Surfaces Definition and 459 Threads

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. L

    Geodesic flows on compact surfaces

    Does a geodesic flow on a compact surface - compact 2 dimensional Riemannian manifold without boundary - always have a geodesic that is orthogonal to the flow?
  2. R

    Determining total charge on the surfaces of spherical conductors

    In the attached picture is all of the information to complete this problem. The picture is of a solid sphere at the center of a hallow sphere, both of which are conductors. The question asks to find the total charge of the exterior and interior surfaces of the hollow conductor, as well as the...
  3. M

    Why Isn't the Cylinder Slice Method Used for Surface Area of Revolution?

    Hello. I just started to study integral calculus not long ago and I have some confusion when it comes to calculating the areas of surfaces of revolution using integral. As from my testbook, when we want to calculate this kind of surface area, we often use the frustum surface area to approximate...
  4. S

    Surfaces of 3 Variable Question

    Homework Statement A cone C, with height I and radius I, has its base in the xz-plane and its vertex on the positive y-axis. Find a function g(x, y, z) such that C is part of the level surface g(x, y, z) = 0. Homework Equations What would be the formula for the cone such that the base of the...
  5. Z

    Numerical double integrals along discontinuous surfaces

    I posted this in the aerospace engineering forum but I think it may get more replies here: I've been trying to compute the bending-torsion coupling constants for a wing, B1, B2 and B3. The expression for this is \begin{bmatrix} B_1 \\ B_2 \\ B_3 \end{bmatrix} = \iint (y^2 +...
  6. M

    What is the angle between the normals to the surface at two given points?

    Homework Statement find the angle between the normals to the surface xy=z2 at the points (1,4,2) and (-3,-3,3) Homework Equations none The Attempt at a Solution del S = 2zy i + 2zx j and at the two points, del S = 16i+4j and del S = -18i-18j using the dot product, i got cos...
  7. Pengwuino

    Can You Turn Your BBQ Into a Flat-Top Grill?

    I was at this japanese tepenaki places or however you spell it where they make the food in front of you and my dad had an interesting though. The stations they have have these big flat cooking areas powered underneath by what I assume were a bunch of gas burners. We bought a new bbq and my...
  8. I

    Equipotential surfaces; finding the radius

    Homework Statement a metal sphere of radius 0.39 m carries a charge 0.55 μC. Equipotential surfaces are to be drawn for 100-V intervals outside of the sphere. Determine the radius of the first, tenth and 100th equipotential from the surface. Homework Equations V = kQ / r Volt =...
  9. L

    Dynamic Friction between surfaces

    Homework Statement We have a body V1, which is in contact over body V2 over a flat (i.e. planar) surface S. (The planar approximation can be considered a good one). Body V1 is moving, V2 is stationary. A normal force F exists between the bodies at surface S, acting in point P. This force...
  10. Pengwuino

    Calculating equipotential surfaces

    So this may be a bit silly, but one thing I've never really learned in all my years is how one actually goes about calculating equipotential surfaces for arbitrary potentials? Let's say I have a potential that goes like \Phi = A_0e^{-\left({{r}\over{r_0}}\right)^2}\cos^2 (\phi) \sin^2(\theta)...
  11. P

    Question about Stokes' Thm and Boundaries of Surfaces

    So I'm going over Rudin's chapter on differential forms in his Principles of Mathematical Analysis and I'm looking at Example 10.36 which gives the 1 form \eta = \frac{xdy-ydx}{x^2+y^2} on the set \mathbb{R}^2-{0} and then parametrizes the circle \gamma(t)=(rcos(t),rsin(t)) for fixed r>0 and...
  12. G

    Linear interpolation between two surfaces

    Hi, I'm trying to create an interpolated volume from two surfaces. Let me explain exactly what I'm doing. I am trying to obtain a rough estimate for the temperature in a certain geographical area and at depth. I have the temperature of the rocks at the surface in the form T1=T(x,y,z) where z...
  13. Z

    Parametric Surfaces and Their Areas

    Homework Statement Find the area of the part of the sphere x^2 + y^2 + z^2 = a^2(a > 0constant) that lies inside the cylinder x^2 + y^2 = ax. Homework Equations double integral of the cross product of the vector Ra and Rb with respect to dA. The Attempt at a Solution I tried to parametrize...
  14. F

    I am a lot of problems with parametrizing surfaces

    Homework Statement Evaluate the surface integral \iint_S xy \;dS S is the boundary is the boundary of the region enclosed by the cylinder x^2 + z^2 =1 and the planes y = 0 and x + y = 2 2. The Solutions [PLAIN]http://img844.imageshack.us/img844/8827/unledje.jpg The...
  15. N

    Electric Field around Conducting Surfaces

    Hey, Completing AP Problems, I ran into a puzzling answer. Basically, there was a conducting sphere with charge Q on it and radius R1, and then around it two hollow hemispheres were placed (forming a spherical capacitor) which had no charge, with Radius R2 at the inner edge and R3 at the...
  16. G

    Volume bounded by four surfaces. Need the solution as a guide.

    Homework Statement Find the volume bounded by the following surfaces: z = 2(x^{2}+y^{2}) z = 18 y = \frac{1}{\sqrt{x}} y = -\frac{1}{\sqrt{x}} x\geq0 The Attempt at a Solution I have no Idea how to attempt it! I mean, I will, somehow. But want to know a straight-forward way. Would...
  17. H

    Parametric Curve from the intersection of 2 surfaces

    Homework Statement Prove that the curve \vec{r}(t) = <cost,sint/sqrt(2), sint/sqrt(2)> is at the intersection of a sphere and two elliptic cylinders. Reparametrize the curve with respect to arc length measured from (0, 1/sqrt(2), 1/sqrt(2)) in the direction of increasing t. Homework Equations...
  18. J

    Sign Conventions for Paths and Surfaces for Electromagnetic Calculations

    Hello all. I'm trying to figure out how to determine the correct sign of paths and surfaces defined for calculating quantities in electromagnetic problems. For example, say there's a wire in the shape of a rectangular prism along the z-axis with some current density, \vec{J} . Then the...
  19. F

    EM waves and conducting surfaces

    Firstly, I'm a bit confused about EM wave propagation. Take the picture you see everywhere illustrating the perpendicularity of E and B in a traveling EM wave (like this http://web.onetel.net.uk/~gdsexyboy/em_wave.jpg) -- does that actually illustrate the magnitudes of E and B at a particular...
  20. W

    What Is the Shape of Equipotential Surfaces Near a Charged Disk?

    Homework Statement Find the equipotential surface at the edge of a uniform electric charged disk ? Homework Equations \nabla^2 V= - \displaystyle \frac{\rho}{\epsilon} \displaystyle V= \iiint_R \frac{ {\rho}r dr d\phi dz}{4 {\pi} {\epsilon}|\vec r -\vec r'|} The Attempt at a...
  21. K

    Volume of solid bounded by 2 surfaces

    Just working on some practice problems. I missed a couple classes due to sickness and just need some extra help. If you could walk me through how to do these types of problems that would be amazing. Homework Statement Evaluate the volume of the solid bounded by the surfaces (x2 + y2)1/2 =...
  22. I

    Electric Field of Infinite Plate: Why Use Cylindrical Gaussian Surface?

    I'm trying to show that the electric field for an infinitely large plate is E = \frac{\sigma}{2\epsilon_o} I was wondering why you can only use a cylindrical gaussian surface and not a cube? Thanks
  23. D

    Gaussian surfaces can someone help walk me through this problem?

    Gaussian surfaces...need help...can someone help walk me through this problem?? A solid insulating sphere of radius 5 cm carries a net positive charge of 2 μC, uniformly distributed throughout its volume. Concentric with this sphere is a conducting spherical shell with inner radius 10 cm and...
  24. F

    What is the volume enclosed by the parabolic cylinder and two planes?

    Homework Statement Find the volume of the solid enclosed by the parabolic cylinder y=10 - a2x2 and the planes z=y and z=2-y, where a > 0 is a constant.Homework Equations I have graphed the 3 surfaces on Maple to visualize the solid enclosed by these surfaces but the problem is there is no...
  25. P

    Rolling Motion on Rough Surfaces

    Friction is necessary to make motion rolling but if the body is already in rolling motion on rough surface without slipping then, is friction necessary to continue the rolling motion ?
  26. C

    Tangent points of two surfaces

    Homework Statement Determine if the surfaces x^3 + 6xy^2 + 2z^3 = 48 and xyz = 4\sqrt{2} have any points of tangency, and if so, find those points. Homework Equations The Attempt at a Solution I'm mostly wondering if anyone can find any problems with my approach. I assume that the geometry...
  27. M

    Equipotential surfaces for the given charge distribution

    Homework Statement Two infinitely long wires running parallel to the x-axis carry uniform charge densities \lambda and - \lambda. a.) Find the potential at any point (x, y, z) using the origin as your reference. b.) Show the equipotential surfaces are circular cylinders, and locate the...
  28. A

    Level Surfaces Problem- Calc III

    I have a couple of questions, but probably only need one worked out to figure out the rest. 1. Find a function f(x,y,z) whose level surface f=8 is the graph of the function 3x+4y => I know that a level surface for f(x,y,z) is the solution to f(x,y,z)=k. However, now I'm stuck. I know how...
  29. M

    Vector help Curves intersecting with Surfaces

    A curve in space is specied by the one parameter set of vectors x(t). Also given is a surface in space parameterised by x(u, v): x(t)= <2+t, -t, 1+3t2> x(u,v)=(u2 - v + u, u+5, v-2> A) Show that the curve intersects the surface in exactly two points. Show that xi = <4 - \frac{\sqrt{46}}{2}...
  30. P

    Drag coefficient for non-static surfaces + diagram

  31. B

    Need help with physics project involving force of friction of various surfaces.

    Hey I have to do a project in which I am going to roll a soccer ball across different surfaces such as turf, grass, concrete, etc... and calculate the force of friction of each of these surfaces. I believe that i need to find the Initial force of the ball, then the final and subtract to find the...
  32. M

    Divergence and surfaces integral, very hard

    Homework Statement A vector field h is described in cylindrical polar coordinates by ( h equation attached ) where i, j, and k are the unit vectors along the Cartesian axes and (er) is the unit vector (x/r) i+(y/r) j Calculate (1) by surface integral h through the closed surface...
  33. C

    Parametrized Surfaces: Evaluating and Integrating f(x,y,z) = yz with u and v

    Express f(x,y,z) = yz in terms of u and v and evaluate \int\int_S f(x,y,z)dS This is supposed to be simple but I really don't know how to do this. I rewrote f(x,y,z) = yz as x = g(y,z) so then \Phi(y,z) = (y,z, x) Tx = (0,0,1) and Ty=(1,0,0) and their corss product, n, is <0,0,-1> Am...
  34. J

    Surface area of the boundary enclosed by surfaces

    Homework Statement Find the area of the surface that is the boundary of the region enclosed by the surfaces x^{2}+y^{2}=9 and y+z=5 and z=0 Homework Equations A(S)=\int\int_{D}\left|r_{u}\times r_{v}\right| \; dA The Attempt at a Solution I am really confused as to what he...
  35. M

    Area vectors of oriented surfaces

    Are area vectors of oriented surfaces always perpendicular to the surface?
  36. J

    Identifying Surfaces in Spherical Coordinates

    Homework Statement \rho = sin\theta * sin\phi Homework Equations I know that \rho^{2} = x^{2} + y^{2}+z^{2} The Attempt at a Solution I tried converting it to cartesian coordinates but I can't seem to get a workable answer that way. I know that the answer is the sphere with radius...
  37. R

    Minimal surfaces and asymptotic curves

    Hi all. I'm wandering about the existence of closed asymptotic curves (curves having zero normal curvature everywhere) on minimal surfaces. I believe that the only minimal surface that admits closed asymptotic curves is the plane, but I cannot find a simple way to prove it. Here is where my...
  38. N

    Why Do We Minimize the Squared Length in Surface Distance Calculations?

    A little clarification is required for the following techniqueTHE TECHNIQUE Given a surface z = f(x,y), and some point Q in R3 (not on the surface) The point P on the surface for which the distance from P(x, y, f(x,y)) to Q is the shortest distance from the surface to Q (i.e. vector PQ has...
  39. C

    Mapping Surfaces: Show F is Differentiable

    Let M be a simple surface, that is, one that is the image of a single proper patch x: D -> R3. If y: D -> N is any mapping into a surface N, show that the function F: M -> N such that: F (x(u, v)) = y(u, v) for all (u, v) in D is a mapping of surfaces. To demonstrate this one does the...
  40. M

    Suggestion for a good book on Riemann Surfaces - your personal experiences

    Hello everyone - I'm a third year student at Cambridge university, and I've recently started taking a course on Riemann surfaces along with a number of other pure courses this year. The problem is, the lecturer of the course is of a rather sub-par standard - whilst I don't doubt he's probably...
  41. A

    Does friction in any way increase when the surfaces get very smooth

    does friction in any way increase when the surfaces get very smooth ...? a teacher told us that it can increase due to electromagnetism...is it true...i searched the net but couldn't get any useful info...
  42. H

    Finding a tangent vector to the intersection of two surfaces

    Homework Statement The surfaces S1 : z = x2 + y2 and S2 : x2 + y2 = 2x + 2y intersect at a curve gamma . Find a tangent vector to at the point (0, 2, 4). Homework Equations i thought about finding gradients of the two functions and plug in the given point in the gradients and cross...
  43. O

    Minimal Surfaces, Differential Geometry, and Partial Differential Equations

    Last night in a lecture my professor explained that some partial differential equations are used to observe events on minimal surface (e.g. membranes). A former advisor, someone that studied differential geometry, gave a brief summary of minimal surfaces but in a diffy G perspective. 1.)...
  44. J

    Parametrize the Right Hyperboloid x^2 + y^2 - z^2 = 1

    Homework Statement The part of the hyperboloid x^2 + y^2 - z^2 = 1 that lies to the right of the xz-plane The Attempt at a Solution Clearly, since it's demanded in turns of xz-plane, we have y = sqrt( z^2 - x^2 - 1) To parametrize it, we can simply use x = x, z = z, and y =...
  45. kandelabr

    Sealing with two hard, flat surfaces

    Imagine the following scenario: There is a hollow steel cube. I cut the top facet off so that I get a box with a lid and then polish both cut surfaces so that geometry fits almost perfectly (no gaps are larger than 1/100 of a milimeter or so - or even smaller). Then I close the box and...
  46. J

    How come light only reflects off surfaces and not internal planes?

    When light approaches and enters a material, why does all the reflection happen at the surfaces? I mean, light is coming at say, some crystalline solid. It hits the first plane of atoms and some is reflected. Some goes into the crystal and passes through the 2nd, 3rd, 4th,..., (n-1)th plane of...
  47. C

    Do things roll on frictionless surfaces?

    I was solving this question in rotational mechanics (not asking for help). The author then goes to solve this using energy conservation, equating the final kinetic and rotational energy to the initial potential energy. Now, he obviously assumed the object rolled on the surface, which, if...
  48. Q

    Multi-surface problem in refraction at spherical surfaces.

    Homework Statement A thick-walled spherical shell made of transparent glass(n=1.50) has internal radius R and external radius 2R. The central cavity and region surrounding the shell are filled with air (n=1.00). The center of symmetry of the object is located at the origin of an xy coordinate...
  49. M

    Identifying Quadric Surfaces: Memorization or Visualization?

    Hi there, I am learning about quadric surfaces in my second year multivariable calculus course. I would like to know how most people would identify (find the name of) a quadric surface if they had the equation. We only need to know 6 different quadric surfaces, so should I ... (a)...
  50. jegues

    Curve of intersection of surfaces

    Homework Statement See first figure attached Homework Equations The Attempt at a Solution I was able to sketch the two curves individually to get an idea of what I'm looking at, but I still can't really visualize how the two curves would intersect each other in the first octant...
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