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. M

    Differential geometry of surfaces in affine spaces

    I'm looking for a book or two that details affine spaces and transformations, then differential geometry of surfaces in affine spaces, starting at a level suitable for a year 1-2 undergraduate. In particular, I'd like to understand a few properties (e.g. what's the gradient and curvature at a...
  2. T

    Electric Field Zero on Gaussian Surface with No Charges

    why is the electric field zero , if the closed gaussian surface doesn't enclose any charges. but,, if the charges are present at a distance outside the conductor?? and why isn't the electric field zero inside the gaussian surface, if there are charges distributed on the surface of the...
  3. Salvador

    Find E Field of a Torus: Inside and Outside Surfaces

    Hello , please help me out , I can find the E field of a sphere on google and read that there is no field on the inside of the sphere , but what is the e field of a torus ? Not on the inside but on the outside surfaces also in the inner loop or the middle ?
  4. Prashasti

    Electric field, Equipotential surfaces

    Question : The top of the atmosphere is at about 400 kV with respect to the surface of the Earth, corresponding to an electric field that decreases with altitude. Near the surface of the Earth, the field is about 100 V/m. Why then we do not get electric shock as we step outside our house into...
  5. Z

    What is the separation between the two surfaces

    Homework Statement You measure a uniform electric field of 13.3 x 10^3 N/C between two equipotential surfaces. One surface is at a potential of 1543 V and the other is at 951 V. What is the separation between the two surfaces? Homework Equations Eave = - deltaV/delta x The Attempt at...
  6. A

    Why do grease, cheese, butter, jam, etc., stick to smooth surfaces?

    Why do things like grease, cheese, butter, jam, etc. stick to smooth surfaces like a butter knife or teflon? What are the ways in which they would not stick and be allowed to release without being heated?
  7. vanceEE

    Drawing Surfaces & Space Curves

    Homework Statement Drawing surface ##f(x,y) = ...## or ##r(t) = <f(t),g(t)>## etc. The Attempt at a Solution I've been working on drawing space curves lately, by breaking into separate planes and by level curves. I'm struggling w/ this topic. (1) If I'm not mistaken, this is fundamental in...
  8. Z

    Parametrization of a Parallelogram: Mapping Rectangles onto Planar Regions

    Homework Statement (a) Show that the four points r1 = (1, 0, 1), r2 = (4, 3, 5), r3 = (6, 4, 6) and r4 = (3, 1, 2) are coplanar and the vertices of a parallelogram. Let S be the closed planar region given by the interior and boundary of this parallelogram. An arbitrary point of S can be...
  9. W

    Understanding Disks-and-Bands Surfaces: Genus and Boundary Components

    Hi all, I was reviewing some old material on the representation of orientable surfaces in terms of disks and bands , in page 2 of: http://www.maths.ed.ac.uk/~jcollins/Knot_Theory.pdf Please tell me if I am correct here. Assume there is a horizontal line dividing the surface into an...
  10. R

    Does This Equation Represent a Family of Surfaces in 3-D Space?

    Hi I am currently reading a book where this showed up: The author gave a ##3## parameter equation (note ##Y## and ##y## are two separate variables): Y_1dy_1 + Y_2 dy_2 + Y_3dy_3 = 0 and states that this does not necessarily represent a family of surfaces in 3-D space and that only if the...
  11. Daaavde

    Optics: Why not polished surfaces reflect less than polished ones?

    Hello, I would like to ask why, when a surface is not polished, it reflects less. I understand that when the surface is not polished, microscopically it presents a lot of irregularities so that when the light strikes the surface it gets reflected in all directions and instead of getting...
  12. L

    MHB Optimization of the sum of the surfaces of a sphere and cube

    If the sum of the surfaces of a cube and a sphere as is constant, deierminar the minion of the diameter of the sphere to the edge of the cube in cases in which: 272) The sum of the volumes is minimal 273) The sum of the volumes is maximum And the answer are 272 = 1 and 273 = infinit Ok Vs =...
  13. Demon117

    Understanding Triply Periodic Surfaces: Translations and Invariance?

    So I understand that a surface is triply periodic when the surface is invariant under three tanslations in R^{3}. When looking at the primitive for example, how is that translation defined? Say that the primitive is a set defined by the equation cos(x)+cos(y)+cos(z)=0 My guess is that...
  14. J

    1st and 2nd moment of interia of curves and surfaces

    If I can to calculate the 1st and 2st moment of inertia of areas and volumes, I can compute for curves and surfaces too?
  15. WannabeNewton

    Is Clock Synchronization Throughout a Time-Like Congruence Transitive?

    "Global simultaneity surfaces" For a long time I had basically taken for granted the usual interpretation of one-parameter families of orthogonal space-like hypersurfaces relativized to a time-like congruence as "global simultaneity surfaces" (c.f. MTW section 27.3), and left it at that. See...
  16. J

    MHB Exploring Non-Flat Surfaces and Euclid's Fifth Postulate

    Hi. I'm from singapore. I'm now interested in maths. I only studied till secondary school(high school) I wasn't interested in maths then. Now I am. I read maths book written by David Berlinski , John allen paulos and others to try to understand. I do not have mathematician friends so I couldn't...
  17. N

    Exploring the Impact of Z on Quadratic Surfaces: Hyperbolas in the XY Plane

    4x^2-y^2+2z^2+4=0 x^2-y^2/4+z^2/2+1=0 -x^2+y^2/4-z^2/2=1 In the xy trace -x^2+y^2/4=1+k^2/2 taking k=0 will yield the hyperbola but what affect will z have on the resulting surface as it tends to +- infinity It appears to me that as z to +- infinity the hyperbola in the xy plane becomes...
  18. N

    Drawing Quadratic Surfaces: Tips for Rendering Sheets by Hand

    z=y^2-x^2 Trying to render these sheets by hand is very difficult for me. I can conceptualize the sheet in general by observing that the trace in z=y^2-k^2 follows the trace of z=k^2-x^2 as z tends to negative infinity. The opposite is also true as z tends to infinity. This information...
  19. J

    Are There Surfaces of Force in Addition to Lines of Force in Fields?

    I think everyone has heard about equipotential lines and equipotential surfaces; such as lines of force too. My question is: exist surface of force too? If yes, what is it?
  20. B

    Help finding direction of area vector on curved surfaces

    Homework Statement I understand how to determine the direction on rectangular surfaces, however when it comes to a cylinder or sphere or, in the case of the problem i am working on now, for a long rod that has been bent around in a circle i.e a a rectangular ring, I don't know what to do.
  21. M

    Exploring the Classics: Studying Curved Surfaces

    I wonder if one should study books like Gauss's General Investigations on Curved Surfaces or Euler's works or there are more modern texts that are state of the art?
  22. ShayanJ

    Magnetoresistance and Fermi surfaces

    There seems to be relation between Fermi surfaces and magnetoresistance, but I guess because I don't have a clear picture of fermi surfaces,I have problem understanding this relationship. Also I have heard about open and close fermi surfaces and saturation of magnetoresistance which I can't...
  23. J

    Moment of Inertia of Curves and Surfaces

    Greetings! I enjoyed the definition of moment of inertia for a volume and for an area in the form of matrix. It's very enlightening! I = \int \begin{bmatrix} y^2+z^2 & -xy & -xz\\ -yx & x^2+z^2 & -yz\\ -zx & -zy & x^2+y^2 \end{bmatrix}dxdydz '->...
  24. MarkFL

    MHB Adwt's question at Yahoo Answers regarding surfaces of revolution

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  25. L

    Curves on surfaces (differential geometry)

    A few topics we are covering in class are: Gauss map, Gauss curvature, normal curvature, shape operator, principal curvature. I am having difficulty understanding the concepts of curves on surfaces. For example, this problem: Define the map ##\pi : (\mathbb{R}^3-\{(0,0,0)\})\to S^2## by...
  26. Q

    A GR question about null surfaces, vectors and coordinates

    I wondered anyone can explain the significance of the above as applied to metrics in the context of general relativity. This came up when the video lecturer in GR mentioned that r for example, was null or this or that vector or surface was null, say in the context of the eddington finkelstein...
  27. tom.stoer

    Riemann surfaces over algebraic surfaces

    Suppose we have one polynom ##P(r_1, r_2, \ldots, r_n) = 0## in n complex variables. This defines a n-1 dimensional complex algebraic surface. Suppose that for each variable we have ##r_i = e^{ip_i}## with complex p. In the case n=1 of one variable r this results in the complex logarithm...
  28. T

    Solving Laplace's Equation Between Two Conical Surfaces

    "Two co-axial conducting cones (opening angles ##\theta_{1} = \frac{\pi}{10}## and ##\theta{2} = \frac{\pi}{6}##) of infinite extent are separated by an infinitesimal gap at ##r = 0##. If the inner cone is held at zero potential and the outer cone is held at potential ##V_{o}## find the...
  29. K

    [Waves] Wavelength and surfaces penetration

    Hi everybody! Just 2 important notes: I don't study physics and so I'm not sure if this question belongs here (but I don't know where should I put it..); second I'm not english native, so ask me if I did not explain somthing enough. My electronic professor told once in my class, that you...
  30. C

    Parametric Equation for Intersection of Parabola and Ellipsoid

    I need to find a parametric equation of the vector function which is the intersection of y = x2 and x2 + 4y2 + 4z2=16. I know the graph of the first equation is a parabola which stretches from negative infinity to infinity in the z direction. I also know that the second equation is that of an...
  31. J

    Gaussian Surfaces and infinite lines of charge

    So I've been wondering.. from my previous post: https://www.physicsforums.com/showthread.php?p=4500082#post4500082 if we have a plane of infinite charge, then electric field does not depend on distance however, for a infinite line of charge: If we use a cylinder with radius 'r' as our...
  32. 7

    What is the Electric Field at Position D on the Equipotential Surfaces?

    Homework Statement The drawing shows a graph of a set of equipotential surfaces in cross section. The grid lines are 2.0 cm apart. Determine the magnitude and direction of the electric field at position D. Specify whether the electric field points toward the top or the bottom of the drawing...
  33. J

    Why do Gaussian Surfaces work?

    I was doing a problem: An infinite conducting plane has a uniform surface charge density of 30 μC m‾². Find the electric field strength 7.0 mm from the plane. so we can use a gaussian surface (e.g a cylinder), and come to the conclusion that E = 30 μ / ε but that got me thinking...
  34. C

    Good treatments on the differential geometry on surfaces.

    Hi! I'm trying to read up on the subject of hypersurfaces related to GR; First and second fundamental form, Theorema Egregium etc.. Does anyone know any good treatments? (Books or notes)
  35. B

    Straight lines and flat surfaces

    Suppose I have a parameterized line ##\phi:\mathbb{R}\to\mathbb{R}^n## given by ##\phi(t) = (x^\mu(t))|_{\mu=1}^n##. How can I tell that the line is straight. My best answer so far is that at every time ##t## the acceleration (2nd derivative) is parallel to the velocity (1st derivative), i.e...
  36. Mandelbroth

    Understanding Riemann Surfaces and Their Applications in Complex Analysis

    I'll preface this by saying that this just isn't getting through to me. I know the material, but my brain feels like it doesn't, for lack of a better word, "fit." Looking back at Riemann surfaces in complex analysis after familiarizing myself with some differential geometry really makes them...
  37. L

    Equipotential Surfaces physics problems

    Homework Statement Homework Equations - Charge conservation - Equipotential surfacesThe Attempt at a Solution Let Q1 be the amount of charge on the inner sphere with radius c, and Q2 be the amount of charge on the outer sphere with radius b. Using Gauss's law, I figured out that Q1=+2Q and...
  38. J

    Jumping on a sliding board with no friction between surfaces.

    Homework Statement You stand at the end of a long board of length L. The board rests on a frictionless frozen surface of a pond. You want to jump to the opposite end of the board. What is the minimum take-off speed v measured with respect to the pond that would allow you to accomplish that...
  39. B

    Electrostatics & Quadric Surfaces

    Hey guys, I'm in some serious trouble - I don't know how to solve a lot of problems in electrostatics. The main reason is that once I seen that you had to be able to find electric fields & electric potentials due to charge distributions on quadric surfaces I panicked & gave up - I was still...
  40. I

    I don't understand gauss's law for sheet surfaces

    hi all, gauss's law states that for any surface, the total electric flux coming through an open surface is always q/ε0. okay.. i understand this... so for a sphere.. the simplest case... http://imgur.com/bwJzf7t,2KFO8Xw,O9Mgafg \Phi=\int\vec{E}\bullet\hat{n}dA=EA= \frac{q}{4(pi)ε}...
  41. K

    Velocity of a bee moving along the curve of intersection of 2 surfaces

    Hi, I've attached the problem and the solution. I understand the solution except for one thing. I've circled the part I'm having problems with. How do I decide if the circled part should be f2(x,y,z)=x2-y2-z or f2(x,y,z)=z-x2+y2 I'm sure it has something to do with the fact that the problem...
  42. alyafey22

    Mathematica Plotting Riemann Surfaces with Wolfram Mathematica 8.0

    Does anybody know how to plot the Riemann surface of complex functions on wolfram Mathematica 8.0 ?
  43. C

    Finding a surface form the intersection of two surfaces- Stokes' Thrm.

    Homework Statement Let \vec{F}=<xy,5z,4y> Use Stokes' Theorem to evaluate \int_c\vec{F}\cdot d\vec{r} where C is the curve of intersection of the parabolic cylinder z=y^2-x and the circular cylinder x^2+y^2=36 Homework Equations Stokes' Theorem, which says that \int_c\vec{F}\cdot...
  44. C

    The Modern Geometric View of Black Holes

    Im trying to understand an argument from Hobson and Estathiou's book on GR where they argue that null-surfaces; i.e surfaces with a null-vector as a normal are in general horizons. Their argument goes as follows (page 316) in the book: "Before discussing the particular case of a stationary...
  45. J

    Surfaces without an explicit representation.

    Ok this question may be kinda stupid, but here goes. Do any surfaces exist for which a parametric form is possible, but cannot be described explicitly due to their highly irregular shape? (Or vice-versa)
  46. D

    2-dimensional differentiable surfaces

    What is a good book on 2-dimensional surfaces (3-spheres, etc.)? I need to know about geodesics, etc.
  47. C

    Help with double integral - volume between 2 surfaces

    Homework Statement find volume of the solid bounded by the surfaces z = 1- \sqrt{\frac{x}{4}^2 + \frac{y}{2 sqrt{2}}^2} and x^2/4 -x +(Y^2)/2 = 0 and the planes z = 0 and z = 1 Homework Equations z = 1- \sqrt{\frac{x}{4}^2 + \frac{y}{2 sqrt{2}}^2} and x^2/4 -x +(Y^2)/2 = 0...
  48. J

    More on super-reflective surfaces.

    So, I'm just going to suggest a circumstance, and leave it up for discussion. I recently read an interesting forum on super reflective material in which all energy is reflected (all theoretical, of course) and none of it is absorbed by the material. Now, say we construct a one way mirror where...
  49. S

    How are Riemann surfaces graphed

    I am creating a python application to graph riemann surfaces, from the wikipedia article it says the functions are graphed as x real part of the complex domain, y, imaginary part of the complex domain, z, the real part of f(z), but how does one represent the imaginary part with the color, would...
  50. R

    Radii of eqiuipotential surfaces of a charged metal sphere

    Homework Statement A metal sphere of radius r0 = 0.29 m carries a charge Q = 0.90 µC. Equipotential surfaces are to be drawn for 100 V intervals outside the sphere. Determine the radius r of the following equipotentials from the surface. a) first:___m b) tenth:___m c) 100th:_____m...
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