  16. A

    Describe surfaces of equal pressure in a rotating fluid

    Hi, I am trying to solve a basic question from a Fluid dynamics textbook. Could you help me with the answer? The question is as follows: A closed vessel full of water is rotating with constant angular velocity \Omega about a horizontal axis. Show that the surfaces of equal pressure are...
  17. F

    Flux through two parametrized surfaces

    Homework Statement Show that the flux through a parametrized surface does not depend on the choice of parametrization. Suppose that the surface \sigma has two parametrizations, r(s,t) for (s,t) in the region R of st-space, and also r(u,v) for (u,v) in the region T of uv-space, and suppose that...
  18. N

    Superconductors and Fermi surfaces

    Hi The dispersion of Bogolyubov quasiparticles in a d-wave superconductor is E(\mathbf k) = \pm \sqrt{\varepsilon (\mathbf k)^2+\Delta (\mathbf k)^2}, where ε(k) is the normal-state dispersion and ∆(k) is the gap dispersion. My question is: The Fermi surface (FS) of the normal...
  19. J

    3D printing of Riemann Surfaces

    Hello. Does anyone know of a group that has used 3D printing techniques such as laser sintering to create Riemann surfaces of some simple functions? For example, just \sqrt{z}? Actually I would be interested in more complex function and preferable color-code various components of the surface...
  20. A

    Graphing Surfaces with Non-Linear Equations: What Are My Options?

    Im trying to check my answers to a problem, and in the past I've used a 3d grapher to graph functions like f(x,y) = whatever. but now i need to find a tangent plane to a surface at a point. the surface is: x2y+y2z-z2x=1but i don't know how to go about graphing something expressed that...
  21. matt_crouch

    Describe geometrically the level surfaces of the functions

    So the question is as titled i) f=(x^2 +y^2 +z^2) ^1/2 if I can figure out the method I can solve the other equations but I'm not really sure where to start I know that a function f(x,y,z) of a level surface well be constant so do I just find del f ?
  22. B

    Generic Intersection of non-planar Surfaces in R^4

    Hi, everyone: How do we show that 2 planar surfaces in R^4 intersect at points (possibly empty sets of points, but not in lines, etc.). I am curious to see how we justify the Poincare dual of the intersection form in cohomology being modular, i.e., integer-valued...
  23. G

    Plotting bounded surfaces with conditions

    Homework Statement Attached question Homework Equations The Attempt at a Solution I tried rearranging S1 for Z then using Maple to plot it, which gave me a cone extending from the point z=1. For S2, would I have to plot it twice? once for <1 and once for =1? I have no...
  24. A

    DFT: Investigating Change in States at Surfaces and Interfaces

    Dear all, In Density Functional Theory (DFT) the Kohn-Sham eigenvalues are used to construct the band structure and the density of states (DOS). For a 3D extended system the eigenvalues are determined up to a constant since there is no absolute energy reference, while for a 2D extended...
  25. Z

    Residue calculus and gauss bonnet surfaces

    I am not a mathematician but I have noticed how strangley similar the treatments of curvature and residues are when you compare the residues of residue calculus and the curviture of the gauss bonet forumlation of surfaces. Is there some generalization of things that contains both of these...
  26. I

    Equipotential Surfaces and Electric Field

    Homework Statement A given system has the equipotential surfaces shown in the figure . http://session.masteringphysics.com/problemAsset/1122530/1/Walker.20.39.jpg A)What is the magnitude of the electric field? B)What is the direction of the electric field? (in degrees from the +x...
  27. T

    Maxima and Minima on surfaces in three dimensional space

    Homework Statement Find the maximum and minimum values of f(x,y) = (xy)2 on the domain x2 + y2 < 2. Be sure to indicate which is which Homework Equations I am not sure what to put here. I solved this problem a different way, and I am not confident I did it correctly. The Attempt at a...
  28. I

    Patches and Surfaces (Differential Geometry)

    I'm completely confused with patches, which were introduced to us very briefly (we were just given pictures in class). I am using the textbook Elementary Differential Geometry by O'Neill which I can't read for the life of me. I'm here with a simple question and a somewhat harder one...
  29. C

    BRS: euclidean surfaces a la Cartan

    In this thread, I plan to try to explain (with some apropos ctensor examples) in a simple and concrete context some basic techniques and notions about Riemannian two-manifolds which also apply to general Riemannian/Lorentzian manifolds. Suppose we have a euclidean surface given by a C^2...
  30. W

    Equipotential surfaces electric field problem

    Homework Statement A given system has the equipotential surfaces shown in the figure What is the magnitude of the electric field? What is the direction of the electric field? (degrees from + x axis What is the shortest distance one can move to undergo a change in potential of 5.00...
  31. A

    Mathematica Mathematica better resolution of surfaces

    Hi I've been drawing surfaces in Mathematica but some of the images come out jaggy and very unsmooth. Is there a command that can be added that will increase 'resolution' or smoothness of the surfaces. Thanks
  32. P

    Tangent vector to curve of intersection of 2 surfaces

    Homework Statement Find the tangent vector at the point (1, 1, 2) to the curve of intersection of the surfaces z = x2 + y2 and z = x + y. Homework Equations The Attempt at a Solution I haven't started the problem, because I'm not sure what the first thing to do is. Do I have to parametrize...
  33. P

    Tangent line of the curve of intersection of two surfaces

    How do I find this?
  34. I

    Ellipsoids and Surfaces of Revolution

    My textbook notes that if: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}}=1 and a \neq b \neq c Then the ellipsoid is not a surface of revolution. It seems to me though that one can always find a curve in the plane, which when rotated around a line will produce the...
  35. J

    Comparing Bound Charges on Cylindrical Dielectric Surfaces

    Homework Statement A conducting wire carrying a charge \lambda per unit length is embedded along the axis of the cylinder of Class-A dielectric. The radius of the wire is a; the radius of the cylinder is b. Show that the bound charge on the outer surface of the dielectric is equal to the...
  36. K

    Calculation of electric field from a set of equipotential surfaces

    Homework Statement A set of concentric hemispherical surfaces is given, each of which is an equipotential surface. These concentric surfaces do not, however, have the same value of potential, and the potential difference between any two surfaces is also not constant. The surfaces are spaced...
  37. K

    Dark Surfaces and Global Warming

    Do dark surfaces exposed to the sun really contribute to global warming?
  38. R

    Find some vector function whose image is the intersection of two surfaces

    Hi all, I'm quite new here, but it's been a while since I've been browsing through these forums for past answered questions for calculus and physics, but now comes the time where I'm the one needing help that's not been questioned yet. Homework Statement Find some* vector funcion r with...
  39. M

    Volume of bounded by 2 surfaces

    Homework Statement I need to find the volume of the body bounded by the following surfaces: z = x2 + y2 z = 1 - x2 - y2 Homework Equations Volume of a body between z=o and the upper surface: \iint_{D} z(x,y) dA The Attempt at a Solution Ok, this is something I need to do with...
  40. I

    Finding Equations for Plane Containing Intersection of Quadric Surfaces

    Here is the problem exactly how it is written on my paper... Consider the surfaces x^2+2y^2-z^2+3x=1 and 2x^2+4y^2-2z^2-5y=0. a. What is the name of each surface? b. Find an equation for the plane which contains the intersection of these two surfaces. That is the question. For...
  41. I

    Why do surfaces get darker when they are wet?

    A couple of pictures to get started: I'm not a physicist, but a professional artist, so an overly technical explanation may not have any meaning (especially if it's maths!). However I am very interested in how light interacts with matter and I am puzzled as to what is going on here...
  42. M

    Exploring Level Surfaces of a Multivariable Function

    Homework Statement Homework Equations f(x,y,z,)=(x-2)2+y2+z2 The Attempt at a Solution
  43. S

    Differential Geometry Theorem on Surfaces

    Homework Statement I am having difficulty understanding the proof of the following theorem from Differential Geometry Theorem S\subset \mathbb{R}^3 and assume \forall p\in S \exists p\in V\subset\mathbb{R}^3 V open such that f:V\rightarrow\mathbb{R}^3 is C^1 V\cap S=f^{-1}(0)...
  44. K

    Equipotential Surfaces: Understanding & Conceptual Problems

    I need help with developing a good understanding of equipotential surfaces corresponding to regions of three dimensional electric fields. I would appreciate if someone could refer me to a site or sites where this is comprehensively explained along with illustrations and with related conceptual...
  45. C

    Calculating Surface Area Using Parametrization: Tilted Plane Inside Cylinder

    Homework Statement Use parametrization to express the area of the surface as a double integral. tilted plane inside cylinder, the portion of the plane y+2z=2 inside the cylinder x^2+y^2=1 Homework Equations the area of a smooth surface r(u,v)=f(u,v)i+g(u,v)j+h(u,v)k a<=u<=b...
  46. E

    Vizualizing Infinity: Exploring Non-Flat Surfaces

    Try drawing this mentally: Start with a circle of radius r, draw n number of points spaced evenly on the circle. at each point on the circle draw another circle of radius r, once again with n number of points. What sort of a picture would one get repeating this process a million times, and as n...
  47. N

    Newton's laws and incline surfaces

    Homework Statement A box is given an initial velocity of 5m/s up a smooth 20 incline surface . The distance the box travel before coming to rest is? Homework Equations I can't solve it correctly , I can't get the idea of this question The Attempt at a Solution x= ? vi=5 v=0...
  48. D

    Ball Bearing Surfaces: Does the Ball Touch Both Races?

    Do the balls in a bearing actually touch both races? If so, how does the thing turn? Won't the balls be moving in different directions at each race, and therefore dragging against one?
  49. Y

    Calculus 3, dealing with tangent planes and surfaces.

    okay i came up with doing the gradient of the ellipsoid. Then set that equal to the vector, <4,-4,6>. I solved and got x,y,z = 1,-2,1 I looked at the answer key and it said (1/3) (1,-2,1) Does anyone know where the 1/3 came from?
  50. K

    Parametrizing Surfaces and Curves

    Homework Statement Given the surface: x^2 + y^2 + z^2 = 1 but x + y + z > 1 (actually greater than/equal to) I'd like to parametrize both this portion of the sphere and I'd like to find a parameterization of the boundary of the surface (that is, the intersection of the above sphere and...
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