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

    Capacitor with connections on inner surfaces

    A capacitor C is made from large disks with a very large R and the gap between the plates s is very small (s<<R). The connections to the plates are made "inside" the capacitor (on the inner surfaces) at the center of the plates. The capacitor is hooked up to a battery and switch. The circuit...
  2. X

    Area-Minimizing Surfaces

    Homework Statement "If an area-minimizing surface can be given by the graph of a function, that function satisfies the minimal surface equation: (1+{f_y}^2)f_{xx}-2f_xf_yf_{xy}+(1+{f_x}^2)f_{yy} = 0 1.)Determine if the graphs of the following functions may be area minimizing: a.) z = 2x+4y+10...
  3. L

    Calculating Area of Ellipsoid: Surfaces of Revolution

    Consider the ellipse: (\frac{x}{2})^2 + y^2 = 1 We rotate this ellipse about the x-axis to form a surface known as ellipsoid. Determine the area of this surface. Start off by solving for y. y = \sqrt{1-\frac{x^2}{4}} Then find the derivative. y' =...
  4. A

    Finding the angle between surfaces

    Homework Statement http://img526.imageshack.us/img526/4890/asdny3.jpg *Length of each side of the square is 3. *Length of DP = Length of BQ = Length of GR = 1 Calculate \theta, the angle that is made between the surface of PQR and the surface of CGHD Homework Equations No...
  5. H

    3d-graphing freeware that accepts three variables (quadric surfaces)

    I'm looking for a 3D-graphing-freeware that's takes x, y, z variables (axis dependence) because I want to see how the different coefficients and constants change the graph. I've tried two open-source programs thus far and none of the accepts the equation for ellipsoids.
  6. D

    3D Surfaces - Equation Formed When Rotating a 2D Line About an Axis?

    Hi everyone, I'm pretty new to Physics Forums but it seems like a fairly friendly community. :) Homework Statement Determine the equation of the surface formed when the line x=3y is rotated about the x-axis. Homework Equations x=3y is the given line.The Attempt at a Solution First I...
  7. M

    Proving Areas, Surfaces, and Volumes Using Integral Methods

    hi how can the following be proved using integral methods: a) prove surface area of sphere, radius a, is 4 \pi a^2 b) prove area of a disk, radius a, is \pi a^2 c) prove volume of ball, radius a, is \frac{4}{3} \pi a^3 d) prove volume of axisymmetric cone of height h and base with radius...
  8. W

    Finding Shortest Paths on Surfaces

    Ok, so I was eating an apple and I asked myself: If I had two points on the surface and I wanted to find the shortest path between those two points, how would I go about that mathimatically. How about if I had an ellipsoid? Thanks
  9. P

    Concentration of Caco3 in water flow on leaned flat surfaces with rain channels

    Homework Statement On flat, leaned, surfaces, there are channels (rillenkarren), which are made by dissolution by rainwater (http://mafija.fmf.uni-lj.si/seminar/files/2005_2006/rillenkarren.pdf - not very important). They are 2-3 cm wide and around 0.5 m long. The formation is not yet known...
  10. K

    What is the formula for defining curvature on three-dimensional surfaces?

    How do you define curvature for curves on three-dimensional surfaces when the surface is given in the form z=f(x,y)? The resulting formula should be a lot simpler than the one for parametric curves of the form r(t)=(x(t),y(t),z(t)), like it becomes for two-dimensional curves given by y=f(x)...
  11. D

    Solving Quadric Surfaces: Reducing, Classifying & Sketching

    Homework Statement Reduce the equation to one of the standard forms, classify the surface, and sketch it: 4x = y^2 - 2z^2 Homework Equations The Attempt at a Solution I really don't know what to do for this one because most of the equations I've seen like this involved x^2. Unrelated to...
  12. J

    Friction between two surfaces

    Homework Statement hi, i am almost done my lab on the coefficient of friction on an inclined plane. I was attempting to prove that the coefficent of static friction would be the same no matter the weight of the mass that was static on the incline. However, i plotted a coefficient versus mass...
  13. O

    A problem about vectors and surfaces

    Homework Statement Find the set of all points on the surface (y + z)^2 + (z − x)^2 = 16 where the normal line is parallel to the yz-plane. Describe this set. The Attempt at a Solution I find the gradient vector of the surface then I said that the f at f*i should be zero when it is...
  14. D

    Help with paper on gradient descent evolution of surfaces

    Hi all, I'm trying to understand someone's PhD thesis on the topic of variational surface evolution and its application in computer vision, and I'm having trouble working out how he evaluates some particular types of expressions involving the gradient. I think it'll be easier if I specify the...
  15. A

    Please some help With integrall over surfaces

    Homework Statement Please read the attached Homework Equations In the image attached The Attempt at a Solution Thanks a lot
  16. A

    Solving for Surface Equations and Area: Parametrized Surfaces Explained

    Homework Statement Find an equation describing a surface, and find the surface area Homework Equations Please se the picture attached The Attempt at a Solution Thank you so very much for any help---
  17. quasar987

    Fundamental polygons and surfaces

    The theorem of classification of closed surfaces says that any closed surface is homeomorphic to a fundamental polygon in the plane. I was wondering if any fundamental polygon can be made into a closed surface by adjoining an appropriate atlas to it. The topological requirements of a closed...
  18. I

    Are the Helicoid and Catenoid Conjugate Minimal Surfaces?

    Homework Statement Show that the helicoid and the catenoid are conjugate minimal surfaces Homework Equations the helicoid is given by the parameterization X(u,v) = (asinh(v)*cosu, asinh(v)*sinu, au) = (x1, x2, x3) the catenoid is given by the parameterization Y(u,v) = (acoshv*cosu, a...
  19. P

    Gragging a board across surfaces with different frictions

    "A uniform board of length L and mass M lies near a boundary that separates two regions. In region 1, the coefficient of kinetic friction between the board and the surface is (mu1), and in region 2, the coefficient is (mu2). The positive direction is from the region with mu1 to the region with...
  20. B

    Understanding Electric Field & Equipotential Surfaces

    Homework Statement If the electric field at a point in space has a magnitude of 300 volts/meter, about how far apart are the equipotential surfaces that differ by 10 volts? well, i think that they are 30 meters apart. All I did was 300 volts/meter/10 volts since that will cancel the...
  21. T

    Treatment makes surfaces self-cleaning

    "Treatment makes surfaces self-cleaning" I read this article, a few weeks ago http://www.engineerlive.com/european-design-engineer/instrumentationelectronics/17068/treatment-makes-surfaces-selfcleaning.thtml Do anyone know more about this? Can it actually be applied to metal? i.e...
  22. O

    Finding Area Vectors for Paraboloid Surfaces

    Homework Statement I am wondering what the set of area vectors for a surface would be. For a plane on the xy-plane, I know the set of area vectors is <0,0,dx*dy>. Homework Equations So, for a set of points (x,y,z) that make a paraboloid, if F(x,y,z)=0 then [grad(F)•<dy*dz, dx*dz...
  23. K

    What are the level surfaces of the function f(x,y,z) = z + sqrt(x^2 + y^2)?

    Homework Statement Describe the level surfaces of f(x,y,z) = z + sqrt(x^2 + y^2) The Attempt at a Solution First of all, what is actually a level surface? Just a normal surface in space? I followed an example I found on the internet, and this is my attempt at a solution: First...
  24. B

    Finding Parametric Equation of Tangent Line to Intersection of Surfaces?

    Homework Statement Hi, I need help with the following. I'm asked to find the parametric equation of the tangent line to the curve of the interrsection of the paraboloid z = x^2 + y^2 and the ellipsoid 4x^2+y^2+z^2 = 9 at the point (-1,1,2). Homework Equations I think I'm asked to find...
  25. P

    Net Flux Through Closed Surfaces & Point Charges

    "The net flux through ANY closed surface surrounding a point charge q is given by q/"permittivity of free space" and is independent of the shape of that surface." I'm having a little trouble understanding this... when my book derived this formula, it did so by using the surface area of a...
  26. D

    Proving the Independence of Connected Sum on Open Discs: A Topological Approach

    I am trying to show that the connected sum of two topological surfaces does not depend on the open discs removed. Any hints?
  27. L

    Solving a Block's Motion on Inclined & Horizontal Surfaces

    A block of mass 3.0 kg starts at a height 0.60 m on a plane that has an angle of 30 degrees from the horizontal. Upon reaching the bottom, the block slides along a horizontal surface. If the coefficient of friction on both surfaces is 0.20, how far does the block slide on the horizontal surface...
  28. T

    Identifying Surfaces for Vectors: k, l, m, n, \hat{u}

    The question reads: Identify the following surfaces given that k, l, m, n are fixed values and \hat{u} is a fixed unit vector. a) |\overrightarrow{r}|=k b) \hat{r}\cdot \hat u=l c) \overrightarrow{r} \cdot \hat{u} = m|\overrightarrow{r}| for -1 \leq 1 d)|\overrightarrow{r} -...
  29. quasar987

    Smoothness of surfaces

    According to Presley (Elementary differential geometry), "A smooth surface is a surface \mathbf{\sigma} whose atlas consists of regular surface patches." (The atlas of a surface is a collection of homeomorphisms that "cover" it. A surface patch is just another word for an homeomorphism in the...
  30. T

    Fluid Mechanics Forces on Curved Surfaces

    Not really a homework problem, but I'm having a hard time with this section. It's mostly dealing with the horizontal component of the force. This is from my book, Fluid Mechanics by frank m white 5 th ed, "The horizontal component of force on a curved surafce equals the force on the plane...
  31. D

    Book on tensor symmetry, geometries, surfaces

    I'm looking for a good tensor reference book that provides insights into the following areas of tensors: 1. Symmetries & how to extract them. 2. Full tensor visualisation eg. how to visualise the stress tensor in its 9 component form without resorting to a split into 3 simultaneous...
  32. I

    What is the flux though each of the surfaces

    What is the flux though each of the surfaces: I got: Surface 1: -q/ε Surface 2: -q/ε Surface 3: 0 Surface 4: 0
  33. F

    Electric Field and Potential Change in a System with Equipotential Surfaces

    Problem: A given system has the equipotential surfaces shown below, where Vo = 12.0 V. (a) What are the magnitude and direction of the electric field? (b) What is the shortest distance one can move to undergo a change in potential of 5.00 V? I am not too sure on what equations I...
  34. V

    If the second equation is correct, then the volume is infinite.

    Volume between two surfaces? hi guys, i hope you can help with this, how to find the volume between those surfases: x^2+y^2+z^2=0 z=\sqrt{x^2+y^2} thanks in advance
  35. B

    Parametric Surfaces: Integral of S = x^2 + y^2 + 2z^2 = 10

    I need to take a surface integral where S is x^2 + y^2 + 2z^2 = 10. I need help with the parametrization of the curve. Letting x=u and y=v makes the problem too complicated. Can you let x=cos(u), y=sin(u) and z=3/sqrt(2)?
  36. B

    Property of integrals over surfaces

    If f:S \to R is a continuous function on a surface S, we define \int\limits_{}^{} {\int\limits_S^{} {fdS} } = \int\limits_{}^{} {\int\limits_D^{} {\left( {fo\Phi } \right)} } \left\| {\frac{{\partial \Phi }}{{\partial u}} \times \frac{{\partial \Phi }}{{\partial v}}} \right\|dudv...
  37. E

    What is the Equation for Finding the Minimum Area of a Non-Euclidean Surface?

    If we have that for geodesic they satisfy... S=\int_{a}^{b}ds\sum_{a,b}(\dot{x_{a}\dot{x^{b}) then minimizing the functional we get the geodesic equation..my question is if for the Area of a Surface is: A=\iint_{S}dA\sum_{a,b}(f_{a})(f_{b}) wherewehave defined f_{a}=df/dx_{a}...
  38. C

    How do two-dimensional surfaces vibrate under forced oscillations?

    Hi, I'm looking into the subject of "Chladni plates": http://www.physics.montana.edu/demonstrations/video/3_oscillationandwaves/demos/chladniplates.html For a lecture I'm supposed to prepare, and I'm looking for information on how exactly a two-dimensional surface vibrates under forced...
  39. P

    How can I graph cylinders and quadratic surfaces?

    I desperately need some help with sketching cylinders and quadratic surfaces. We did this in first year and I understood it then but now that I look at it, I have no idea where to start. Oh and yes I have been into talk with my ta quite a few times but I still don't quite understand; I'll go...
  40. maverick280857

    Gaussean Surfaces [Can they pass through charge distributions?]

    Hello. My textbook says that a Gaussean surface must be carefully chosen so that a point charge (or point charges constituting a discrete charge distribution) does not lie ON it, as otherwise the electric field at the location of the charge would be infinite and hence, it would not be possible...
  41. Q

    Curve of intersection between surfaces

    I looked through some books and couldn't find how to find curves of intersection between surfaces. My question asks: explain why the curvature between surfaces z=x^2 and x^2+y^2=4 is the same of intersection between the surfaces z=4-y^2 and x^2+y^2=4. please help i feel really dumb right...
  42. F

    E-Field Lines and Equipotential Surfaces

    Hey all, I'm a bit confused over an experiment in which we mapped Equipotential Surfaces to then use to draw Electric Field lines using electrodes placed into water to act as a dipole. Along with this, we also had to place a circular conductor and insulator (both uncharged) into the water as...
  43. M

    Ranking E field in equipotential surfaces, confused

    Hello everyone I'm confused on this topic. I read about it in the book and it made sense though. The question is: Figure 24-25 shows three sets of cross sections of equipotential surfaces; all three cover the same size region of space. Diagram: http://www.webassign.net/hrw/25_29.gif (a)...
  44. Q

    Quadric Surfaces: Definition & Examples

    why are they called quadric surfaces?
  45. P

    Questions about forces on surfaces

    I've been trying to go through the problems in our assigned textbook, College Physics, but like any other Serway book the material is not in-depth enough for me to be able to solve the problems easily. This one particular problem has got me thinking about how blocks would move across a surface...
  46. RadiationX

    Volume of a Region bounded by two surfaces

    Find the volume of the solid region R bounded above by the paraboloid z=1-x^2-y^2 and below by the plane z=1-y The solution to this problem is: V=\int_{0}^{1} \int_{-\sqrt{y-y^2}}^{\sqrt{y-y^2}} (1-x^2-y^2)dxdy -\int_{0}^{1} \int_{-\sqrt{y-y^2}}^{\sqrt{y-y^2}}(1-y)dxdy I thought that...
  47. M

    Solving E-Fields using Gauss' Law: Choosing Surfaces

    I wonder if i could compute resultant E-fields using Gauss' law and finding the field from the flux. I have a few difficulties, the first is of course, finding the E-field from the flux and the second is regarding the closed surface. how should i choose what surface to use, especially if the...
  48. R

    Need Help with Nodal Surfaces in Chemistry

    Hello, New to chemistry need help with this question and I don't understand what a nodal surface is How mnay nodal surfaces are there for a) a 2s orbital b) a 3px orbital If anyone can help me that would be great. Thank you
  49. L

    Surface Integral of Two Surfaces

    Hello! This is a question from one of our past exams, and it's had me stumped for the past hour. The question states: The cylinder x^2+y^2=2x cuts out a portion of a surface S from the upper nappe of the cone x^2+y^2=z^2. Compute the surface integral: \int\int (x^4-y^4+y^2z^2-z^2x^2+1)...
  50. B

    How Can We Accurately Characterize Rough Surfaces in Epitaxial Systems?

    Does anybody know any neat measurements one can do to characterise a surface? I'm thinking generally of rough surfaces such as the ones you come across in epitaxial systems. In other words: If I have two ensembles of surfaces, what is the best method to use to determine whether the same...
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