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. jonathanm111

    Vector Calculus, setting up surface area integral.

    The question goes like: find the SA of the portion S of the cone z^2 =x^2 +y^2 where z>=0 contained within the cylinder y^2+z^2<=49 this is my attempt using the formula for SA, I could switch to parametric eqns, but even then I'd have hard time setting up limits of integration.
  2. A

    How high does the ball rise on the non-slip surface?

    Homework Statement A solid ball with radius of 9.7 cm is released from the height of hs=88.1 cm on a non-slip surface. After reaching its lowest point the ball begins to rise again, on a frictionless surface. How high does the ball rise on that side? Express your answer in cm. Homework...
  3. G

    I Glass -- how to increase the surface area

    I need to increase the surface area of the glass which will be used in a solar still with the intention of keeping the glass as cool as possible. My first thought was bubble wrap because it's transparent and I thought it would not interfere with the light but then I remembered it is a good...
  4. M

    Mathematica Manipulating a Surface and Color Contours

    Hi PF! I'm trying to have color contours of a surface and then manipulate the surface. Using the sample code below, we see that if you step forward Mathematica plots the surface and then also the color contours. However, rather than stepping forward, if we play the video the color contours...
  5. J

    A For MOSCAP: why is surface potential constant after Vg = Vth

    Hi, I'm studying MOSCAP at university and there are 3 regions based on Vg. Accumulation Depletion Inversion For P substrate the surface potential increases as I increase the gate voltage (positive). The books say that at Vg = threshold voltage the surface potential is maximum. But why does it...
  6. T

    How to evaluate a surface integral with three points?

    Homework Statement Let G=x^2i+xyj+zk And let S be the surface with points connecting (0,0,0) , (1,1,0) and (2,2,2) Find ∬GdS. (over S) Homework EquationsThe Attempt at a Solution I parametrised the surface and found 0=2x-2y. I’m not sure if this is correct. And I’m also uncertain about...
  7. T

    Volume of a Parametrised Surface

    Homework Statement Let C be the parametrised surface given by Φ(t,θ)=(cosθ/cosht, sinθ/cosht,t−tanht), for 0≤t and 0≤θ<2π Let V be the region in R3 between the plane z = 0 and the surface C. Compute the volume of the region V .Homework EquationsThe Attempt at a Solution I thought I needed...
  8. ccdani

    Transmission of blue light through ceramics and surface roughness effects

    Hey :) I measured the transmission of blue visible light (350-550nm) through lithiumdisilicate ceramics with an ulbricht ball and an spectrometer. The light source was a led dental curing unit (bluephase style). The light guide was positioned direct on the ceramics. Now I wanted to test...
  9. E

    Gauss' Law problem: determine the electric flow through a square surface due to a nearby charge

    Homework Statement determine the electric flow through a square surface of side 2l due to a load + Q located at a perpendicular distance l from the center of the plane I really don't know how to answer this question .i need help guys Thanks Homework EquationsThe Attempt at a Solution I ended...
  10. M

    Conceptual Question: Block on a wedge on a frictionless surface

    Homework Statement This is more of a conceptual question, but say a block was set on top of an inclined plane, which was set on top of a frictionless level surface. Would the inclined plane move? Why or why not Homework Equations None The Attempt at a Solution My thought...
  11. J

    Forces on a Block and Friction on a Surface

    Homework Statement A 2.5 kg block is initially at rest on a horizontal surface. A horizontal force of magnitude 5.8 N and a vertical force are then applied to the block. The coefficients of friction for the block and surface are μs = 0.43 and μk = 0.24. (a) Determine the magnitude of the...
  12. M

    Calculate the given surface integral [Mathematical physics]

    Homework Statement Calculate \int_{S} \vec{F} \cdot d\vec{S} where \vec{F} = z \hat{z} - \frac{x\hat{x} + y \hat{y} }{ x^2 + y^2 } And S is part of the Ellipsoid x^2 + y^2 + 2z^2 = 4 , z > 0 and the normal directed such that \vec{n} \cdot \hat{z} > 0 Homework Equations All the...
  13. K

    What is the thickness of the drop if its radius is r?

    1. A small drop of fat floats on the surface of a liquid whose surface tension is s. Surface fat tension at the air-fat interface is s1, at the fat-liquid interface is s2. Determine the thickness of the drop if its radius is r.2. ##F=\sigma l## ##\delta P=\sigma (\frac 1 R_1 + \frac 1...
  14. K

    Surface tension trivial problem

    1. The films of the two liquids are separated by a bar of length l. The coefficients of surface tension of liquids are equal to s1 and s2, respectively. What force acts on the bar on the liquid side?(It is a rectangular surface of 2 liquids separated by a bar of length l) 2. Force=(surface...
  15. R

    I Why does Mars have so much iron on it's surface?

    Earth also has iron rich deposits, I think generally they are thought to be remains of meteorites. Same is likely for Mars, but there is a lot more iron (and compounds) on the surface of Mars than there is on Earth. Are there substantial amounts of silicate rocks, as Earth has?
  16. K

    How Does the Cassie-Baxter Model Explain the Lotus Effect?

    1.The lotus effect refers to self-cleaning properties that are a result of ultrahydrophobicity as exhibited by the leaves of "lotus flower". Dirt particles are picked up by water droplets due to the micro- and nanoscopic architecture on the surface, which minimizes the droplet's adhesion to that...
  17. Raihan amin

    The shape of the surface of a soap film

    1. Two coaxial rings of radius R=10 cm are placed to a distance L from each other.There is a soap film connecting the two rings(that looks like a cylinder which have different radii with z coordinate. (The rings lie in xy plane)).Derive a differential equation describing the shape r(z) of the...
  18. mishima

    Surface of a Cylinder inside a Sphere

    Homework Statement This is a problem from Boas, Mathematical Methods of the Physical Sciences chapter 5, section 5, number 6. Find the area of the cylinder x^2+y^2-y=0 inside the sphere x^2+y^2+z^2=1. Homework Equations This section deals with projecting curved areas onto a coordinate plane...
  19. Gene Naden

    I Differential for surface of revolution

    O'Neill's Elementary Differential Geometry contains an argument for the following proposition: "Let C be a curve in a plane P and let A be a line that does not meet C. When this *profile curve* C is revolved around the axis A, it sweeps out a surface of revolution M." For simplicity, he...
  20. M

    MHB Change the form of equation of surface

    Hey! :o We consider the surface $S$ of the space $\mathbb{R}^3$ that is defined by the equation $2(x^2+y^2+z^2-xy-xz-yz)+3\sqrt{2}(x-z)=1$. I want to find (using symmetric matrices) an appropriate orthonormal system of coordinates $(x_1, y_1, z_1)$ for which the above equation has the form...
  21. P

    Free surface charges on concentric cylinders

    Homework Statement Consider an infinitely long cylindrical rod with radius a carrying a uniform charge density ##\rho##. The rod is surrounded by a co-axial cylindrical metal-sheet with radius b that is connected to ground. The volume between the sheet and the rod is filled with a dielectric...
  22. Jozefina Gramatikova

    Fractional uncertainty of g on the surface of the Sun

    Homework Statement Homework Equations The Attempt at a Solution it looks like I got too big numbers for the uncertainty
  23. bob012345

    I Gabriel's Horn, Inside vs. Outside Surface area

    Consider Gabriel's Horn, the mathematical object formed by a surface of revolution of the curve x= 1/x from x=1 to infinity. It is known that one can fill the horn with a volume of Pi cubic units of paint but it would take an infinite amount to paint the surface. I think they usually mean the...
  24. I

    B How to find the surface density for a given linear density

    Given a square with a linear mass density of: λ(x) = a * x (see image below where black is high density and white is low density) how would you deduce what the surface mass density is? I get confused for the following reason: To me it seems that the surface mass density should depend on x...
  25. H

    I Gauss' law applied on a four-dimensional surface

    in a 4D plane or on a four-dimensional surface can gauss law be used?
  26. Death eater

    Does an ideal fluid have zero surface tension?

    Does ideal fluid have zero surface tension? What does zero surface tension signify?
  27. bob012345

    I How to measure the surface area of an arbitrary 3D object

    I'm looking for an easy way to get the surface area of an arbitrary shaped 3D object. Getting the volume is easy by water displacement. What about area? Any neat tricks? We know different shapes can have the same volume and thus different surface areas so it's not a trivial problem. The purpose...
  28. digogalvao

    Slowly oscillating surface current on a solenoid

    Homework Statement From an original surface current ##\vec{K}=K\hat{\phi}## on a finite solenoid, I got ##\vec{B}=\mu_{0}Kf(z)\hat{k}##, for ##r<R##. Assuming that ##\vec{K}## now slowly oscillates in time such as: ##\vec{K(t)}=K_{0}\cos\left(\omega t\right)\hat{\phi}##, so that I still can use...
  29. S

    I What is the proof that the divergence is normal to the surface?

    If I am given a function f( x , y , z , ...) = C then the normal direction to it is simply the (unit vector of the) divergence of the function. How has this been proven?
  30. Dante Meira

    Move a mass on frictionless surface in a vacuum chamber

    How much energy is necessary to move one kilogram of mass horizontally for one meter on a perfectly frictionless surface inside a vacuum chamber? Assuming the initial velocity of the mass is zero, the mass is at rest.
  31. J

    I Which object will hit the surface of a planet first?

    If we have two objects A and B appear on the opposite sides of the equator of a planet like Earth with the same mass as Earth. Object A is a neutron star with the mass of the sun and object B is a iron cube with the mass of one gram. Will A or B hit the Earth at the same time or will one hit...
  32. Krushnaraj Pandya

    Height to which rolling ball rises on a surface

    1. The problem A ball moves without sliding on a horizontal surface. It ascends a curved track upto height h and returns. Value of h is h1 for sufficient rough curved track to avoid sliding and is h2 for smooth curved track, then how are h1 and h2 related (greater, lesser, equal or multiplied by...
  33. Krushnaraj Pandya

    Rolling on a plank which is resting on friction less surface

    Homework Statement A long plank of mass M rests upon a smooth horizontal surface. A thin circular ring (m, R) slips (without rotation) upon the plank with initial velocity v(i). The coefficient of friction between the wheel and the plank is C. at time t, the ring stops slipping and pure rolling...
  34. Felipe Lincoln

    Stokes' Theorem, how to apply for this surface?

    Homework Statement With the stokes' theorem transform the integral ## \iint_\sigma \vec{\nabla}\times\vec{F}\cdot\vec{\mathrm{d}S} ## into a line integral and calculate. ## \vec{F}(x,y,z) = y\hat{i} -x^2\hat{j} +5\hat{k}## ##\sigma(u,v) = (u, v, 1-u^2)## ## v\geq0##, ##u\geq0##...
  35. M

    Flux with an infinitely long surface cutting through a sphere

    Homework Statement Bildschirmfoto 2018-06-19 um 18.50.50.png In the y-z plane there is an infinite long surface with charge density ##\sigma## that slices through a sphere with radius R. Determine the Flux.The Attempt at a Solution I have solved the problem but am stuck at the last part. I used...
  36. joyxx

    Weight of person on Earth, and on a different planet's surface?

    Homework Statement A) What is the weight (in Newtons) of an 80 kg person at the Earth's surface? B) What is the weight of the same person on the surface of the planet with a mass 8.0 x 10^24 kg and a radius of 5.5 x 10^6 m? (G=6.67 x 10^-11 N*m^2/kg^2) C) what is the apparent weight (scale...
  37. E

    A Surface states of 3D topological insulators

    I have a question (more like a curiosity) related to three-dimensional topological insulators, which support Dirac-like states at their surfaces. From the theory, it is well known that these states are immune to scattering from non-magnetic impurities, i.e. impurities that do not break...
  38. srm

    Calculation of the weight of an insect floating by surface tension

    Homework Statement The surface of a liquid is just able to support the weight of a six-legged insect. The leg ends can be assumed to be spheres each of radius 3.2 × 10−5 m and the weight of the insect is distributed equally over the six legs. The coefficient of surface tension in this case is...
  39. CollinsArg

    I Surface area of a revolution, why is this wrong?

    Why is this way of thinking wrong?. can't I assume that when Δx tends to zero is a sufficient approximation of what I want to get? It confuses me with the basic idea of integrating a function to get the area beneath a curve of a function (which isn't also as perfect) . PD: I put Δx tends to...
  40. M

    I What is the Formula for Computing Surface Metric on a PF Surface?

    Hi PF! I'm trying to compute $$ \frac{1}{\sqrt g}\frac{\partial}{\partial u^\mu}\left( \sqrt g g^{\mu v} \frac{\partial \eta(s,\phi))}{\partial u^v} \right) $$ where I found $$ \sqrt g = \csc^2\alpha \sin s\\ g = \begin{bmatrix} \csc^2\alpha &0\\ 0 & \csc^2\alpha\sin^2 s \end{bmatrix}...
  41. Z

    I Calculating the Ricci tensor on the surface of a 3D sphere

    Hello, I'm trying to calculate Christoffel symbols on 2D surface of 3D sphere, the metric tensor is gij = diag {1/(1 − k*r2), r2}, where k is the curvature. I derived it using the formula for symbols of second kind, but I think I've made mistake somewhere. Then I would like to know which of the...
  42. M

    Effect of the surface roughness of a wing

    Homework Statement I am doing a report on the effect of the surface roughness of a wing on the wing's performance. This was done by testing a a wing design in a wind tunnel using a smooth, intermediate and then rough surface. The test involved varying the pitch angle and tunnel power setting...
  43. C

    Unit Normal to a level surface

    Homework Statement Given $$\phi = x^{2} +y^{2}-z^{2}-1 $$ Calculate the unit normal to level surface φ = 0 at the point r = (0,1,0) Homework Equations $$ \hat{\mathbf n} = \frac{∇\phi}{|\phi|}$$ $$ z = \sqrt{x^{2}+y^{2} -1} $$ $$ \mathbf n = (1,0,(\frac{\partial z}{\partial x})_{P})...
  44. R

    Electric Flux Through Spherical Surface Centered at Origin

    1. A uniform linear charge density of 4.0 nC/m is distributed along the entire x axis. Consider a spherical (radius = 5.0 cm) surface centered on the origin. Determine the electric flux through this surface. Homework Equations L=2rπ φ=Q/ε λ=Q/L The Attempt at a Solution I found the charge by...
  45. Telemachus

    Surface plot for a 3D function

    Hi there. I wanted to plot a surface, given implicitly by a function ##f(x,y,z)=c##, with ##c## a constant. My ##f(x,y,z)## is obtained numerically, so I don't know the expression explicitly, I have the values of ##f(x_i,y_j,z_k)## for given integer numbers ##(i,j,k)##, ##i=1,2,3...## same for...
  46. A

    Surface Tension of needle on water

    I attempted the question but it was wrong... I don't understand where i went wrong my working seems logical, can someone please help. When a needle is gently placed on the surface of still water (γ = 0.0730 N.m-1) it can be supported by surface tension if the mass of the needle is small enough...
  47. J

    Equipotential surface and electric field

    Homework Statement Homework EquationsThe Attempt at a Solution I know the relation between electric field and electric potential . I can also find Electric field if expression for potential is given and vica versa . But I do not know how to work with electric field and equipotential...
  48. J

    Ball floating on the surface of water

    Homework Statement Homework EquationsThe Attempt at a Solution When air is evacuated , atmospheric pressure pushing down the cork is eliminated , cork should rise a little i.e option b) . This is incorrect . What am I missing ?
  49. F

    Find the charge on the outer surface

    Homework Statement Homework Equations ##E=\frac{kQ}{r^2}## The Attempt at a Solution The answer is "A".Right ?
  50. F

    Find the charge on the conducting surface

    Homework Statement Homework Equations ##Q_{net}=Q_1+Q2## The Attempt at a Solution ##Q_{net}=Q_1+Q2## ##-15 = +5 + Q_2## ##Q_2=-15-5 = -20nC##
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