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. Addez123

    Unsolvable max/min of surface with constraint

    Trying first with lagrange multiplier ##grad(f) = (2x + y, 2y + x)## ##g = x^4 + y^4 -8 = 0## ##grad(g) = (4x^3, 4y^3)## $$grad(f) = \lambda grad(g)$$ gives us 2 equations (1) ##2x + y = \lambda4x^3## (2) ##2y + x = \lambda4y^3## From (1) we get ##y = \lambda4x^3 - 2x## insert that into (2) and...
  2. greg_rack

    I Exploring Acceleration at Contact Point Between Wheel and Surface

    Hello guys, I am getting more and more confused each time I try to get a definitive answer on this doubt: what's the acceleration at the contact point between a surface and a wheel spinning on it(without slipping). Considering this standard FBD for the above-described situation, (the direction...
  3. Addez123

    Another max value on surface (no boundaries)

    I've double-checked my equations and can't find what's wrong. First I calculate the partials: $$f_x = (2x - 2x^3 -4xy^2)e^{-x^2, -y^2}$$ $$f_y = (-2yx^2 + 4y - 4y^3)e^{-x^2, -y^2}$$ By setting f_x = 0 I get: $$x^2 = 1 - 2y^2$$ Then I calculate f_y = 0 $$-2yx^2 + 4y - 4y^3 = 0$$ I plug in the...
  4. Addez123

    Can't find the correct max on this surface

    I calculate f_x to be $$f_x = (2x - x^2)e^{-x-y}$$ $$f_y = (1 - y)e^{-x-y}$$ f_x = 0 when x_1 = 0, x_2 = 2 f_y = 0 when y = 1 This gives two critical points: (0, 1) and (2, 1) which yields e^-1 and 5e^-3 respectively. I then check the x line and y lines by doing $$f(x, 0) = x^2e^{-x-y}$$ $$f_x...
  5. docnet

    Engineering Can you find a surface from a metric?

    if a metric like ##ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2 ## is given, we know it corresponds to a sphere in spherical coordinates . if you are given an arbitrary metric with two variables for example ##ds^2=\frac{du^2}{u}+dv^2## is ther guarenteed to be a surface embedded in ##R^3##...
  6. A

    Dependence of the stress vector on surface orientation

    According to Cauchy's stress theorem, the stress vector ##\mathbf{T}^{(\mathbf{n})}## at any point P in a continuum medium associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.e., in terms of...
  7. pasta-lord

    Effect of Surface Area on the Drag Coefficient of a Parachute

    Summary:: Does the surface area of a parachute affect its drag coefficient? If so, how? I have been trying to figure out the effect of surface area on the drag coefficient of a parachute. I have designed a lab in which parachutes of different surface areas are dropped and the terminal velocity...
  8. H

    A Surface waves and vorticity in 2D

    The classical free surface profile for the solitary wave for irrotational and incompressible fluids for small amplitude and long wavelength is the classical Korteweg-deVries(KdV) equation given by:\frac{\partial\eta}{\partial t}+\frac{\partial \eta}{\partial x}+\eta\frac{\partial\eta}{\partial...
  9. E

    I Help with FLP argument of non-uniformly distributed surface charges

    Hello, I'm reading FLP vol II, and I would appreciate some help to understand the argument supporting Figure 6-6. Basically they claim if a sphere has non-uniform charge distribution whose surface density is proportional to the cosine of polar angle, then this surface charge distribution is...
  10. Addez123

    Find all points where surface normal is perpendicular to plane

    a. I solved a but I don't fully understand how it works. $$z = f_x'(1, -1)(x -1) + f_y'(1, -1)(y+1) = 2(x-1) + 3(y+1)$$ Eitherway it's b that's my issue. I can find the gradient of both plane and surface, but trying to do "dot-product of both normals = 1" will give an equation involving two...
  11. patric44

    Checking if a function is an equipotential surface

    hi guys I came across that theorem that could be used to check if a surface represented by the function f(x,y,z) = λ could represent an equipotential surface or not, and it states that if this condition holds: $$\frac{\nabla^{2}\;f}{|\vec{\nabla\;f}|^{2}} = \phi(\lambda)$$ then f(x,y,z) could...
  12. dRic2

    I Stokes' theorem and surface integrals

    Hi, So my goal is to compute the integral of the "curl" of the vector field ##A_i(x_i)## over a 2-dimensional surface. Following a physics book that I am reading, I introduce the antisymmetric 2-nd rank tensor ##\Omega_{ij}##, defined as: $$\Omega_{ij} = \frac {\partial A_i}{\partial x_j} -...
  13. Dario56

    I How Does Surface Tension Balance Small Objects on Water Surface?

    When small object such as needle is put on the surface of water it displaces small amount of water which creates a depression under the object. Such depression increases surface area of the water because of which surface tension tends to decrease it. Explanation why surface tension balances the...
  14. Dario56

    I Du Nouy Method for Measurement of Surface Tension

    This method calculates surface tension based on force balance acting on the ring placed on the liquid surface: $$ F = G + \gamma L $$ where ##G## is weight of the ring, ##L## is wetted length of the ring which is equal to its circumference, ##F## is outer force of tensiometer acting on the ring...
  15. Dario56

    Owens - Wendt Model for Surface Energy of Solid - Liquid Interface

    Owens - Wendt model is used for calculating surface energy on liquid - solid interface and it is given by following equation: $$ \gamma_{sl} = \gamma_s + \gamma_l -2(\sqrt {\gamma_l^d \gamma_s^d} + \sqrt {\gamma_l^p \gamma_s^p}) $$ So, if we use liquid and solid of known surface energy as well...
  16. T

    B Delta-v from Lagrange points to lunar surface?

    What is the delta-v requirements from each of the Earth-Moon lagrange points to landing on the lunar surface? What would be the best software I could use to visualise and calculate that kind of thing? Thanks.
  17. V

    Volume density vs Surface density of charge distribution

    This doubt is confusing to me. I know it's something to do with conductors and insulators, but cannot explain. Conductors have mobile/free electrons unlike insulators. Having free electrons doesn't seem to explain this difference of charge distributions.
  18. BillTre

    How surface water flows in the US

    Found this map website. Click on the map somewhere, and it will figure out the path surface water, from that point, would take, as it flows to the sea (or where-ever). It also makes a 3D fly over if you click the path. It doesn't show drainage basins, which would have been nice, but there...
  19. Amaterasu21

    B Charged Particle on Earth's Surface: Will It Emit Radiation?

    General relativity tells us that an object in free-fall is actually inertial, following a geodesic through curved spacetime, and not accelerating. Instead, it's objects like us, on the surface of a large body, that are accelerating upwards. Maxwell's equations also tell us that accelerated...
  20. P

    A Condition for a spacelike surface to be achronal

    A hypersurface being spacelike (a local condition - every tangent to the surface being spacelike) does not preclude that points on it cannot be causally connected (one is in the future or past light cone of the other). A classic example is a spacelike spiral surface. Typically, for foliating a...
  21. S

    Gato / Balao Submarine "Can't" vs "Must" Surface on Dead Battery

    I've read a lot about Gato / Balao / Tensch class submarines, the ones America used in WWII, and I can't seem to sort out the specific consequences of a dead battery. A lot of you are diesel experts, so maybe someone here knows? Historical accounts are vague. Dead batteries are certainly a big...
  22. Shreya

    What is the Young-Laplace equation and its relationship to surface tension?

  23. V

    Electric Field on the surface of charged conducting spherical shell

    When I look at the relevant equations, then there is no mention of field for a point on the surface of the shell, so it gets confusing. On the other hand, I feel the radial E will get stronger as we approach the surface of shell and magnitude of E will approach infinity.
  24. G

    Plug the Closed Ends of a Pipe at Depth and Bring it to the Surface

    Hi Guys - I have very simple question but I cannot get my head around. Say if we cut pipeline into one section (12m) then close each cut end with temporary plugs Water depth is 100m. Before the cut there is seawater inside the pipeline. Now I understand there is no differentiated pressure...
  25. A

    MHB Calculate Volume & Surface Area of a Cylinder Without Lid

    how to find volume and surface area of this without using the upper lid
  26. pairofstrings

    B Equation of Circle: Boundary or Surface?

    Equation of circle: ##x ^2+y ^2=1##. Is this equation describing boundary or a surface? Thanks.
  27. K

    What is the surface of fridge door made from?

    I think most refrigerators have their cases and doors attracting to magnet, but their surface look like plastic plate. what are they actually made of?
  28. B

    From weatherstation data to solar irradiation on specific surface

    So, at home I have a weatherstation which measures solar irradiation in the east, south and west in lux. I can read in these measured sensordata. From these measurements, I would like to calculate the solar irradiance on my windows. The solar energy formulas could help me with that. The work of...
  29. M

    Bearing (surface) pressure versus load rating

    Hi, From suppliers of bearings, some of them specify the maximum bearing (surface) pressure or the maximum load rating (static or dynamic). What's the difference between this two? And which is most commonly used during selecting and calculating bearings in a mechanical design? Thanks in advance,
  30. cwill53

    Capacitor and Surface Charge Density Question

    When I plug in the numbers I get ##2.9513\cdot 10^{-5}C/m^2##, not ##17.6\cdot 10^{-6} C/m^2##. Can someone point out where I'm going wrong?
  31. M

    MHB Points of the surface with minimum distance to the point (3,0,0)

    Hey! :giggle: We have the function $$f(x,y)=(x-3)^2+y^2+(x-y)^2$$ and I have shown that at $(2,1)$ we have a minimum and so $f(2,1)\leq f(x,y)$ for all $(x,y\in \mathbb{R}^2$. I did that in this way: I calculated the gradient and set this equal to zero and found that the only critical point...
  32. G

    Optics question: Looking at mirrors from parallel to the surface

    I did a little experiment recently where I took a plane mirror and held it underneath a ceiling light. Then, I began to lower my head so that my view was closer and closer to the surface. When I did this, the image of the light began to drift lower and lower in the mirror until it completely...
  33. P

    Problem on Pressure due to Surface tension

    The method to solving this is to equate forces along a portion of the balloon through which ##\sigma_L## acts, and another portion through which ##\sigma_t## acts. The former potion should be a circular cross section of the cylinder, while the latter will be a rectangular cross section. You will...
  34. P

    Good resources for learning basic surface tension

    Summary:: Hi, I realized that surface tension is not covered in introductory physics textbooks. Where can I get a good introduction on surface tension? *info provided should be about the same depth as topics in Halliday, Young. Hi, I realized that surface tension is not covered in...
  35. LuccaP4

    Solid-state Physics: Fermi surface and necks in an FCC structure

    Does anyone have some bibliography about necks in FCC structure Fermi surface? I have to solve this problem and I have no idea how to start. Thanks.
  36. V

    Find the net outward flux through the surface and the charge density

  37. Drakkith

    B Mass and Surface Gravity of a Dyson Sphere

    I'm re-watching Star Trek TNG and I just started the episode where they encounter Scotty aboard a ship that's crashed into a Dyson sphere. That got me thinking. What would the mass and external surface gravity of a Dyson Sphere be? I've done the math myself, but I'd appreciate someone double...
  38. T

    What is the PV panel surface area?

    Hello everyone, I am trying to do some calculations for the energy output of a solar farm that I am designing as my dissertation. However, when I trie to calculate the following formula: Wp = ηpvGBA from Equation (11) above, where: ηpv is module efficiency (18.4%) GB is solar irradiance (3.8)...
  39. J

    I Is the Surface of a Sphere Locally Flat?

    Given a certain manifold in ##R^3## I've been told that at every location ##p## it is possible to encounter a reference frame from which the metric is the euclidean at zero order from that point and its first correction is of second order. This, nevertheless does not match with the following...
  40. A

    How do surface piercing propellers work?

    We have a new boat, and it has surface piercing propellers. Half of the prop is above (out of) the water, and half is below. I can't quite understand how that actually works or why it would be better.
  41. P

    Do charges always lie on the inner surface of capacitors?

    Parallel plates A, B are 5mm apart, with charges +1C and -1C respectively. Parallel plates C, D are 2mm apart, with charges +1C and -1C respectively. Capacitor CD is slid between capacitor AB. Find the potential difference between AB. The key idea to solving this problem is to suppose that +1C...
  42. AN630078

    Rates of change: surface area and volume of a sphere

    The surface area of the sphere is 4πr^2. dr/dt is given as 3cm^-1. dS/dt=dS/dr*dr/dt Differentiating 4πr^2 is dS/dr= 8πr dS/dt=8πr*3 dS/dt=24πr Given that r=5 dS/dt=24π*5=120 π The volume of the sphere is 4/3πr^3, differentiating which is dV/dr=4πr^2 dV/dt=dV/dr*dr/dt dV/dt= 4πr^2*3...
  43. neilparker62

    Ripple Pattern on Martian surface

    See graphic below. Are these ripples just wind-formed sand dunes or the remnant of an ancient (possibly catastrophic proportion) flood event? The ripples appear to emanate from a structure to the left if you look earlier in the descent video.
  44. A

    Calculating the surface tension using the method of Lecomt de Nouy

    Hello! I am having unexpected difficulties with this problem.A friend showed me an article regarding the Lecomde ring method,and at the bottom was this little problem that to solve.I've done it like this. $$ y = \frac{F}{4\pi r} $$ where F is the given force and r is = 0,0325 in meters. The...
  45. A

    Why does the top surface of a Silicon Die measure close to 0 ohms?

    I have gotten access to a large bare silicon die (almost 1" across, 14nm process) that my company gets from a fab. I've been monkeying around with an ohmmeter placed at the top of the die, with the test leads placed at various points around the die, and I almost always measure the same value...
  46. M

    Build a surface normal vector (I use Mathematica)

    Not HW, but seems to fit here. I compute $$n.S = \frac{(-1+\cos(c s))}{c^2} \sin(c s) \neq 0$$ I use the following in Mathematica: r[s_, \[Alpha]_] := Sin[Cos[\[Alpha]] s]/Cos[\[Alpha]] z[s_, \[Alpha]_] := (1 - Cos[Cos[\[Alpha]] s])/Cos[\[Alpha]] x[s_, \[CurlyPhi]_, \[Alpha]_] := r[s...
  47. F

    Whether charges can stay inside wires or on the surface of a conductor

    There's an isolated capacitor initially carrying nonzero net charge. Then place the capacitor in a circuit connected with ideal wires (no resistance). Where will the excess charge go? Can they stay in wires or on any surfaces of conductors in the circuit? Electric field needs to be 0 everywhere...
  48. H

    Oscillation of a particle on a parabolic surface [equation of motion]

    Hi, I have a particle on a parabolic surface $$y = Ax^2$$ and I have to show that the frequency is $$\omega = \sqrt{2Ag}$$ I don't know how to deal with a parabola. I don't think I can use the polar coordinates like a circle. I don't see how to start this problem and in which coordinates...
  49. J

    Stainless Steel track surface grade to match a 440C roller?

    I have a machine I am designing that has a carriage supported by Track rollers on a machined flatbar track. The load that each individual roller will experience could reach 2000lbs. worst case scenario. The track surface is machined 304L - 1" wide as you can see in the green in the attached...
  50. S

    B Darkest Place on the Surface of the Earth

    Hello All On mid-winter's day, where is the darkest place in the northern hemisphere, and at what time? I'm assuming no moon, no northern lights, and no local light pollution. Would it be at the North Pole? best regards ... Stef
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