# What is Surfaces: Definition and 458 Discussions

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. ### Equipotential surfaces for an electric dipole

The answer is given as (a), but I think it's not correct based on the equipotential surfaces diagram given in our book for an electric dipole as below. The red dashed lines, which are supposed to be the equipotential surfaces, are surely not representing a sphere centred at the dipole center...
2. ### Engineering COMSOL plot of thermal radiation from heated surfaces -- Help please

There is heating of the surface of the material using an electron beam. It is necessary to calculate how much heat will be released and build a graph of dependence. Please tell me how this can be done, which modules in COMSOL can be used?Thank you!
3. ### My Mistake? Understanding Friction Force & Work Done on Snowy & Icy Surfaces

The answer is (D), but I don't understand why. Option (A) is wrong because the work done = 0. Then, I divide the motion into 3 parts: 1) motion on snowy surface Since the sledge is being pulled horizontally (let assume to the right), there will be tension force T to the right and friction...
4. ### A complicated problem of motion on rough surfaces

I couldn't draw the motion after the collision, since the whole angular displacement of the plane got me confused.
5. ### MNCP deleting input surfaces

Hello Tried to model gas cooled reactor with hexagonal fuel elements. MCNP keep deleting surfaces (If you could, run my input and check the .txto file) so the simulations won't run Any advice?
6. ### B Oriented Surfaces & Surface Area: Investigating the Impact

I usually think of a sphere as the set of all points ##P_x##, that have the identical distance r to some point ##C## which is the center of the sphere. I calculate the surface area ##A## of the sphere as $$A=4 \pi (C P_x)^2$$ However, what happens if I think of the distance between the points C...
7. ### Quadratic Surfaces: Substitute (a,b,c) into z=y^2-x^2

Substitute (a,b,c) into z=y^2-x^2: c=b^2-a^2 Substitute the parametric equations of L1 into the equation of the hyperbolic paraboloid in order to find points of intersection: z=y^2-x^2 c+2(b-a)t=(b+t)^2-(a+t)^2=b^2-a^2+(b-a)t c=b^2-a^2
8. ### Use Stokes' theorem on intersection of two surfaces

I parameterize surface A as: $$A = (2cos t, 0, 2sin t), t: 0 \rightarrow 2pi$$ Then I get y from surface B: $$y = 2 - x = 2 - 2cos t$$ $$r(t) = (2cost t, 2 - 2cos t, 2sin t)$$ Now I'm asked to integral over the surface, not solve the line integral. So I create a new function to cover the...
9. ### I Do volume integrals involve bounding surfaces?

In Vanderlinde page 171-172, the author derives the vector potential for the magnetic dipole (and free currents) \begin{align} \vec{A}(\vec{r}) &=\frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{J}\left(\vec{r}^{\prime}\right) d^{3}...
10. ### I Vapour pressure/solubility of small and big grain surfaces and corners

When a condensed phase - solid or liquid - is in an immiscible fluid (gas or liquid), it has surface energy. Several small pieces of condensed phase have bigger combined surface than one bigger piece of the phase of the same volume, and thus bigger energy. In case of liquid, the surface of a...
11. ### Σ free on two dielectric spherical surfaces

I have found the total dipole moment of for this problem but am having trouble finding the electric field. I believe my electric field when r>2R ( I mistakenly wrote it as r<2R on my work, but it is the E with a coefficient of 2/3) is correct as it fits the equation: . I don't believe this...
12. ### Create curve function from intersection of two surfaces

What I do is set the two equations equal to one another and solve for z. This gives: $$z = \sqrt{x^2+2y^2-4x}$$ which is a surface and not a curve. What am I doing wrong?
13. ### B How to Visualize 3D Surfaces for a PVT Diagram

1. Fold in half the long way 2. Fold in half the short way 3. Unfold paper, grasping the left side horizontal crease, bend the paper as shown by making the left crease flush with the bottom vertical crease and crease along the -45 degree bend (not shown) 4. This is the visualization for...
14. ### I Global simultaneity surfaces - how to adjust proper time?

Hi, searching on PF I found this old post Global simultaneity surfaces. I read the book "General Relativity for Mathematicians"- Sachs and Wu section 2.3 - Reference frames (see the page attached). They define a congruence of worldlines as 'proper time synchronizable' iff there exist a...
15. ### I Surfaces wrapped around a cylinder for 3d printing

Hi All, I am looking to determine how these Vases where modeled using maths on this webpage https://www.3dforprint.com/3dmodel/sine-wave-vase-generator/2116. It looks like the surface is parametrically defined and wrapped around a cylinder. Interestingly he mentions "Sine waves combine to...

17. ### I Passive radiative cooling of surfaces below ambient air temperature

Apparently, it's possible to cool surfaces below ambient air temperature by passive (no input of energy required) radiative cooling to harvest water from the atmosphere: https://www.science.org/doi/10.1126/sciadv.abf3978 There's also a Nature paper about this effect, but behind a paywall...
18. ### A Representing flux tubes as a pair of level surfaces in R^3

I am trying to see if Vector fields(I am thinking of electric and magnetic fields) without sources(divergence less) can be represented by a pair of functions f and g such that the level surfaces of the functions represent flux lines. I am trying to solve this problem in ## R^3 ## with a...
19. ### Faraday's law -- How is the RHS required for all surfaces?

∫c Edl =-d/dt∫sBda
20. ### Transparent surfaces that become opaque

Are there any materials that go from transparent to opaque when a bright light is shined on it? In particular I would like something that acts like a window until I project a movie onto it and then is opaque in the regions where the light hits it at certain levels of brightness.
21. T

### B Ways to abstract curves or surfaces

Hello. So, we have curves and surfaces. We already know about generally manifolds and Riemannian manifolds but what i want to produce are ways to abstract curves or surfaces but i am not talking about manifolds. Do you have any ideas? Perhaps the feature of curvature would help? To make an...
22. ### Engineering Dynamics help: Block on a wedge on an incline (frictionless surfaces)

So I’m having trouble with relative motion with moving inclines and I literally can’t find any help online and my prof does a lot of these problems. This is one of my homework problems, can anyone help me with it please.
23. ### Confusion when dealing with loops and surfaces with Maxwell equations

DISCLAIMER: in Italy, we talk about "circuitazione" of a field through a closed loop ##\gamma##, for the physical quantity $$\Gamma_\gamma(\overrightarrow{E}) = \sum_{k}\overrightarrow{\Delta l_k} \cdot \overrightarrow{E_k}$$ but after some research, I haven't managed to find the correspondent...
24. ### Intersection of a few surfaces

Summary:: Describe what the intersection of the following surfaces - one on one - would look like? Cone, sphere and plane. My answers : (1) A cone intersects a sphere forming a circle. (2) A sphere intersects a plane forming a circle. (3) A plane intersects a cone forming (a pair of?)...
25. ### Applying desiccant to surfaces (aluminum sheet or PLA/PETG)

I will either be using aluminum strips wound up, or 3d printing it out of PLA or PETG. How can I apply a coating of silica to the surface of either of these materials so that it can hold and release the moisture of the incoming air/outgoing air? This is to manage heat and humidity in my...
26. ### Typical surface roughness heights for aircraft surfaces

Hello! I am investigating some preliminary aerodynamics on a regional turboprop and for the drag model the surface roughness height of the airplane is required as an input. For this i do not have any data. In which range are the typical values for modern aircraft surfaces? Thanks a lot!
27. ### Pressure force on curved surfaces: vertical component

hi guys, i can't understand why they calculate F yin this way, the part of floor that is the vertical proyection has less water than the floor in the left so i tought Fy would be less, please can someone explain this concept to me?
28. ### I A Zorich proposition about local charts of smooth surfaces

From Zorich, Mathematical Analysis II, 1st ed., pag.163: where the referred mapping (12.1) is a map ##\varphi:I_k\to U_S(x_0)##, in which: 1. ##I_k\subset\mathbb{R}^n## is the k-dimensional unit cube, 2. ##x_0## is a generic point on the surface ##S## and ##U_S(x_0)## is a neighborhood of...
29. ### Appliances Non-stick sprays harm non-stick cooking surfaces?

Are there exceptions to the generality that non-stick cooking sprays (such as "Pam") harm the non-stick surfaces of cooking appliances such as electric skillets? Online, I find this stated as a generality (e.g...
30. ### Particles on Deforming Surfaces: Theory & Analysis

All books in analytical mechanics explain the case of a particle moving on a given static surface. But what happen if, for example, the surface is having some deformation?. I imagine that the principle of virtual work, and hence, D'Alembert are no longer valid since the normal force by the...
31. ### Why isn't copper used for surfaces in hospitals?

If copper kills corona how come we've not spend money on putting it in hospital's etc ? How effective is it would silver be more effective?Our a combination of the both copper disc and a silver disc ?
32. ### I Gradient vectors and level surfaces

Homework Statement:: Wondering about the relationship between gradient vectors, level surfaces and tangent planes Relevant Equations:: . I know that the gradient vector is orthogonal to the level surface at some point p, but is the gradient vector also orthogonal to the tangent vector at that...
33. ### I Do there exist surfaces whose boundary is a closed knot?

I ask this for the condition of application of Stoke's theorem.
34. ### B Why don't solids always stick together?

Hi all, Something I've been wondering - why don't two solid surfaces always stick together when touching each other? As far as I'm aware there are five basic types of solids: Atomic solids: Frozen noble gases containing single atoms held together by London dispersion forces. Molecular...
35. ### Find the Electric potential from surfaces with uniform charge density

I do not have the solutions to this problem so I'm wondering if my attempt is correct. My attempt at solution: We have two surfaces which we can calculate the area of. I think we can use gauss law to find the electric field and then integrate the E-field to find the electric potential. So for...
36. ### Flipping Fidget Spinners on Sloped Surfaces

Water Bottle Flipping Fidget Spinner on Sloped Surface
37. ### Horizontal force on glued surfaces

I am conducting tensile test on a bigger scale. When the material breaks i have the maximum kilograms that can be a applied to the material. I pull horizontal in a rubber patches glued to a rubber sheet and i want to find the maximum tensile strength in the glue. How do i convert my 2000 kg pull...
38. ### I Penrose Singularity: Trapped & Anti-Trapped Surfaces Explained

As I understand Penrose proved there must a be a singularity using certain assumption for black holes and something called a "trapped surface:. Can anyone give a lay person explanation of what this and "anti trapped surface" are? How were they used in the singularity theorems and what was new...
39. ### I Derived metrics on surfaces of positive curvature

Start with a closed surface of positive Gauss curvature embedded smoothly in ##R^3##. At each point, choose two independent eigenvectors of the shape operator whose lengths are the corresponding principal curvatures. By declaring them to be orthonormal one gets - I think - a new metric on the...
40. ### Engineering Determine the capacitance between two surfaces

For my solution I'm skipping writing out all the vectors, I just want to see if I'm in the right way or totally off. Attempt at solution: Qenc = ∫ E(r)*e0 ds = ∫ E(r)*e0 *h* r*dtheta, we integrate from 0 to phi0. This will give us Q = E(r)*e0*h*r*phi0. Now we find V by integrating E from a to...
41. ### Which surfaces get positively charge by grabbing (static electricity)

So is it becouse the material or becouse the fact that the balloon is the object that moves and the hair is static. and does every two objects that been grabed together will nacessrly continues each other. and also why does the minos of a bttary doesn't stick to the flower
42. ### Finding domains of 3d quadratic surfaces

##z^2\leq x^2+y^2, z\geq x^2+y^2## I know the shapes of those inequalities, but the question is: How do i find if the point are external the shape or internal?
43. ### I When two surfaces rub against each other, do they lose atoms?

When two surfaces of any kind rub against each other which happens everywhere every moment, do they lose atoms/particles from the surface theoretically ? Is it what we understand as "wear"? However, looks like in our daily life, something doesn't seem to wear over time (even after many years)...
44. ### Work of a object moving across surfaces with different friction

Homework Statement: Hetsut is the foreman of a construction project in ancient Egypt. He needs to move a giant block of stone, of mass 12 metric tons, from the docks to the temple grounds. He can go along the roads by traveling 295 meters east, then 89 meters north. Along the roads, the...
45. ### Do surfaces ahead of propellers decrease thrust?

Hi! I have had this question for ages, nearly impossible to find anything on the web. My experiments confused me even further! Its concerning the blockage effects of surfaces ahead of a propeller. For example let's consider the usual dual vectoring propellers on the sides of airshipcars like...
46. ### Static Analysis of Bolts on Multiple Surfaces

I am trying to analyze if the bolts in my design are sufficient to support the load on a shaft. Unfortunately, the bolts are not all located on one face, and though I know how to analyze each face by itself, but I would like to be able to combine them all into one problem. I have attached a...
47. ### Trying to better understand sound & vibration on textured surfaces

Hi, First thank you in advance for reading and answering my question. I am very much a newbie when it comes to physics, so while I feel my question in pretty straight forward I'm sure it's more complex than my thinking. OK so here it is... Does the texture of two surfaces change the decibel...
48. ### Calculate combined friction on multiple different surfaces?

Hello, I have a cuboid on two different surfaces. I am applying a changeable force F to it, but it should stay on the same point / in static friction. I now want to calculate the combined (static) friction coefficient μ_combined out of the two known coefficients of cuboid to surface 1 μ_1 and...
49. ### Gradient & Smooth Surfaces: Implicit Function Theorem

Section ##3.8## talks about the gradient and smooth surfaces, defining when the directional derivative ##(\partial f/\partial\mathbf{u})(\mathbf{p})## takes maximum value and that when it equals ##0##, then ##\mathbf{u}## is a unit vector orthogonal to ##(grad\ f)(\mathbf{p})##.It also says that...
50. ### 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...