What is Surface: Definition and 1000 Discussions

A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is the portion with which other materials first interact. The surface of an object is more than "a mere geometric solid", but is "filled with, spread over by, or suffused with perceivable qualities such as color and warmth".The concept of surface has been abstracted and formalized in mathematics, specifically in geometry. Depending on the properties on which the emphasis is given, there are several non equivalent such formalizations, that are all called surface, sometimes with some qualifier, such as algebraic surface, smooth surface or fractal surface.
The concept of surface and its mathematical abstraction are both widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface. The concept also raises certain philosophical questions—for example, how thick is the layer of atoms or molecules that can be considered part of the surface of an object (i.e., where does the "surface" end and the "interior" begin), and do objects really have a surface at all if, at the subatomic level, they never actually come in contact with other objects.

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  1. U

    Surface density diamond structure [1 1 1]

    Homework Statement Calculate the surface density (planar density) of the Silicon diamond structure in the [1 0 0],[1 1 0],[1 1 1] planes. Given a = 5.431 A (latice constant) Hint: For the (111) plane, it should help you to think of the diamond lattice as two interpenetrating FCC lattices.2...
  2. K

    Normal Vector for Null Surfaces: How to Define and Fix it Completely?

    I am trying to use the Israel junction conditions for a null surface, but I am running into complications with defining a normal vector for a null surface. As I understand it the normal vector is defined to be perpendicular to the surfaces tangent vectors n\cdot e_i=0, as well as satisfying...
  3. C

    Maxwell equation at the surface of a conductor - paradox?

    Assume that we have a conductor of any shape, say a ball of copper. At electrostatic equilibrium, it is well known that the potential inside this conductor is constant, for otherwise free charges would move from points of highest potential to points of lowest potential (this includes the surface...
  4. Jimbob999

    Surface Charge Density (Electric Fields)

    A disk with a uniform positive surface charge density lies in the x-y plane, centered on the origin. The disk contains 2.5 x 10-6 C/m2 of charge, and is 7.5 cm in radius. What is the electric field at z = 15 cm? I have used the formula...
  5. P

    Parametric Equation of Surface

    Homework Statement Find parametric equations for the portion of the cylinder x2 + y2 = 5 that extends between the planes z = 0 and z=1. Homework Equations I can't really find any connection but I do have x=a*sinv*cosu y=a*sinv*sinu z=a*cosv The Attempt at a Solution I...
  6. S

    Is mass relevant on a frictionless surface?

    I was looking at a problem: http://gyazo.com/c872ea999197823a42568809f9d97d3f and I understood that the reason that the force would have to be greater on a surface with friction because the equation for the force of friction is dependent on mass (μk * mg) and with two masses it essentially...
  7. norlesh

    Surface treating mild steel for hard vacuum

    I understand mild steel is very bad at out gassing so is never usually considered for hard vacuum applications. But if the decision was constrained by other factors would it be possible to apply a surface treatment or coating to the internal walls of the vessel - am I correct in assuming that as...
  8. SpiderET

    Why is higher density causing higher surface gravity?

    Normal density of Earth is 5,5 tons/m3 and surface gravity is 9,8 m/s and. But if we would compress mass of Earth to 50 km diameter, we would get density 1,4 mil tons/m3 and surface gravity would be 158 860 m/s. I know the math, in basic gravity equotation F=Gm1m2/r2 with decreasing r you get...
  9. H

    Molecular desgin for a narrow potential surface curve

    Rather a short question, but what kind of design can be used for a narrow potential surface curvature in molecules?
  10. B

    Could a particle beam reach the surface of the moon?

    Could a particle(proton, gold ion, electron) beam reach the surface of the moon, given current technology? Would the beam diffuse too much too scenter the regolith? Or would the beam have to be focused or continually refocused in order to achieve that? If so why can they do it with laser...
  11. G

    What happens on external surface of the Faraday cage?

    Hello. Let's say we build the Faraday cage which surround strong radiation such that no radiation can escape to outside world for safety. What happens on external surface of the Faraday cage in voltage during shielding? Is it fluctuated?
  12. D

    Question about surface area of faraday cage

    Hi, I am making a faraday cage to be used in reactive ion etching of silicon. I was wondering if the size of the cage, or the surface area has any impact on it's effectiveness or how it works? I know the size of the mesh and the material I use, as well as whether or not it is grounded all...
  13. I

    What does it mean for a charge to be uniformly distributed on a spherical shell?

    Surface current density, K is defined as: K = σv where σ is surface charge density and v is velocity. Given a uniformly charged spherical shell with radius R, spinning at constant angular velocity ω, find the current. So, I start with this formula: dI = K dl dI = σ Rω dl and I placed the...
  14. kostoglotov

    2 cylinders intersect, area of resulting parametric surface

    Homework Statement I want to know if I got the answer correct and if my reasoning is sound. The text answers and solutions manual only gives answers/solutions for odd numbered problems. Here is the problem: And a direct link to the imgur page: http://i.imgur.com/Tko1xFh.png Homework...
  15. S

    Conformal Transformation: Fluid flow over surface waves

    I would like to obtain the conformal map from a uniform rectilinear fluid flowing in the x-direction, where the field is bounded below by the x-axis, to the flow in the w-plane. In the w-plane the flow is correspondingly bounded from below by a trochoid. (A trochoid is a continuous waveform...
  16. R

    Kinetic friction on smooth then rough surface

    Homework Statement Given a 2.0 kg mass at rest on a horizontal surface at point zero. For 30.0 m, a constant horizontal force of 6 N is applied to the mass. For the first 15 m, the surface is frictionless. For the second 15 m, there is friction between the surface and the mass. The 6 N force...
  17. Sobe118

    Point movement on a surface of a sphere

    I'm trying to find the resulting location of a point on a sphere in spherical coordinates or Cartesian. Based on velocities from the perspective of an object on the sphere. So given the: location on the sphere (in spherical or Cartesian) zy - plane rotation of the point up direction of the...
  18. M

    Intersection of line and surface

    A straight line in 3 space can be described as A + Bt, where A is a position, B a direction, and t a scalar parameter. CAD surfaces can be represented in terms of polynomial functions of two variables (u and v) with the highest degree term being u^nv^n. The intersections can then be obtained as...
  19. M

    FInding distance from top of diving bell to lake surface

    Homework Statement A cylindrical diving bell with open bottom and closed top 12.0m high is lowered into a lake until water within the bell rises 8.0m from the bottom end. Determine the distance from the top of the bell to the surface of the lake. Homework Equations I actually solved this...
  20. T

    Calculate gravitational field strength above surface of Mars

    1. Calculate the gravitational field strength at 500km above Mars' surface. Mass of Mars: 6.39 x 1023 kg Radius of Mars: 3.39 x 103 km Constant G: 6.67 x 10-11 2. I used the equation g = GM / r23. To begin with I added the 500km height above ground to the radius, giving 3.89 x 103 km. I then...
  21. S

    How to know which surface represents equation Q (x,y,z) =0?

    The equation. Q(x, y,z) = -5/2:X2 - y2 + 4z2 + 7xy - 2xz - 2yz. Find its axis and draw its intersection with the plane x + y + z = 0 .
  22. X

    Induced Metric on Surface t=const

    Homework Statement Let g_{\mu\nu} be a static metric, \partial_t g_{\mu\nu}=0 where t is coordinate time. Show that the metric induced on a spacelike hypersurface t=\textrm{const} is given by \gamma_{ij} = g_{ij} - \frac{g_{ti} g_{tj}}{g_{tt}} . Homework Equations Let y^i be the coordinates...
  23. E

    Electric Field of a Uniformly Charged Cylindrical Surface

    Homework Statement An infinitely long cylindrical surface of circular cross-section is uniformly charged lengthwise with the surface density σ = A cosΦ where Φ is the polar angle of the cylindrical coordinate system whose z axis coincides with the axis of the the given surface. Find the...
  24. alex.pasek

    What is the total flux of F across the given surface S?

    Homework Statement Considering the vector Field F(x,y,z))(zx, zy, z2), and the domain whose boundary is provided by S=S1∪S2 with exterior orientation and S1={(x,y,z)∈ℝ3 : z=6-2(x2+y2), 0≤z≤6}, S2={(x,y,z)∈ℝ3 : z=-6+2(x2+y2, -6≤z≤0}. Compute the total flux of F across S. Homework Equations...
  25. D

    Surface joining two points in a family of concentric spheres

    Hi, What's the surface joining two points in a family of concentric spheres? Shown below is the general idea; it's actually optical. Two rays meet at P from P1 and P2, respectively, where each point comes from a different sphere. How do I find surface S if I know the coordinates of P1 and P2...
  26. V

    MHB Differential form surface integral

    Question: Evaluate the surface integral $$J = 2xzdy \land dz+2yzdz \land dx-{z}^{2}dx \land dy$$ where S \subset {\Bbb{R}}^{3} is the rectangle parametrised by: $$x(u,v) = 1-u,\ y(u,v) = u,\ z(u,v) = v,\ \ 0\le u, v \le 1$$ so far I have: \begin{array}{}x = u\cos v, &dx = \cos v\, du -...
  27. S

    Does GR reduce gravitational force on a planet's surface?

    Am I correct in thinking that the force of gravity between 2 test objects at rest on a planet's surface is less than it would be for the same objects at rest in deep space? I understand that this occurs because in GR gravitational potential has a mass value which is lost on the surface, while...
  28. D

    How to compute the surface height based on normal vectors

    Suppose I have already found the surface normal vectors to a set of points (x,y), how do I compute the surface height z(x,y)? Basically what I have are the normal vectors at each point (x,y) on a square grid. Then I calculate the vectors u = (x+1,y,z(x+1,y)) - (x,y,z(x,y)) and v =...
  29. A

    Charge distribution on an irregular conducting surface

    I recently read that the charge density is less on surfaces with greater radius of curvature on the surface of a charged irregular conducting body . if anyone can provide a proof or explanation , please help!
  30. SurajBahuguna

    How to determine shape of surface of a fluid?

    I have been told by my teacher that the surface of a fluid is always perpendicular to the net force acting on it. The reason being a fluid can not withstand tangential stress and if a shear stress is applied to it, it will slip until the surface becomes perendicular to the net force. So my...
  31. P

    Projectile Motion- Incline vs. Horizontal Surface

    Homework Statement Two identical metal blocks, H and A, are placed at equal heights on frictionless ramps as shown above. The blocks are released at the same time, travel down the ramp, and then slide off their respective ends of the table. Block H leaves the table from a horizontal surface...
  32. J

    Gravity in General Relativity and Earth's Surface

    Is there something in General Relativity whose value is 9.8 m/s2 at the surface of the Earth?
  33. R

    Contact pressure of piston rod onto a surface?

    I have two images below. This is a single piston with pressure inside, the variables are as listed. P_in = 2700 Pa. P_out = 0 Pa. R1 = Radius 1 = 10 cm. R2 = Radius 2 = 5 mm. P_contact = ? My solution to P_contact is as follows. Area A1 of piston at R1 = .0314m^2 and area A2 at R2 =...
  34. S

    Period of an object orbiting near the surface of the Earth

    During a physics lecture, the professor demonstrated how to find the period of an object that was dropped through a hole drilled straight from one end of our planet to the other. He finished by saying "an object orbiting the Earth near the surface will have a period of the same length as that of...
  35. M

    Thermofluids (Determine surface temperature)

    Homework Statement Air flows at a mass flow rate(m) of 0.05kg/s in a thin pipe with a diameter of 0.15m. The air enters the pipe at (Ti)103 deg celc and cools to(To) 77 deg celc after traveling 5 m in the pipe. The heat transfer coefficient between the duct outer surface and the ambient...
  36. S

    Need help finding the minimum surface area of a cylinder?

    Member warned about posting without showing an effort 1. Homework Statement 'Using the graphing function on your Graphics Calculator, or otherwise, determine the radius for a minimum Surface Area.' I HAVE NO IDEA WHAT TO DO? Homework EquationsThe Attempt at a Solution
  37. P

    How much charge inside a Gaussian surface?

    Homework Statement The electric field has been measured to be horizontal and to the right everywhere on the closed box shown in the figure. All over the left side of the box E1 = 80 V/m, and all over the right, slanting, side of the box E2 = 400 V/m. On the top the average field is E3 = 260...
  38. AdityaDev

    Surface temperature of a planet revolving a sun

    Homework Statement Find the surface temperature of a small planet having circular orbit around the sun with time period T,assuming sun and planet to be black bodies. Take radius of sun = R, its mass = M and its surface temperature as ##\theta_0##. Homework Equations ##P=eA\sigma T^4## Total...
  39. U

    Pressure of a sphere on a regular surface

    Since the pressure a sphere exerts on a surface tends to infinity, how do you actually calculate it? My guess would be trying to see how many atoms of the surface (a straight line) and of the sphere collide. But this is very dependent on the materials and exterior factors. I have searched...
  40. H

    Electric field outside a conductor and its surface charge

    Imagine a surface charge ##\sigma##. The boundary condition on ##E## is ##E_{above}-E_{below}=\frac{\sigma}{\epsilon_0}\hat{n}##, where ##\hat{n}## points outwards perpendicularly to the surface. Because the field inside a conductor is zero, it requires that the field immediately outside is...
  41. Hanyu Ye

    Is surface tension conservative?

    Hello, everybody. Is surface tension a conservative force? I think so, because it is related to surface energy. But I am not 100% sure. Thanks a lot.
  42. Yam

    A wheel rolling on a horizontal flat or inclined surface

    Homework Statement A wheel rolling on a horizontal flat surface at a constant velocity experiences no friction force. Why? A wheel rolling on an inclined surface at a constant velocity experiences friction force. Homework EquationsThe Attempt at a Solution A wheel rolling on a horizontal flat...
  43. P

    Surface Area of y=sin^2(x)+x^2 from 0 to 1 about x axis

    Homework Statement Set up a definite integral for the surface area generated by rotating the curve ##y= \sin^2x+x^2## from ##x=0## to ##x=1## about the a-axis. Homework Equations Surface Area about x axis=##2 \pi y \cdot ds ## The Attempt at a Solution I found ##\dfrac{dy}{dx} = 2 \sin x\cos...
  44. Tarpie

    Surface area of revolution about y

    Homework Statement [/B] Find the surface area obtained by rotating the curve y = x^2/4 - ln(x)/2 1 \leq x \leq 2 Homework Equations 2π \int f(x)\ \sqrt{1+(f'x)^2} dx The Attempt at a Solution I can't seem to isolate for x in terms of y. I raised both sides to e and separated the exponents...
  45. Tarpie

    How to Rotate a Surface Area about the Y-Axis?

    Greetings, y=x2/4 - ln(x)/2 from 1=<x<=2 rotated about the y-axis. I did the equation rotating about the x-axis via 2pi* integral (f(x)*sqrt(1+f'(x)^2)) dx with dy/dx = x/2 - 1/2x but the question calls for rotation about y and i can't seem to rearrange the equation to isolate for...
  46. R

    Minimal Surface between two different coaxial circules

    Dear All, I am trying do find the minimal surface of revolution between two coaxial circular rings of DIFFERENT diameter. I could not find it solved in the net. So I tried to solve numerically system (13-14) Minimal Surface of Revolution -- from Wolfram MathWorld to determine "a" and "b" for...
  47. darida

    Integration of Ricci Scalar Over Surface

    Does this integration of Ricci scalar over surface apply in general or just for compact surfaces? ∫RdS = χ(g) where χ(g) is Euler characteristic. And could anybody give me some good references to prove the formula?
  48. T

    Ball rolling on a larger ball on a surface

    I've just been wondering about this kind of problem. Let m be the mass of the smaller ball, and M be the mass of the larger ball. Assuming the ball does not slip and that the surfaces are frictionless, what is the time that it takes for the smaller ball to reach the bottom/floor if the the radii...
  49. TESL@

    Parametrizing a Complex Curve on a Torus Surface

    Hello, I am currently trying to parametrize a surface constructed by thickening a rather complicated curve, defining its normal, binormal and tangent vectors. Even using Mathematica simplification, the resulting vectors are page long expressions and the reason for it is because I have four...
  50. M

    Electric field on the surface of charged conducting sphere?

    just above the surface it's (kq/r^2) where r is the radius of the sphere and just below the surface it's zero, so is the electric field zero also exactly on the surface ? (as the q enclosed then will be zero since the flux is coming from the surface and not actually penetrating it) and...
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