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. D

    Fresnel zone and reflection of light on surfaces

    In a Scientific American article from 1968 in which he explains classically how light interacts with matter, Victor Weisskopf states that "the reflection of light on the surface of a solid or liquid involves only the oscillators (electrons) located in a small, pillbox-shaped volume at the...
  2. DaveC426913

    Followup: pi on curved surfaces

    Is pi = 3.14159... only true in our flat universe? We talk about whether our universe is open or closed - positive or negative curvature - and scientists have concluded that it is very nearly flat. If the universe did indeed have a large positive curvature, would that result in a different...
  3. B

    Rotation of wheel between two non-slip surfaces

    Homework Statement A certain machine can be modeled as a wheel between two translating bodies. Point P is on the upper translating body and is moving to the left at 6m/s and Point Q is on the lower translating body and is moving to the right at 3 m/s. The radius of the wheel is .3m. Find...
  4. DaveC426913

    What is the relationship between Pi and curvature on different surfaces?

    I was lying awake the other night and thinking about Pi and flatlanders. I haven't done a lot of topology reading, so forgive my naivete. Pi on a flat surface is a number we know well, but what happens to the ratio of a circle's diameter to its circumference on curved surfaces? First question...
  5. MexChemE

    Diffusion -- Tarnishing of metal surfaces

    Homework Statement Problem 18B.13 from Transport Phenomena, BSL. Tarnishing of metal surfaces. In the oxidation of most metals the volume of oxide produced is greater than that of the metal consumed. This oxide thus tends to form a compact film, effectively insulating the oxygen and metal from...
  6. amind

    Charge distribution on the surfaces of parallel conducting s

    Problem: Consider two parallel and large sheets with a surface area . One has a charge and the other is uncharged. q | | | | | | | | | | What would be the electric fields on the three regions as divided by the sheets ? General solution to problems like as told...
  7. qq545282501

    Volume bounded by two surfaces, what am I missing?

    Homework Statement Find the volume of the solid bounded by z=x^2+y^2 and z=8-x^2-y^2 Homework Equations use double integral dydx the textbook divided the volume into 4 parts, The Attempt at a Solution [/B] f(x)= 8-x^2-y^2-(x^2+y^2)= 4-x^2-y^2 i use wolfram and got 8 pi, the correct...
  8. K

    Mass sliding on rough and smooth surfaces

    Homework Statement [/B] Mass m starts sliding down on a rough surface with coefficient of friction μ. it reaches point B and starts sliding frictionlessly till it reaches point D without velocity, i.e. without escaping the arc. What is the maximum length AB=x0 not to escape the arc. What is...
  9. M

    Calculating D field in a region between two charged surfaces

    1. Question: There is a wire with charge, surrounded by a metal cylinder with opposite charge. There is a dielectric surrounding the wire, going out half way to the outer cylinder. Calculate the D field in the region between the cylinder and wire.Homework Equations : [/B] Gauss's law for...
  10. T

    Double integral to find volume between two surfaces

    Homework Statement set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations y = 4 - x^2 z= 4 - x^2 first octant The Attempt at a Solution I am fairly confident in my ability to evaluate double integrals , but I am having a problem figuring...
  11. Conservation

    Volume of a solid bound by four surfaces

    Homework Statement Compute the volume of the solid bounded by the four surfaces x+z=1,x+z=−1,z=1−y2,z=y2−1 Homework Equations Fubini's theorem? The Attempt at a Solution I have tried to visualize this solid and define the limits; when I attempted to integrate by dxdzdy (in that order), I set...
  12. Adrian B

    How much do surfaces tilt due to tidal forces?

    I've read that tides deform the Earth's crust by about 40cm. When I try to visualize the tidal bulge approaching me and then receding away from me, it seems like the local surface under my feet would tilt slightly one way as the bulge approaches, then level out, and then tilt slightly the other...
  13. goonking

    Parametric Surfaces Homework Help

    Homework Statement Homework Equations The Attempt at a Solution so to start this off, I choose a random point, by setting u and v = 0 giving me the point (0,3,1) but I have no idea how what to do next. how do I find ua and vb?
  14. M

    MHB Find all solutions (integral surfaces)

    Find all solutions u(x,y,z) to V \cdot \triangledown u = 0 where V=(1,1,z) and u(r(t)) = constant where r(t)=<x=t, y=0, z=sin(t)>. What are the constants? It has been a really long time since I've done Diff Eq and just trying to prepare to take a grad level course in the Spring. From the...
  15. Alan I

    Distance between equipotential surfaces

    Homework Statement A non-conducting sphere (radius 11.3 cm) has uniform charge density ρ = 0.596 μC/m3. Find the distance, in meters, between equipotential surfaces V1 = 16.2 Volts and V2 = 42.3 volts. (Distance is always positive.) Homework Equations V=kq/r ρ=Q/V The Attempt at a Solution...
  16. diegzumillo

    Curved surfaces that do not preserve area

    Hi all Making this title was harder than I thought. It certainly makes the topic look more advanced than it actually is. I studied differential geometry during my masters but never went much in depth, just enough so I could apply basic concepts to my specific problems at the time. Now I'm...
  17. MTd2

    Holographic and trapped surfaces

    This the breaktrhough by bousso, which was a student from Hawking. It sees it was accepted under peer review magazines in 2 months. You can find the versions at: 1. arXiv:1504.07660 [pdf, other] Proof of a New Area Law in General Relativity Raphael Bousso, Netta Engelhardt Comments: 15 pages...
  18. S

    How do I parameterize the intersection of these two surfaces?

    Homework Statement Parameterize the curve of intersection of the two surfaces: x^2+y^2+z^2=14 z=y^2-x^2 Homework EquationsThe Attempt at a Solution I tried manipulating the equations above but can't seem to get a nice parameterization which I can use to do the rest of the (calculus) problem.
  19. M

    Volume bounded by 3 surfaces, did I do this correctly?

    Homework Statement Find the volume of the solid bounded by the surfaces ## (x^2 + y^2 + y)^2 = x^2 + y^2 ## ##x + y + z = 3 ## and ##z = 0## Homework EquationsThe Attempt at a Solution I begin by converting to polar coordinates to do a cylindrical integration with 3 variables. ## (x^2 + y^2 +...
  20. electrodruid

    A simple physics model for vehicles on different surfaces?

    Hi, Just to let you know my level of knowledge/ability, I studied a degree that included some dynamics, but that was nearly 15 years ago, so I'm rusty. I'm a games programmer, and I tend to understand code (or things that can be translated into code) more easily than hardcore maths equations...
  21. K

    Distance between surfaces in an assembly of sliding parts

    I have been working on designing something in SolidWorks for the first time, which involves several parts with surfaces that slide into one another. I am wondering if there is a table or a standardization in terms of the distance that should be left between metal surfaces with sliding parts. I...
  22. W

    Transversal Intersection of More than 2 Surfaces

    Hi, There is a result that if two manifolds ## M_1, M_2 ## ( I don't know to what extent this generalizes to other topological spaces) intersect transversally, say in ##\mathbb R^m ## , then the dimension of the intersecting set is given by m - ##\Sigma Cod(M_i ) ; i=1,2##, where ##Cod(M_i):=...
  23. R

    Help with Proving Ruled Surface Equation

    Hello, I am studying for an analytic geometry final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on how to do it. If anyone could help I would appreciate it. Question: Prove that the...
  24. S

    Interference on paper, ground glass etc. surfaces

    With laser pointers being so ubiquitous, everyone is familiar with the sight of interference patterns on paper, ground glass and other surfaces (not to mention more subtle experiments like this one): Quantum Eraser -- which has been discussed recently in other threads. We take it for granted...
  25. P

    Pressures indenting surfaces that are stone, metal or skin

    Hello everyone, thank you for helping me in the other post, this one is different but includes pressure and how to calculate how much of it is present when an object is being "indented" or dug into. I come from a forum that discusses fictional characters a lot and we find it highly enjoyable to...
  26. NaturalSymphony

    Geophysics: Dynamic form factor and Equipotential surfaces

    I've got the following problem which I need help with. I'm used to calculating coefficients when the problem is about ellipsoids and first order approximations. But when it comes to spheres and coefficients J_n I really don't know how to approach the problem. Can somebody help me out? Consider...
  27. homer

    SOLVED: Equipotential surfaces for finite line of charge

    Homework Statement Purcell 2.10 [/B][not the problem I'm asking about, but needed for Purcell 2.11 which I am asking about] A thin rod extends along the z axis from z = -d to z = d. The rod carries a charge uniformly distributed along its length with linear charge density \lambda. By...
  28. B

    Parametrization of a curve(the intersection of two surfaces)

    Homework Statement I am looking to find the parametrization of the curve found by the intersection of two surfaces. The surfaces are defined by the following equations: z=x^2-y^2 and z=x^2+xy-1 Homework EquationsThe Attempt at a Solution I can't seem to separate the variables well...
  29. brianeyes88677

    Why Can't We See Diffraction from Metal Surfaces?

    The atoms in a metal (ex. Cu) are arranged as a 3-D grating. But to our common sense ,smooth metal surfaces only reflect lights. Why can't we see diffraction from metal surfaces?
  30. K

    Extended Surfaces (Fins) Adiabatic Tip

    Homework Statement Please see images - full problem statement given. Summary: I have to calculate TL for an adiabatic tip extended surface I found all the equations for points 1-6 but cannot figure out 7. Homework Equations 1st BC: θ(0) = Tb - T∝ 2nd BC: x=L The Attempt at a Solution If it is...
  31. Q

    Help please -- Problem of hydrostatics force in flat surfaces

    Homework Statement Homework EquationsThe Attempt at a Solution I could know the pressure in point B If I calculate the heigh of pressure I got: But I don't know where is my free surface in the container 3, is it down?? I don't know how to keep doing it...
  32. M

    MHB Level Surfaces & Intersection of a Graph: Exploring $f(x,y,z) = x^2+y^2$

    Hey! :o Draw or describe the level surface and an intersection of the graph for the function $$f: \mathbb{R}^3 \rightarrow \mathbb{R}, (x, y, z) \rightarrow x^2+y^2$$ I have done the following: The level surfaces are defined by $$\{(x, y, z) \mid x^2+y^2=c\}$$ - For $c=0$ we have that...
  33. M

    MHB What Surfaces Can Be Described Using Cylindrical Coordinates?

    Hey! :o I am looking at an exercise that asks to describe the surfaces r=constant, θ=constant and z=constant in the cylindrical coordinate system. The cylindrical coordinates are $(r, \theta , z)$, that are defined by $x=r \cos \theta , y=r \sin \theta , z=z$ $r=\sqrt{x^2+y^2}, z=z ...
  34. Calpalned

    Level curves, level surfaces, level sets

    Homework Statement I know that the equation ##z = f(x,y)## gives a surface while ##w = f(x, y, z) ## gives an object that has the same surface shape on top as ##z = f(x,y)## but also includes everything below it. If these statements are correct, what is the level surface of a function of three...
  35. C

    Visualizing a Parametric Equation in 3D Space

    Homework Statement Given the eqn x=2, y=sin(t), z=cos(t), draw this function in 3-space. Homework Equations ABOVE^ The Attempt at a Solution I did this: x^2+y^2+z^2=2^2+(sin(t))^2+(cos(t))^2=5 Therefore we get x^2+y^2+z^2=5 Which is the eqn of a sphere with radius root5. My friend said it's...
  36. RJLiberator

    How to identify the curve at intersection of level surfaces

    Homework Statement Sketch a picture of the cone x = sqrt(y^2+z^2) , and elliptic paraboloid x = 2−y^2−z^2 on the same grid. Although the picture does not have to be perfect, indicate clearly the orientation of both figures relative to coordinate axes. Identify the curve at the intersection of...
  37. V

    Flux through various Gauss' surfaces

    Homework Statement I have uploaded a file that shows the question.[/B]Homework Equations I believe the only relevant equation is: flux = Q(enclosed)/E(knot)[/B]The Attempt at a Solution Well I have some questions first. The problem statement says that the sphere on the left has a net...
  38. R

    Motion between two charged surfaces, find initial speed

    Homework Statement Two charged, parallel, flat conducting surfaces are spaced d = 1.3 cm apart and produce a potential difference ΔV = 625 V between them. An electron is projected from one surface directly toward the second. What is the initial speed of the electron if its comes to rest just at...
  39. B

    MHB Parameterisation of quadric surfaces of order 2

    Hi Folks, 1) Can anyone provide some online sources on how to parameterize quadric surfaces of order 2 as shown in this link Algebraic Surface -- from Wolfram MathWorld Is there a standard technique? I did a google search with no useful info. Thanks
  40. J

    Surfaces of constant gradient-magnitude

    In other words, when we take for potential function instead of F the square root of (6F/6x)²+(6F/6y)² (in the particular case of two-dimensions). Does this lead to anything interesting?
  41. E

    Frank-Condon Principle - Potential Energy Surfaces

    Theory and my Understanding: So I understand how the frank condon principle let's us effect electronic transitions instantaneously, since the motion of nuclei (on the timescale of such electronic transitions) is quite slow. Consequently, when a photon of light is absorbed you can have an...
  42. W

    Evaporation of water from non polar surfaces

    I'm puzzled by a phenomenon that my daughter pointed out to me. If you have no plastic ware in the dishwasher, your glass and ceramic dishes will dry faster. Slow evaporation from plastic is easy to understand; the water beads up and presents a smaller surface area. What I'm not clear on is why...
  43. moriheru

    Recommended books on Riemann surfaces in context of bosonic stringtheory

    I am looking for a introductiory book on Riemann surfaces in context of bosonic String theory, or heterotic String theory for that matter. Prices should be affordable but should not matter, of I nead guide books this also does not matter...Help is seriusly apreciated.
  44. W

    Complements of Curves in Closed Surfaces: Homeomorphic?

    Hi, let ## \alpha, \gamma ## be non-isotopic curves in a compact, oriented surface S. There is a result to the effect that ## S-\alpha## is homeo. to ## S- \gamma ## . This is not true as stated; we can , e.g., remove a disk (trivial class) in a copy of S and then remove a meridian ( a...
  45. P

    Hydro static forces on curved surfaces

    Homework Statement http://postimg.org/image/4fhu5k9r9/ Can someone explain what is meant by 'missing water' in this solution diagram The original question diagram had no such water above the gate AB Homework Equations The Attempt at a Solution
  46. L

    Color through a prism on black and white surfaces.

    When looking through a triangular prism, I found that a black shape on a white back ground causes the blue end of the spectrum to be on the top of the black shape, and the red/yellow end is directed towards the bottom. The reverse is true for a white shape on a black background. Why is this?
  47. PhysicsKid0123

    Vector calculus, surfaces, and planes.

    I have attached an image... Okay, so I have been stuck on this problem for like 2 hours now and I have no idea how to find r(x). I know the trace is the intersection of the plane and the surface. My first attempt was to substitute the plane y+2x=0 equation for the surface equation by solving...
  48. I

    MHB Cylinders and quadric surfaces

    ok so this is the part I am really struggling with. we need to be able to recognize an ellipsoid, cone, elliptic paraboloid, hyprboloid of one sheet, hyperbolic paraboloid, hyperboloid of two sheets given an equation. he's going to give us an equation of one and ask us to identify and sketch the...
  49. G

    Static Charge distribution along textured surfaces

    How does the texture of a surface affect the concentration of charge on that surface? Say we compare a balloon (smooth) and a football (textured) (ignoring material differences) and give them the same total charge. Then we introduce dust particles. How do the two surfaces attract dust...
  50. 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...
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