Surface Definition and 1000 Threads

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

    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...
  2. 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...
  3. 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.
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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...
  9. 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?
  10. 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...
  11. 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...
  12. 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...
  13. C

    Light Refraction on the Surface of a Sphere

    Hello All, Using Snell's Law, it is pretty obvious how to calculate the angle of refraction when both index of refractions are known. My question is how would I apply this to a 3 dimensional situation, such as light refraction in a sphere? Since there are two angles in relation to the normal...
  14. H

    Where to place Gaussian surface

    Homework Statement I'm trying to calculate the electric field through two oppositely charged plates with the same magnitude: http://imgur.com/uXCQqtW Homework Equations Flux = ∫E⋅dA = Qenclosed/ε0 The Attempt at a Solution I understand how the electric field is calculated using S1 and S4...
  15. spareine

    Aluminium oxide surface and conductivity?

    Aluminium surfaces are protected by a thin aluminium oxide layer. The oxide layer supposedly does not conduct electricity. However, when measuring the contact resistance between the test pen of a multimeter and the aluminium, this resistance is always small. Why? Is the oxide layer easily punctured?
  16. M

    EM wave interacting with refelcting surface

    When trying to explain reflection through the EM treatment of light waves, how do we account for the fact that the electric/magnetic field of the incoming light would penetrate into the medium from which it is reflecting off of? Diagrams like these: show the 'reflection point' on the axis of...
  17. mattyboson12

    Why Does Crank-Nicolson Become Unstable for Large Time Steps?

    Hi, I have written a MATLAB program to calculate the surface temperature on a tile and I am trying to find the most stable/accurate differenceing technique. I have plotted the surface temperature (where its actual value is ~122.5) against the time step for the differencing techniques shown...
  18. W

    Coriolis Force Along the Surface of the Earth

    Homework Statement I don't want to post the actual question because I want to understand the situation in a general case. Basically, there is a bullet that moves south along the surface of the Earth as in this diagram: http://abyss.uoregon.edu/~js/images/coriolis_effect.gif. You have to find...
  19. U

    Hydrostatics on plane surface (basic fluid mechanic Q)

    I don't understand how they got the "position of individual forces". For F1, I'm guessing they used a pressure diagram and that's why it's 2/3 from the tip of the triangle. But why wouldn't the force act on the centroid for this square section (thus its 3.5/2)? But for position of F2 (y2)...
  20. U

    Fermi Surface squashed by potentials

    Taken from my textbook: My understanding is that: One valence electron, 2 spin states -> Half-filled Brillouin zone Seeking inspiration from "Nearly Free Electron Model": gaps open up at zone boundaries States nearer to zone boundaries get pushed down in energy further Since a fermi...
  21. G

    Determining the brightness on a projected surface

    I'm trying to determine how bright my projection image will be on a projected surface. I'm struggling to find a formula for this. I'm using (for example) the Canon Rialis projector which is 4500 lumens. I would be using the zoom lens with maximum zoom so the throw ratio would be 7:1. At a...
  22. U

    Finding parametric representation of a surface

    Homework Statement I am trying to find parametric representation of the right surface of a sphere which was cut along the line y=5. x^2 + y^2 + z^2 = 36 Homework EquationsThe Attempt at a Solution x^2 + y^2 + z^2 = 36 This is an equation of a sphere with radius given by: r^2 = 36 r=6...
  23. D

    Surface Barrier in Type 2 Superconductors

    If a field line is moved into a superconductor it meets resistance. This resistance is modeled by adding an image flux thread outside the superconductor which then attracts the flux thread being moved into the superconductor. Why is this attractive force an accurate model of the resistance a...
  24. F

    Muons reaching Earth's surface problem

    Question: Muons created in the upper atmosphere are sometimes able to reach the Earth's surface. Imagine that one such muon travels the 60km from the upper atmosphere to the ground (in the Earth's frame) in one muon half-life of 1.52us (in the muon's frame). How thick is that part of the Earth's...
  25. S

    Vector Potential A: Discontinuity at the surface current

    Homework Statement Prove Eqn. 1 (below) using Eqns. 2-4. [Suggestion: I'd set up Cartesian coordinates at the surface, with z perpendicular to the surface and x parallel to the current.] Homework Equations I used ϑ for partial derivatives. Eqn. 1: ϑAabove/ϑn - ϑAbelow/ϑn = -μ0K Eqn. 2: ∇ ⋅ A...
  26. T

    Increase Catalyst Surface Area & Reaction Rate: Experiment

    Hi there, I am doing an experiment in increasing the mass of MnO2 when it is added to H2O2 decomposition, and I'm measuring the rate of temperature change. I chose increments of 0.050, 0.100, 0.150, 0.200, and 0.250 g to put into H2O2 when it is decomposed, i.e. the MnO2 is a catalyst. I...
  27. gracy

    Where Am I Going Wrong in Calculating Minimum Coefficient of Friction?

    Homework Statement :[/B] block M slides down on frictionless incline .Find the minimum co efficient of friction so that m does not slide with respect to M.Homework Equations :acceleration of the system=total driving force/total mass: [/B] The Attempt at a Solution m does not slide with...
  28. R

    Electrostatic Problem at point on a conical surface

    Homework Statement I am trying to understand a solved problem which is about finding electrostatic potential at point b of the following conical surface with a given surface charge: I have attached the worked solutions to this post. In the solutions, I don't understand how they have got the...
  29. G

    Can a Gaussian Surface Confirm Zero Electric Field Between Two Charges?

    1. We have two positive point charges (+q) at a distance from each other 2. Goal is to Find electric field at point A halfway between the point charges 3.By logic and summation of fields we should get ZERO 4. But how would I use a gaussian surface (sphere i assume ) to prove at point A...
  30. T

    Angle to lift a body off the surface

    Homework Statement A rectangular box is being pulled by a rope with a force T on a horizontal surface, with a friction coefficient μ. What is the minimum angle α at which it will lift of the surface? See attached image. Homework Equations \begin{equation*} \tau = rF\sin\alpha \end{equation*}...
  31. G

    Finding the kinetic friction between the block and surface

    Homework Statement [/B] The block shown is pulled across the horizontal surface at a constant speed by the force shown. If M = 5.0 kg, F = 14 N and Theta= 35°, what is the coefficient of kinetic friction between the block and the horizontal surface? Homework Equations Fx = Fcos(theta) Ff = Fn...
  32. Graeme M

    Greenhouse effect and Earth's surface temperature

    I posed this question in the Earth section of the forums, but I think I wasn't clear enough in how I posed my question, plus I think the question is more physics than Earth science. I have decided to try posing my question as a series of questions building to the main point. This is because I...
  33. M

    Clarification on force on frictional surface constant speed

    for the following 2 examples from: http://www.physicsclassroom.com/class/energy/Lesson-1/Calculating-the-Amount-of-Work-Done-by-Forces A 10-N force is applied to push a block across a frictional surface at constant speed for a displacement of 5.0 m to the right. Would it be more accurate to...
  34. Graeme M

    What is meant by 'Earth's surface temperature'?

    Seeking clarification here. As a climate change sceptic, I loosely follow several blogs that offer critical views of the science behind the idea of global warming, but I am not especially across the breadth of the science. One of the things I struggle to understand is the idea of 'surface...
  35. F

    Normal vector of curved surface

    Homework Statement Homework EquationsThe Attempt at a Solution I can understand it intuitively, but can't prove mathematically...Can you help me??
  36. B

    MHB Why does the equation of the level surface not have ln?

    Could someone please help me with the following question: Consider the function g(x,y,z)=ln(x^2−y+z^2). Find an equation of the level surface of the function through the point (−1,2,1) which does not have ln. First find g(−1,2,1). When I sub in the points I get...
  37. S

    Find the surface of the generated volume

    Homework Statement The region bounded by the graphs of the curves y=4√x , y=0, x=5, x=12 is revolved about the x-axis. a) Find the religion lateral surface area of the generated volume. b) Find the total surface area of the generated volume, including both ends. Homework EquationsThe Attempt at...
  38. A

    Heat transfer and surface area of a pipe

    So a friend of mine is living in an apartment with central heating but the radiators are removed, so there are just the pipes. He seems to think that if you cut aluminium jar caps and cover the pipes with them, it will increase the surface area of the pipe and therefore heat the room more. The...
  39. P

    The surface of the Earth as a reference frame

    At about 05.30, Dr. Lewin makes a small calculation to find the centripetal acceleration one experiences due to Earth's spin (at the equator). He then says that because of how small the magnitude of this acceleration is, we could consider the Earth's surface to be an inertial reference frame...
  40. Hyo X

    Mobility of 2D material, sigma=N*e*mu with N surface density

    I have measured the conductivity of a 2D material. I want to use σ=N⋅e⋅μ to find the mobility μ=σ/(N⋅e). https://en.wikipedia.org/wiki/Electron_mobility#Relation_to_conductivity I have measured N using a spectroscopic/imaging technique to get a Surface Density with units [cm^-2]. Mobility...
  41. Alettix

    Angle of Inclination of a Water Surface

    Homework Statement So the full problem is the following: "On the top of a truck, there is a cubic container with a side-lenght of 1,00 m and it's filled with 500 liters of water. The truck accelerates with 2.0 m/s2 in the x-direction during the whole experiment. When the watersurface has...
  42. R

    Simple (?) question regarding the Fermi Surface

    We all see diagrams of the Fermi surface, the representation of the occupied states, but I can't seem to find a precise diagram of when an electric field is applied. Most diagrams show that the surface has moved, BUT they do not say in which direction relative to the electric field, and when...
  43. goonking

    Calculating the Ratio of d/r for an Object Above a Planet Surface

    Homework Statement An object is located a distance d above the surface of a large planet of radius r. At this position, its true weight is one percent (1.000 %) less than its true weight on the surface. What is the ratio of d/r? Homework EquationsThe Attempt at a Solution so if something...
  44. J

    Statics, moment on submerged surface problem

    Homework Statement [/B] Shown below is a slew gate controlling the flow of salt water flowing into fresh water in a submarine depressurising chamber (used to measure the pressure depth and which is compared to a control signal that has been calibrated). On the inlet side, the depth of...
  45. B

    Heat equation, periodic heating of a surface

    Homework Statement The temperature variation at the surface is described by a Fourier series \theta(t)=\sum^\infty_{n=-\infty}\theta_n e^{2\pi i n t /T} find an expression for the complex Fourier series of the temperature at depth d below the surface Homework Equations Solution of the...
  46. B

    Derivation of Surface Area and Torque for a Helical Plate Attached to a Shaft

    Hello. I would like to know the derivation of surface area of the helical plate, of single turn with a pitch p, diameter D, attached to a shaft of d. It will look like a circle in plan Also, I would like to get the equation relating the torque resisted by the surface, if it is made to enter a...
  47. patrickbotros

    Why does a Charge Outside a Surface Produce No Net Flux?

    My textbook says "the net flux is zero through a closed surface that does not contain any net charge," and justifies it by saying that everything flowing in has an equal thing flowing out. This makes ABSOLUTELY NO sense to me.
  48. samjohnny

    Surface integral in Spherical Polar

    Homework Statement Attached. Homework Equations The Attempt at a Solution Hi, Ok, so for the first part of this question it asks to evaluate the integral of the dot product between A and dS. The magnitude of dS is as shown above, and it is in the radial direction in spherical polar...
  49. ZARATHUSTRA

    Surface integral for scalar quantities

    so for surface integral for scalar quantities. Why do we use cross product not dot product in the integral? but can we just add an unit normal vector n to make the direction the same? My question seems really stupid too a lot people, but this is really my confusion to surface integral. please...
  50. I

    Area of Surface: x^2+y^2+z^2=R^2, z>=h, 0<=h<=R

    Homework Statement Calculate the area of the surface ##x^2+y^2+z^2 = R^2 , z \ge h , 0 \le h \le R## Homework Equations ##A(S_D) = \iint_D |\mathbf r'_s \times \mathbf r'_t|dsdt## where ##S_D## is the surface over ##D##. The Attempt at a Solution We write the surface in parametric form using...
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