What is Surfaces: Definition and 458 Discussions

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

    Does matter warp space time in the form of minimal surfaces?

    Hello guys. Is this even a valid question? Just curious. Thanks,
  2. J

    Light and Surfaces: What's the Story?

    light doesn't require a medium to travel ...but , then why can't light penetrate all surfaces??
  3. I

    Magnetic fields across lines and surfaces

    I know that \int_{S}^{}\int_{}^{}\vec{B}\cdot d\vec{A} = 0 because \textbf {div} \vec{B}=0 IE, because \Phi_{B} leaving a closed surface must equal \Phi_{B} entering. Yet how is it then that \int_{C}^{}\vec{B}\cdot d\vec{l} isn't also equal to zero? Shouldn't it be true for any...
  4. E

    Surface Homeomorphism Between Cubes and Spheres?

    Is it true that the surface of a (hyper)cube in Rn is homeomorphic to Sn-1? Or only for particular n?
  5. A

    Spherical Refraction Surfaces. (I'm confused as to how you get the answer.

    A beam of parallel light rays from a laser is incident on a solid transparent sphere of index of refraction n. (a) If a point image is produced at the back of the sphere, what is the index of refraction of the sphere. Here's the equation you use to find the index of refraction of the sphere...
  6. W

    Magnetic monopoles, electric field lines and equipotential surfaces

    Homework Statement (i) Explain why it would not be possible to write the magnetic Field (B-field) in terms of a vector potential (A) IF magnetic monopoles existed. (ii) For an electrostatic field (E-field), define the electrostatic potential (Fi), and explain CONCISELY what is meant by a...
  7. L

    Surfaces in Space / Vector-Valued Functions

    Homework Statement Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function. Homework Equations Surfaces: z=x2+y2, x+y=0 Parameter: x=t The Attempt at a Solution So, I sketched the space curve represented...
  8. N

    Arc length and surfaces

    HELP
  9. R

    Do Normal Forces Always Exist When Surfaces are in Contact?

    Homework Statement Is it necessary to have a normal reaction whenever 2 surfaces are in contact with each other? The Attempt at a Solution I thought the answer was "no". But my book says "yes". I can support my answer with an example: the normal force between the surface of a block...
  10. N

    How do you parameterize a surface and visualize it in 3-space?

    I'm trying to finish reading/understanding the textbook we used in Calculus III (multivariate), as we only covered chapters 12-18, but I'm stuck on something. We used McCallum/Hughes-Hallett/Gleason, and I'm referring to section 19.3 (if you have the text) which is about flux integrals over...
  11. A

    Information about missile control surfaces.

    Hi, I need some good amount of information regarding the types of control surfaces being used to control and maneuver the missiles. i couldn't find this stuff on my own. so, please help me. thanks in anticipation Abhishek
  12. P

    Mathematica Coloring surfaces in Mathematica

    Hi, all, I have a question about coloring a 3D parametric surface in Mathematica. Setup: Take as given a surface M in R^3 and a parameterization of that surface p:[a,b] x [c,d] -> R^3. Let f:M -> R be a function defined on M. Question: How can I plot this surface so that points p...
  13. D

    Stereographic Projection for general surfaces

    Stereographic Projection for "general" surfaces First off, sorry if this is in the wrong forum. I came across this while studying computer vision, but it's of a somewhat mathematical nature. Please move it if it's in the wrong place. In the book I'm reading*, stereographic projection is used...
  14. S

    Understanding Dangling Bonds and Hydrogenation of Surfaces

    hi all, I have read papers in which they specify the dangling bonds on the surfaces and tey saturate it with hydrogen forexample; 1. should it be hydrogen atoms or ghost hydrogen? 2. how to specify the dangling bonds? 3. any suggestion for specifying the position of these hydrogens in...
  15. J

    Find the tangent line between two surfaces

    Homework Statement Let C be the intersection of the two surfaces: S1: x^2 + 4y^2 + z^2 = 6; s2: z = x^2 + 2y; Show that the point (1, -1, -1) is on the curve C and find the tangent line to the curve C at the point (1, -1, -1). Homework Equations partial derivates, maybe the gradient...
  16. K

    Frictional Moment produced by contact between rotating and non-rotating surfaces

    [b]1. This problem is part of an engineering model I am working on for a class. I am ultimately trying to model the torque applied to a bottle as a function of the static/kinetic coefficient of friction between it and the rubber cone it is being torqued by (The reason for this being bottle caps...
  17. R

    Finding Nodal Surfaces in Wave Function of H-Like Atom

    Homework Statement One wave function of H like atom is \psi=\frac{\sqrt{2}}{81\sqrt{\pi}a_{0}^{3/2}}(6-\frac{r}{a_{0}})\frac{r}{a_{0}}(e^{\frac{-r}{3a_{0}}})cos \theta How many nodal surfaces are there? 1)1 2)2 3)3 4)none of these The Attempt at a Solution Its an objective...
  18. S

    Surfaces and geodesics in General Relativity

    Hi all. This is one of the problems that I was asked to do for my General Relativity class. I know this may look a little long, but if anyone can help me out with ANYTHING about this problem, I will greatly appreciate it. Homework Statement Consider the family of hypersurfaces where each...
  19. P

    Can Hydrophobic Surfaces Repel Both Water and Oil?

    Hello, I have a quick question regarding an http://www.sciencenews.org/view/generic/id/38466/title/Blueprint_to_repel_oil_and_water" that I just read on ScienceNews regarding hydrophobic surfaces. In the second to last paragraph it's quoted that "although hydrophobic surfaces readily shed...
  20. M

    Geodesic Curves Covering Surfaces

    Are there surfaces that have a geodesic curve which completely covers the surface, or (if that's not possible) is dense in the surface? In other words, if you were standing on the surface and started walking in a straight line, eventually you would walk over (or arbitrarily close to) every...
  21. I

    Exploring Complex Seashell Surfaces with Maple

    I've been doing some experimentation with plotting parametrized surfaces in maple, and I would like to get some ideas for more things I could do. I'm not very clever at figuring out new parametrizations, but I'd like to do some things with seashells. The plots I'm coming up with are very...
  22. M

    Finding Tension and Acceleration on Frictionless Surfaces

    Homework Statement For the following system 1) Find the tension in the string 2) find the acceleration of each of the masses The diagram of the system looks like this ...O-----|200g|-----O ...|.\___|____|___/.| __|__....__|__...
  23. J

    Sketching Hyperbolas in Quadric Surfaces

    In general, say: we have a surface: y^2/4 - x^2/3 - z^2 = 1 I know that this is a hyperboloid of 2 sheets, since the xz trace: x^2/3 + z^2 =-1 doesn't exist, But for the other traces: xy trace: y^2/4 - x^2/3 = 1 and yz trace: y^2/4 - z^2 = 1 Which are both hyperbolas - how do...
  24. S

    Multivaraible Calculus. surfaces in R2 andR3

    Homework Statement In class, we studied functions ⃗r : I → R^3, where I ⊂ R is some interval. Let us now consider a function ⃗r : U → R^3, U ⊂ R^2 That is, we have a function, ⃗r, which sends a point (u, v) in the plane to a point (vector) in R3 . You may call it a “vector function of...
  25. J

    Equation of tangent plane at (2, -1, ln 7): z = ln 7 + (4/7)(x-2) - (6/7)(y+1)

    Just when I thought I got the hang of tangent planes and surfaces there comes a question I haven't quite seen before z = ln (x^{2}+3y^{2}) Find a normal vector n and the equation of the tangent plane to the surface at the point (2, -1, ln 7) So keeping the cartesian equation in mind: z =...
  26. J

    Finding Tangent Planes and Normal Vectors to Surfaces

    Suppose that F(x,y) = x^{2}+y^{2}. By using vector geometry, find the Cartesian equation of the tangent plan to the surface z = F(x,y) at the point where (x,y,z) = (1,2,5). Find also a vector n that is normal to the surface at this point...
  27. E

    Determining Electric Flux for cylindrical surfaces

    Homework Statement Cylin. surfaces \rho = 1 , 2 , 3, cm have uniform surface charge density of 20, -8, 5 nC/m^2. What is the total electric flux that passes through the closed surface rho = 4 cm and z(from 0 to 1 m)? And what is \vec{D} at the point \rho= 4 cm , \phi= 0 , z = .5 cm...
  28. C

    Cylinders and Quadric Surfaces

    Homework Statement Consider the equation below. x^2 = 3y^2 + 5z^2 Reduce the equation to one of the standard forms. I believe its surface is a cone, but I'm not sure how to get it into the form z^2/c^2 = x^2/a^2 + y^2/b^2 thanks!
  29. C

    Angle Between two surfaces and a point

    Find the angle between the surfaces defined by r^2= 9 and x + y + z^2= 1 at the point (2,-2,1)? --- I know this should be extremely simple but it is blowing my mind for some reason. Any help would be greatly appreciated.
  30. Topher925

    Finding the intersecting point of three surfaces.

    I'm faced with a problem where I need to determine the intersecting point of three different surfaces. Normally I would do this by using the incredibly painful method of Lagrange multipliers. However, this computation needs to be done relatively quickly and I can afford some error, in the...
  31. D

    Frechet distance between surfaces

    I am wondering whether or not anybody has any ideas of how to visualize and calculate the Frechet distance between two surfaces, or the sets that they encompass. Let M be an m-dimensional finitely triangulated manifold (with or without boundary). Let f1 and f2 be continuous maps M---->R^n...
  32. B

    Mathematica Calabi-Yau surfaces and algebraic geometry for mathematicans

    Hi, I posted this under General Physics but I thought I should post this under the math section too in case there are mathematicians who do not read posts under General Physics. I read some papers online about Calabi-Yau surfaces but I have some basic questions about them if you can answer...
  33. B

    Calabi-Yau surfaces and algebraic geometry

    Hi, I read some papers about Calabi-Yau surfaces but I have some basic questions about them if you can answer them for me. 1. Who proved that Calabi-Yau mfds fits into string theory? Why does C-Y surface fit into string theory, instead of any other surfaces? 2. I googled C-Y surfaces and...
  34. B

    Book recs please - complex analysis, riemann surfaces, multi-valued functions

    Hi everyone, hope this is the right place to put this :) I have just finished "Theory of Functions" Vol. 1 & 2 by Konrad Knopp. I'd like to continue with a book that picks up where the second volume it left off. (Especially would be nice is a more "modern" book) The second volume is about...
  35. W

    Voltage difference between two surfaces

    How would you find voltage difference between two surfaces? If you are given a capacitor with an internal electric field that varies E. You also have the charge (sigma) and the radius of the inner surface r1 and the radius of the outer surface r2.
  36. T

    Family of surfaces (Diff Geometry)

    Here is a question i am not sure how to tackle, I am not familiar with how to deal with family of curves and don't really have much time to look around for the definition as i am sitting the exam in two days. Homework Statement (this link has an image of the problem)...
  37. T

    Potential of two surfaces coming together

    [SOLVED] potential of two surfaces coming together lets say we have two water droplets(assume them to be spherical shape) with a each having charge q and each has the potential at its surface as kq/r.What is the potential if the two water droplets combine? I thought tht the radius and charge...
  38. T

    Classification Theorem of Surfaces

    I am having difficulties grasping the consequences of this theorem, would really appreciate a little enlightenment. A: Well, the statement of the theorem is clear, that Every closed Surface is homeomorphic to: 1) a sphere, 2) the connected sum of g tori 3) or the connected sum of g...
  39. N

    Surface Integral of a Vector Field on a Paraboloid Above a Square

    Homework Statement Hi all. Please take a look at the following problem: Evaluate the surface integral \int{F \cdotp d\vec{S}} for the following vector field: F(x;y;z) = xyi + yzj + zxk, where i, j and k are unit vectors. S is the part of the paraboloid z = 4-x^2-y^2 that lies above the square...
  40. R

    Finding the charges on surfaces (hollow spheres)

    Dear all, Two hollow conducting spheres have a common center O. The dimensions of the spheres are as shown (attached above). A charge of −200 nC is placed on the inner conductor and a charge of +80 nC is placed on the outer conductor. The inner and outer surfaces of the spheres are...
  41. S

    Two or more different equipotential surfaces to intersect?

    I have two questions about electric field and equipotential surface. Here is first one: For an arrangement of two point charges, --Is it possible to find two points (neither at infinity) where E = 0 ? Secondly, --Is it possible for two or more different equipotential surfaces to intersect?
  42. A

    Explaining the Sun's Effects on Rough Surfaces Covered in Water

    Rough (not smooth) surfaces often appear to be darker in sunlight when they are covered by water.How can we explain this physical phenomen?
  43. B

    Parametric equation of the intersection between surfaces

    [SOLVED] Parametric equation of the intersection between surfaces Homework Statement Given the following surfaces: S: z = x^2 + y^2 T: z = 1 - y^2 Find a parametric equation of the curve representing the intersection of S and T. Homework Equations N/A The Attempt at a Solution The...
  44. D

    Equipotential Surfaces: Calculating Work Required to Move a Charge

    [SOLVED] Equipotential surfaces Homework Statement http://personalpages.tds.net/~locowise/test/equipot1.jpg Fig. 1 -- Some equipotential surfaces In the figure above, you see a set of equipotentials representing an electric field in the region and some labeled points (A..G). What is...
  45. P

    Equipotential Surfaces Question

    [SOLVED] Equipotential Surfaces Question If the radius of the equipotential surface of point charge is 14.3 m at a potential of 2.20 kV, what is the magnitude of the point charge creating the potential? have V = [1/4πεo] [q/r] Given that, V = 2.20kV = 2200V r = 14.3m...
  46. J

    Can we get a negative value for areas of surfaces of revolution?

    the question is.. Find the areas of the surfaces generated by revolvin the curves about the indicated axes. x = (1/3)y^(3/2) - y^(1/2), 1≦y≦3; revolved about y-axis so i use the general formula "S = Integral 2π (radius)(dS)" and the radiu in this case is x which is (1/3)y^(3/2) -...
  47. G

    Electric Field between two positivly chared surfaces

    Homework Statement Figure 24-34 shows cross-sections through two large, parallel, nonconducting sheets with identical distributions of positive charge with surface charge density σ = 2.62 x 10-22 C/m2. What is the y component of the electric field at points (a) above the sheets, (b) between...
  48. A

    How Do I Find the Intersection of the Surfaces z=x^2+y^2 and x^2+y^2+z^2=2?

    z=x^2+y^2 and x^2+y^2+z^2=2...I need to find the intersection of these two surfaces. Would I just substitute z=x^2+y^2 into the equation of the sphere to find the curve of intersection? But when I do that I get an equation with fourth powers and I don't know what kind of curve that makes.
  49. K

    Curves and surfaces, Transformations

    1) http://www.geocities.com/asdfasdf23135/advcal13.JPG Let F1 = x^2 - y^2 + z^2 -1 = 0 F2 = xy + xz - 2 = 0 F3 = xyz - x^2 - 6y + 6 = 0 My thought is to compute the gradients, grad F1 and grad F2. Then by taking their cross product, I can get a tangent vector v for the curve. Now, I can feel...
  50. M

    Please explain equipotential surfaces/ contour maps

    Homework Statement Can someone please define contour map for equipotential surfaces in really simple terms. I understand that the potential diff is from the neg side I don't understand what it means when the lines are closer together? I thought the lines are just divisions of the voltage...
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