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. micromass

    Geometry Algebraic Curves and Riemann Surfaces by Miranda

    Author: Rick Miranda Title: Algebraic Curves and Riemann Surfaces Amazon Link: https://www.amazon.com/dp/0821802682/?tag=pfamazon01-20 Prerequisities: Complex Analysis, Differential Geometry, Abstract Algebra Level: Grad Table of Contents: Preface Riemann Surfaces: Basic Definitions...
  2. I

    Can you triangulate a triangle? (also, odd sided polygons to represent surfaces)

    EDIT: My guess to the below question is that no you can't triangulate a triangle because a legitimate triangulation each edge can only be linked up to exactly two distinct faces, so if you just have one triangle, each edge would be linked up to one face (the face of the triangle) I'm really...
  3. S

    Thermal conduction between difference surfaces

    Homework Statement from this video: It showed that a better conductor of heat will transfer heat as well as absorb heat more quickly than a lousy conductor. However, during the transfer the heat is passed from eg metal at 30 degrees to a solid at 20 degrees . So why would the metal being a...
  4. O

    Multiple integrals: Find the volume bounded by the following surfaces

    Homework Statement Find the volume bounded by the following surfaces: z = 0 (plane) x = 0 (plane) y = 2x (plane) y = 14 (plane) z = 10x^2 + 4y^2 (paraboloid) Homework Equations The above.The Attempt at a Solution I think it has something to do with triple integrals? But...
  5. U

    Understanding Riemann Surfaces in Complex Analysis

    I don't mean to sound ignorant, but when reading up on complex analysis in the broad sense, I don't really see the point of introducing Riemann surfaces. It's a way of making multivalued functions single valued, but so what? I don't see the utility of such an idea, which isn't to argue there is...
  6. P

    How i find friction between surfaces without using COF tester?

    In the morning i get this idea ...the simplest way to determine friction force between two sliding surfaces. http://www.youtube.com/watch?v=T2jWT9r2Z-U&feature=plcp
  7. R

    Polarization of Light After Reflection from Surfaces

    I was wondering what happens with linearly polarized light when it is reflected from a surface such as paper? Since it undergoes diffuse reflection, it is scattered in all directions, but does it become randomly polarized, as well? I can't really find an answer to that anywhere, so I'd be...
  8. O

    News Damning evidence surfaces about Libyan attacks

    So apparently the 2 prior Navy Seals who died broke order to go and save unarmed personnel at the scene of the attacks and then were killed 7 hours later by a mortar round. http://www.foxnews.com/politics/2012/10/26/cia-operators-were-denied-request-for-help-during-benghazi-attack-sources-say/...
  9. T

    Bernoulli's Equation Applied to Two Free Surfaces

    Consider two cylindrical containers of the same diameter. Fill them with a liquid to different levels. We all know that the liquid can be "siphoned" from one container to the other using a thin hose, which we will consider to be thin and of uniform diameter. Bernoulli's Equation can be...
  10. T

    Error incurred from approximating fermi surfaces to be a sphere

    I read somewhere that the error incurred from approximating the Fermi surface to be a sphere in k-space goes as 1/N where N is the number of electrons. So, N is generally of the order 10^23. I couldn't figure out how they came up with the value. I was trying to say that the actual shape of the...
  11. M

    Why are non-intersecting branch cuts necessary for multiple-valued functions?

    Why is it necessary that branch cuts for multiple-valued functions are non-intersecting? Does this have to do with needing each sheet for one value (ex. for positive/negative square roots)?
  12. H

    Parametric Surfaces: rectangular and polar coordinates

    Homework Statement I'm not grasping how to convert a surface with known rectangular graph to a parametric surface (using some polar techniques, I assume). I would appreciate it if someone could clarify the conversion process. One of the examples is as follows: A sphere...
  13. T

    Analyzing surfaces and curves using Implicit Function Thm

    for each of the following maps f: ℝ2-->ℝ3, describe the surface S = f(ℝ2) and find a description of S as the locus of an equation F(x,y,z) = 0. Find the points where \partialuf and \partialvf are linearly dependent and describe the singularities of S(if any) at these points f(u,v) = (2u + v...
  14. J

    Understanding Minimal Surfaces: QFT Math Prerequisites

    What mathematics are necessary for understanding and using minimal surfaces particularly in quantum field theory? As of now I have a very limited mathematical background as I will be taking Calc III, Diff Eq, and Linear Algebra next semester but I hope to get into a quantum field theory research...
  15. S

    Finding the volume between two surfaces

    Homework Statement Find the volume between z = x^2 + y^2 and z = 3 - x - y Homework Equations None The Attempt at a Solution I must use a double integral. Using polar coordinates I find that the volume is equal to: V = ∫∫(3 - rcosθ - rsinθ) r dr dθ - ∫∫r^2 r dr dθ I'm...
  16. C

    Which Gaussian surfaces have an electric flux of +q/εo through them

    Which Gaussian surfaces have an electric flux of +q/εo through them? I think is b since in c, +q and -q cancel each other out, d is only -q is that the right approach? thanks!
  17. S

    Exploring 2D Fermi Surfaces and CDW Effects

    why cdw occurs at low dimensional solids with anisotropic fermi surfaces that have prominent nesting vectors?/ what it means : cdws are also common at the surface of solids??where they are more commonly called surface reconstruction or dimerization . PLZ tell me about two dimensional Fermi...
  18. I

    Question about regular surfaces

    Hello, I have been trying to solve the following problem about regular surfaces from Do Carmo's book of differential geometry of curves and surfaces, section 2-3, exercise 14. Homework Statement Problem: Let A\subsetS be a subset of a regular surface S. Prove that A is a regular surface...
  19. S

    Vortex Surfaces Explained - 65 Characters

    Hi! I hope I'm posting this in the right forum... I was wondering if someone could quickly explain what a "vortex surface" is. I've tried to google it, but I only find papers, which won't help me with a simple explanation. I understand that a vortex surface is an object made from parallel...
  20. B

    Finding the Volume Inside Two 3D Surfaces Using Integration Techniques

    Homework Statement Find the volume of the solid inside the surfaces x^2+y^2+z^2=1 and z=x^2+y^2Homework Equations With x=0: z=y^2 ; z=√(1-y^2) or y as a function of z then gives: y=√z ; y=√(z^2-1)The Attempt at a Solution First, I attempted to sketch a cross section of the region by setting...
  21. T

    Fluid mechanics - submerged triangular surfaces

    Hi, i am having difficulty with a question because i cannot seem to get the right answer. i don't think i am far off i just know i go wrong somwhere and if you could point that out it would be great. the question is attatched. for the first part i find the equations of the lines of the two sides...
  22. P

    Question About Flux of Surfaces: Need Explanation

    I have attached a file with my question. From what i see the flux for both surfaces will be 0. I am unsure and need a little of explaining
  23. B

    Finding volume between two surfaces.

    I am having trouble finding the volume between the two surfaces. Well I did the problem and got an answer but I am not sure if I am approaching this type of problem correctly. So can you take a look at how i approached the problem and tell me if i am on the right path or if it is incorrect...
  24. J

    Mapping contours over normal Riemann surfaces

    Hi, Can someone here help me understand how to illustrate maps of analytically-continuous paths over algebraic functions onto their normal Riemann surfaces? For example, consider w=\sqrt{(z-5)(z+5)} and it's normal Riemann surfaces which is a double covering of the complex plane onto a single...
  25. K

    What's it called when a 3D shape can be made of 2D surfaces of all the same size

    What's it called when a 3D shape can be made of 2D surfaces of all the same shape and dimensions? To make a cube, I can use 6 4-sided-squares (of course they're 4 sided) To make a pyramid (3 sided), I can use 4 3-sided-triangles I can do this with pentagon as well (i don't know what the...
  26. G

    Evaluating integral on surfaces

    Hello everyone! Could someone tell me how to evaluate the integral \int_{B_{\delta}(p_0)}{\frac{1}{(1 + d^2(p_0, p))^2}\mathrm{d}v_g(p)} where B_{\delta}(p_0) \subset M and (M, g) is a generic compact surface and \delta > 0 can be made small as one wishes (as usual, d is the intrinsic...
  27. L

    Meromorphic Functions on Riemann surfaces

    In Felix's Klein's pamphlet, "On Riemann's Theory of Algebraic Functions and their Integrals" he describes ways to construct divergence free irrotational flows on a compact Riemann surfaces such as the torus. One method is simply to cover the surface with a conducting material and place two...
  28. D

    Why are liquid surfaces at constant pressure?

    This is probably a stupid question with a simple answer, but please bear with me. Suppose we have a bucket of water that we begin to spin at a constant angular speed. My textbook asked to find the shape of the surface of water, with the hint that the surface would be a surface of constant...
  29. Y

    Intercepts in quadric surfaces?

    Intercepts in quadric surfaces?? Homework Statement How many intercepts can an ellipsoid have? Homework Equations The Attempt at a Solution First of all, I don't understand what exactly "an intercept" means in quadric surfaces. in two dimensions, intercepts are the points...
  30. S

    Curve of intersection of surfaces problem (Answer included).

    Homework Statement "Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g, differentiable. (a) What are the derivatives f'(1), g'(1)? (b) What is the...
  31. D

    Are spheres isotropic and cubes anisotropic?

    Hi, I have the following definition for an isotropic surface: the normals to the measured surface are randomly distributed. Am I right in thinking the following: 1) The surface of a sphere is isotropic? 2) The surface of a cube is anisotropic? I think (2) is correct but not sure...
  32. S

    Sketching Surfaces: Sphere, Circle & More

    Homework Statement I'm also having trouble with these: provide the names and sketch the following surfaces: x2+y2+z2=16 x2+y2=9 x2+2y2+4z2=16 z=-√(9-x2-y2) z=√x2+y2 z=x2+y2 Homework Equations The Attempt at a Solution So for the first one it's a sphere with radius of 4...
  33. K

    The radii of the curvature of the spherical surfaces which is a lens

    the radii of the curvature of the spherical surfaces which is a lens of required focal length are not same. it forms image of an object. the surfaces of the lens facing the object and the image are interhanged. will the position of the image change?
  34. T

    Net Force: Identifying and Summing Forces for 2D Surfaces at Angles

    See attachement for required information. Please and thank you!
  35. A

    Parameterize the intersection of the surfaces

    Parameterize the intersection of the surfaces z=x^2-y^2 and z=x^2+xy-1 What's getting me stuck on this problem is the xy. I set x=t z=x^2-y^2 z=t^2-y^2 z=x^2+xy-1 t^2-y^2=t^2+ty-1 y^2=1-ty Thats as far as of come, I'm stuck on this
  36. R

    Gauss's Law - Irregular Surfaces

    Gauss's Law -- Irregular Surfaces I don't fully understand why Gauss's Law holds for any Gaussian surface. My textbook clearly derives Gauss's Law from Coulomb's Law using a spherical surface, but it then extends the result to any Gaussian surface without sufficient explanation. Why does...
  37. L

    Calculating viscosity force of oil between 2 surfaces

    I'm working on a project at work that requires some tribology/fluids calculations that I don't know if are applicable to me or not. For this scenario, one of the tests will require me to assume hydrodynamic lubrication where oil is completely separating the 2 surfaces. I have a sliding...
  38. U

    Oblique collisions between non-sliding surfaces

    Good day all. First of all, I apologize if this has been asked a million times before. I have not been able to find a straight-forward answer to my wonderings; maybe because they do not yield a straight-forward question. As part of a large exercise of thinking for the joy of thinking*, I am...
  39. A

    Calculating Velocities on Airfoil Surfaces with Vortex Sheet

    First post here on pf! I wanted to ask if it's feasible to determine velocities at the top / bottom surface of an airfoil due to the vortex sheet placed on the camber line (due to thin airfoil theory). I'm attempting to do this because the big picture is to determine the Cp distribution...
  40. T

    Quote on icing/superhydrophobic surfaces

    Hi, I'm writing a Master's thesis on superhydrophobic surfaces aimed at preventing ice accumulation, and I'm looking for an appropriate quote to include in my introduction. Any suggestions?
  41. H

    Exploring Exoplanets: Could We Zoom In on Their Surfaces?

    I keep seeing reports about how all the planets similar to Earth in other nearby planetary systems are now being discovered. Since we won't have the technology to send probes or go there ourselves for some time, all we can do is look at them. My question is: Could we ever get better resolution...
  42. N

    Laser Measurement Repeatability on Various Surfaces

    Hello Everyone, Here is my situation: I am using a laser to take distance (height) measurements on a variety of surface types (e.g. mirror, machined surface, plastic, etc.) and I would like to quantify the repeatability of the laser measurements on each surface type. Please note that I care...
  43. D

    Parametric Surfaces and their Areas

    Homework Statement Find the area of the part of the plane 3x + 5y + z = 15 that lies inside the cylinder x^2 + y^2 = 25. Homework Equations A=∫∫(√1+(dz/dx)^2+(dz/dy)^2) dA The Attempt at a Solution my bounds were r=0 to 5 and theta=0 to 2pi ∫∫√1 + (-3)^2 + (-5)^2 dA =∫∫√35 dA...
  44. L

    Cal 3 Surfaces: Finding Derivatives at (1,1,1)

    Given that the surface x^8y^7+y^3z^6+z^8x63+6xyz=9 has the equation z=f(x,y) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives fx(1,1)= fy(1,1)= Attempt: I tried plugging (1,1,1) in fx but it wasnt rigth. We haven't seen this in class and I don't...
  45. TrickyDicky

    Riemannian surfaces as one dimensional complex manifolds

    If we consider a Riemannian surface as a one-dimensional complex manifold, what does that tell us about its intrinsic curvature? I mean for one-dimensional curves we know they only have extrinsic curvature so it depends on the embedding space, this doesn't seem to be the case for one-dimensional...
  46. L

    Identify these surfaces- quick vector question

    Homework Statement Identify the following surfaces: i) r.u=L ii) r.u=mlrl for -1\leqm\leq1 where k, L, m are fixed scalars and u is a fixed unit vector. Homework Equations The Attempt at a Solution The first one is in the same form as the equation of a plane, but u is not...
  47. G

    Radiation heat exchange between two surfaces.

    Homework Statement Prepairing for a test and this one came up that's confusing me. A spherical object, water cooled, with a diameter of 10 milimeters and ε = 0,9 is kept at 353 degrees Kelvin, when placed in a very large furnace where a vacuum is formed and which walls are kept at 673 degrees...
  48. A

    Can tubes be minimal surfaces?

    Homework Statement Prove that there are no tubes that are minimal surfaces Homework Equations F(u, v) = γ(u) + R(cosuN(v) + sinuB(v)) The Attempt at a Solution A tube is defined to be the surface formed by drawing circles with constant radius in the normal plane in a space...
  49. J

    Quadratic surfaces question

    Homework Statement Write the following quardatic surface equation as a sum of multiples of squares of independent linear functions x^{2}+4y^{2}+56z^{2}+2xy+4xz+28yz Homework Equations The Attempt at a Solution Please see attachment. nb. there is no answer provided by...
  50. L

    Geodesic flows on compact surfaces

    Does a geodesic flow on a compact surface - compact 2 dimensional Riemannian manifold without boundary - always have a geodesic that is orthogonal to the flow?
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