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. A

    I Variation of geometrical quantities under infinitesimal deformation

    This question is about 2-d surfaces embedded inR3It's easy to find information on how the metric tensor changes when $$x_{\mu}\rightarrow x_{\mu}+\varepsilon\xi(x)$$ So, what about the variation of the second fundamental form, the Gauss and the mean curvature? how they change? I found some...
  2. Pencilvester

    I Parameterized surfaces from coordinates

    For all parameterized (hyper)surfaces that form smooth manifolds of dimension ##n-1## embedded in Euclidean ##\mathbb {R}^n##, will there always exist a coordinate system ##\partial_{\bar \mu}## on ##\mathbb {R}^n## that yields the same manifold when the right coordinate (say ##\partial_1##) is...
  3. A

    I Integral equation for large surfaces

    We often neglect the terms of a surface integral ##\int_v(\nabla•A)dv=\int_s(A•ds)## for ##s## to be very large or ##v## to be very large, What is actually the reason behind this to neglect??
  4. E

    Pressure in Forces on Submerged Surfaces

    Why does the pressure we take into account is the gage pressure and not the absolute pressure? Reading Fundamentals of Momentum Heat and Mass transfer by Welty in chapter 2 it says "the magnitude of the force on the element dA is PgdA ,where Pg is the gage pressure" my question is why the gage...
  5. GabrielCoriiu

    I Finding all valid surfaces that go through a vector field

    Hi, I'm trying to find all the valid surfaces that go through a vector field so that the normal of the surface at any point is equal with the vector from the vector field at the same point. The vector field is defined by the function: $$ \hat N(p) = \hat L(p) \cos \theta + \hat R(p)...
  6. EEristavi

    Describing an object made by the intersection of 2 surfaces

    Homework Statement Describe and sketch the geometric objects represented by the systems of equations Homework Equations x2 + y2 + z2 = 4 x + y + z = 1 The Attempt at a Solution I can sketch both objects: 1) sphere with center (0,0,0) and radius 2 2) "simple" plane with intersection...
  7. G

    I Photoelectric Measurements & the Nature of Surfaces

    Why are photoelectric measurements sensitive to the nature of the photoelectric surface?
  8. Srihari05

    I Exploring Feynman's Theory on Reflection of Light from Glass Surfaces

    I recently started reading Feynmans book QED. There are a couple of questions I have regarding his theory on the percentage of light that is reflected of two surfaces of glass. My question is as follows, A piece of glass in fact has four surfaces. The front of the glass the back side of...
  9. G

    Use of tally with surfaces and macrobodies in MCNP

    Hi. I need some help with the use of tally card in MCNP. I have been trying to use the f1, f4 and f2 tally to calculate surface current, average flux on a cell and avergage flux on a surface respectively, my question is: It's possible use those kind of tallies with macrobodies and surfaces...
  10. Gene Naden

    I How to prove that compact regions in surfaces are closed?

    This is problem 4.7.11 of O'Neill's *Elementary Differential Geometry*, second edition. The hint says to use the Hausdorff axiom ("Distinct points have distinct neighborhoods") and the results of fact that a finite intersection of neighborhoods of p is again a neighborhood of p. Here is my...
  11. Faizan Samad

    Finding the equipotential surfaces of a finite line of charge

    Homework Statement Consider a line of charge stretching along the z-axis from -L to +L. Find the potential everywhere. What are the surfaces of constant potential. (The next question answers the previous question and says its a prolate ellipsoid. Homework Equations I will upload an image of...
  12. T

    I Proof of The volume under surfaces formula

    Hello everyone, Is there a proof that takes us from the sum idea of the volume: $$\sum_{i=1}^m \sum_{j=1}^n f(x_i,y_j) \Delta x \Delta y$$ To the integral idea: $$\iint_R f(x,y) dxdy$$ Or something that relates the volume to the integral just like The Fundamental Theorem of Calculus?
  13. J

    Decide if the energy surfaces in phase space are bounded

    Homework Statement From Classical Mechanics, Gregory, in the chapter on Hamilton's equations of motion: 14.13: Decide if the energy surfaces in phase space are bounded for the following cases: i.) The two-body gravitation problem with E<0 ii.) The two-body gravitation problem viewed from the...
  14. M

    Statics problem -- Rectangular plate lying on two inclined surfaces

    Homework Statement https://www.img.in.th/image/VNaqVa https://www.img.in.th/image/VNa3k9 This is my home work. Homework EquationsThe Attempt at a Solution I have a problem with the force at A and B. I don't know how to use NA NB in the term of α and β to use moment calculation. I got that...
  15. Pencilvester

    I Find Solutions to Helical Worldline's Surfaces of Simultaneity

    For an inertial frame in flat spacetime with cartesian coordinates, and a particle in that frame whose worldline is a helix (moving in a circle at constant speed in x-y plane), given an arbitrary event with coordinates ##t##, ##x##, and ##y##, (we won’t worry about ##z##) how would I go about...
  16. N

    A Difference Between Event Horizons & Trapped Surfaces - Explained Simply

    Can anyone please explain to me in simple terms what the difference is between a trapped surface and an event horizon. I'm not familiar with global general relativity so cannot understand any of the results I've looked up by googling. I think that Tipler has said that when the future C-boundary...
  17. Irfan Nafi

    Conceptual Equipotential Surfaces Problems

    Homework Statement True or False: 1.Equipotential surfaces intersect: 2.Electric field lines are found within equipotential surfaces: Homework Equations E=Vd The Attempt at a Solution 1. I think this is false since the same reasoning describes why electric field lines don't intersect 2. I'm...
  18. D

    I On the Gaussian Curvature of time-like surfaces

    Firstly, I am asking for your patience and understanding because my maths formalism is not going to be rigorous. In another thread here in this forum, I set an example for which now I am asking further instructions. I am going to ask about time-like surfaces immersed in Minkowskian space-time...
  19. Joppy

    MHB Translation Surfaces: Geometric Definition & Billiard Systems

    In this Wiki article, a geometric definition of a translation surface is given. I'm lost in at the first line were it is stated that a given collection of polygons need not be convex. How is this possible? I am trying to understand translation surfaces from the perspective of dynamical...
  20. A

    MCNPX Mesh Tally Problem: Offset of proton flux at surfaces

    Dear all, we are using MCNPX for the simulation of proton beam interactions at our proton therapy facility. But now we found a very strange behavior within a RMESH1:h flux tally: In an even very simple geometry we see a constriction or rather offset of the Proton flux at surfaces (surface 1118...
  21. T

    Heat radiation between two non-parallel surfaces

    Hi everybody, How do you find the heat radiation between two surfaces that aren’t perfectly parallel or perpendicular to each other? I know that the view factors play a part, however, I can only dig up view factors for parallel and perpendicular surfaces...
  22. esha

    Apparent depth when two or more refracting surfaces are present

    I know the concept of apparent depth as such: It is the depth at which an object is seen when viewed from a different medium. But i want to know what happens when two refracting surfaces are kept one after the other. In the given diagram let the object be placed at the bottom of the vessel...
  23. Sunbodi

    When to Integrate Charge Enclosed for Gaussian Surfaces?

    Hello, I was looking over my notes and I was trying to figure out when we integrate Q enclosed when Q = ρ*d(volume). If there's one thing I've learned from physics II you only integrate when a field is non-uniform. I'm just wondering how we know when it's uniform (usually the problem will tell...
  24. U

    Find the Radius and Center of a Sphere, Quadric Surfaces

    Homework Statement [/B] Find the radius and center of sphere ρ = 28 cos ϕ. Homework Equations Relevant equations would be the spherical and rectangular coordinate equations. The Attempt at a Solution I started off by multiplying both sides of the equation by ρ to get ρ^2 = 28 ρ cosϕ Then...
  25. F

    I Closed surfaces and closed curves

    How to know if a surface is closed ? Surfaces such as x^(2/3) + y^(2/3) + z^2 = 1 or x^6 + y^6 + z^6 = 1. Is there a way to know if these surfaces are closed ( without the help of a computer program) ? Is there a general way to know if a surface is closed ? What about curves ? How to know if a...
  26. lonelypancreas

    [Electromagnetics] E-Fields & Equipotential Surfaces

    Homework Statement This is from the book Engineering Electromagnetics by Hayt & Buck.[/B] Homework Equations E = - (ΔV/ΔL)[/B]The Attempt at a Solution At part (a), I took the potential difference between point A and the point directly above at the higher surface (106 V) and plugged in the...
  27. Arman777

    Gauss' Law: Charge distribution on concentric spherical surfaces

    Homework Statement A metallic sphere of radius a is placed concentrically with a metallic spherical shell with inner radius b and outer radius c. The sphere has a total charge of 2Q and the shell has a total charge of 3Q. (a) What is the charge distribution? Specifically, what is...
  28. V

    A Surfaces that present expansion instead of relaxation?

    Since there are no bonds at the other side of the surface, external layers of solids are usually closer to the next layer. This process is called relaxation. (Example in picture a here). However, at a lecture I attended the other day it was mentioned that some surfaces present expansion...
  29. Mr Davis 97

    Tangent vector on the intersection of surfaces

    Homework Statement The surfaces ##x^2+y^2 = 2## and ##y=z## intersect in a curve ##C##. Find a unit tangent vector to the curve ##C## at the point ##(1,1,1)##. Homework EquationsThe Attempt at a Solution So I'm thinking that we can parametrize the surfaces to get a vector for the curve ##C##...
  30. Daniel Petka

    Red laser erases on glow in the dark surfaces?

    Hey guys, So recently I came across a weird phenomenon... Basically, I shined a red laser on a glow in dark surface and the laser could erase it. The fun fact is that before the laser erases the are, it makes it glow for a short time. You need good goggles to see the glow despite the intense...
  31. G

    A Methods to interpolate surfaces from gradient field?

    I have a 2D regular grid of vectors representing average headings on a 2D spatial domain. These are generated by stochastic simulation of chemical-sampling and gradient-estimation techniques for a robotic search algorithm seeking a chemical source. Without going into a lot of detail, I would...
  32. G

    Interaction between two charged surfaces in contact

    Dear all, I'm curious to know how to calculate an interaction. I'm a chemist and I'm not really used to practice Maxwell equations, so I don't have the complete background for that, but I think it may be trivial even for a physicist student. Let's say we have two surfaces, one has a total...
  33. M

    Quadratic surfaces standard form help

    Homework Statement [/B] Suppose a quadratic equation in 3 variables is put into a standard form represents a hyperboloid of one sheet. This hyperboloid has the property that: • the cross section through z= 0 is a circle of radius 1; • the cross section through x= 1 is the two straight lines...
  34. Jarvis88

    Areas of Surfaces of Revolution

    Homework Statement Find the area of the surface generated by revolving the curve y=√x+1, 1≤ x ≤5, about the x-axis. I'm stuck trying to figure out how I can use substitution...if I am even able. I was trying to rewrite 1 as 4(x+1)/4(x+1) but still can't seem to get the right terms to cancel...
  35. S

    How do I parameterize these surfaces?

    Homework Statement Parameterize ##S={ S }_{ 1 }\bigcup { { S }_{ 2 } } ##, where ##S_1## is the surface with equation ##x^2+y^2=4## bounded above by the graph of ##2y+z=6## and below by the ##xy## plane. ##S_2## is the bottom disk Homework EquationsThe Attempt at a Solution ##{ S }_{ 1...
  36. S

    How to parameterize these surfaces?

    Homework Statement Calculate ##\iint { y+{ z }^{ 2 }ds } ## where the surface is the upper part of a hemisphere with radius a centered at the origin with ##x\ge 0## Homework Equations Parameterizations: ##\sigma =\left< asin\phi cos\theta ,asin\phi sin\theta ,acos\phi \right> ,0\le \phi \le...
  37. Carlos de Meo

    A Selective Surfaces for Solar Thermal Conversion

    Hi Guys I was reading about selective surfaces for solar thermal conversion and, according to the literature, an ideal material for that would have high absorptance in the 0.2-2.5 μm (due to the Planck distribution for a 5000 K black body, i guess) and also low emittance to suppress the losses...
  38. ltkach2015

    A Theory of Surfaces and Theory of Curves Relationship

    Hello I am interested in the Frenet-Serret Formulas (theory of curves?) relationship to theory of surfaces. 1) Can one arrive to the Frenet-Serret Formulas starting from the theory of surfaces? Any advice on where to begin? 2) For a surface that contain a space curve: if the unit tangent...
  39. F

    I Geodesics on S2 Surface: Arc-Length Parametrisation

    Consider the metric of ##S^{2}##: $$ds^{2}=d\theta^{2}+\sin^{2}(\theta)d\phi^{2}$$ Then in order to determine the geodesics on this surface one can minimise the integral $$s=\int_{l_{1}}^{l_{2}}\sqrt{\left(\frac{d\theta}{dl}\right)^{2}+\sin^{2}(\theta)\left(\frac{d\phi}{dl}\right)^{2}}dl$$ where...
  40. T

    I Proper Distance Between Surfaces of Constant R in Curved Spacetime

    How about proper length in curved spacetime? Let's consider the radial distance between two spherical shells in Schwarzschild spacetime. The proper distance between them follows from the spacelike form of the metric with ##dt=0## for simultaneity. So I think having the r-values of the shells the...
  41. toforfiltum

    Describing level surfaces of ##g##

    Homework Statement a) Suppose ##g## is a function such that the expression for ##g (x,y,z)## involves only ##x## and ##y## (i.e., ##g (x,y,z)=h (x,y)##). What can you say about the level surfaces of ##g##? b) Suppose ##g## is a function such that the expression for ##g (x,y,z)## involves...
  42. toforfiltum

    Sketching surfaces described in cylindrical coordinates

    Homework Statement The surface is described by the equation ## (r-2)^2 + z^2 = 1 ## in cylindrical coordinates. Assume ## r ≥ 0 ##. a) Sketch the intersection of this surface with the half plane ## θ= π/2 ## Homework Equations ## r= psin φ ## ## p^2 = r^2 + z^2 ## The Attempt at a Solution...
  43. R

    Visualizing Volume Between Surfaces to Using Double Integrals

    Hey guys. So I've been trying to learn Double Integrals on my own and I'm at Volume between surfaces...so googling some worksheet problems I came across the one and I'm a bit confused. 1. Homework Statement Let U be the solid above z = 0, below z = 4 − y^2, and between the surfaces x = siny...
  44. Jianphys17

    I Do Carmo's book, chap2 Regular surfaces, definition 1.2 -- question

    On chapter over regular surfaces, In definition 1 point 2. He says that x: U → V∩S is a homeomorphisms, but U⊂ℝ^2 onto V∩S⊂ℝ^3. I am confused, how can it be so!
  45. M

    I The "real" angle between two triangular surfaces

    Hello everyone, i'm new to the forum so hope it is the right place for my question :) i need to know the angle between two triangular surfaces, the easiest way would be extract the normal for each surface(u,v) and then using the dot product we can easily compute the cosine for the angle I'm...
  46. Jianphys17

    Courses Gd of curve and surfaces or functional analysis before?

    Hello everyone, i just finished a course of analysis(2)\vector calculus.Now iI'm interested in doing Gd of curves and surfaces(Do Carmo), and functional analysis(Rudin'sbook), but do not know what may have precedence between the two, on which i should start before you think?
  47. mr.tea

    I Divergence theorem and closed surfaces

    Hi, I have a question about identifying closed and open surfaces. Usually, when I see some exercises in the subject of the divergence theorem/flux integrals, I am not sure when the surface is open and needed to be closed or if it is already closed. I mean for example a cylinder that is...
  48. K

    How does light travel through semi-opaque surfaces?

    So i was wondering...if you have 2 surfaces of different areas..one bigger and one smaller...and the one with a smaller area disperses light through a lens..does more light travel through the one with the lens or the other or does the same amount of light travel through both? To get further...
  49. K

    Why does the formula for calculating fin surface area differ in different cases?

    I am working on a project and I am having difficulty understanding a concept I have to analyze a rectangular fin in 2 cases (Adiabatic tip AND Convective tip) and I am having difficulty understanding which surface area to use. For the first case, I want to find Afin which according to my book...
  50. Chaos

    Electric flux on non-orientable surfaces

    How to define and calculate electric flux on mobius strip or klein bottle?These surfaces are non-orientable, so I feel confused about that. Thanks for discussion and help.[emoji4]
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