# surfaces

1. ### Calculus derivation of quadric surfaces

Can somebody provide a derivation of the general equations for quadric surfaces, in particular that of a hyperboloid : x^2/a^2 + y^2/b^2 - z^2/c^2 = 1 ? Thanks!
2. ### Find the volume of the solid bounded by the surfaces

(x^2 + y^2 + y)^2 = x^2 + y^2 x + y + z = 3 z = 0 Any help with this problem would be very nice. I have no idea where to start and would appreciate even the slightest hints and nudges. I'm new to the community and I hope I'm not breaking any rules. Thanks all. :)
3. ### curves, surfaces and tangent lines

Could someone please assist with the following questions: Consider $f(x,y) = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and take $C$ to be the curve of intersection of $z = f(x,y)$ with the plane $y=x$. Show that the curve $C$ has a tangent line at the origin? I have tried showing that the directional...
4. ### simple closed curve is nullhomologous iff is separable

A simple closed curve $\gamma$ in an orientable genus $g$ surface $M$ is nullhomologous if and only if $M \setminus \gamma$ consists of two connected components, one of which is a surface $N$ with $\partial N = \gamma$. Could anybody prove this or show a book where it's clearly explained? P.S...
5. ### use triple integral to find volume of solid enclosed between the surfaces

surfaces x=y^2+z^2 and x = 1-y^2. I drew them below: I know I need to find the region projected on the xy plane and integrate over that area, as well as integrate the height on the z-axis; however I am not sure how to proceed. I think I need some equations, possibly setting the two equal to...
6. ### Surface-knots (knotted surfaces)

Hello I am doing research in surface-knots (Embedded surfaces in 4-dimensional Euclidean space. If there is someone interested in this topic I will be very glad to contact him and share ideas. Thank you in advance
7. ### surfaces and solids

Hello people, I am finding it hard to recognize the difference between paraboloids,cylinders,cones and etc..(any 3d surfac) How do I tell the difference between them? for example y^2+4z^2=9 where does this curve open up..how do i tell? I know I am asking stupid questions .. thanks
8. ### Spheres surfaces maximum

The distance between centers of two spheres of radii 4cm and 9cm, respectively, is 35cm. How far from the center of the smaller sphere is a point P along the line of centers of the spheres from which the sum of the areas of the visible spherical surfaces is maximum? Any kind of help would be...
9. ### Curve of intersection with 2 surfaces

Show that the curve with vector equation r(t)=<2cos^{2} t, sin(2t), 2sint> is the curve of intersection of the surfaces and . Attempt: (1) (x-1)^2+y^2=1 y^2 = 1-(x-1)^2 (2) x^2+y^2+z^2=4 plug (1) into (2) for y x^2+1 -(x-1)^2+z^2=4 Expanding and canceling...
10. ### Curve of intersection of the surfaces

Show that the curve with vector equation is the curve of intersection of the surfaces and . Use this fact to sketch the curve. I am completely lost. Would i take z = 2sin(t) and plug it into x^2 + y^2 + z^2 = 4 x^2 + y^2 + (2sin(t))^2 = 4 then x^2 + y^2 + (4sin^2(t)) = 4 I...
11. ### compact surfaces question

Let M be a closed, orientable, and bounded surface in R3 a) Prove that the Gauss map on M is surjective b) Let K+(p)= max{0, K(p)} Show that the integral over the surface M, ∫ K+ dA ≥ 4π. Do not use the Gauss Bonnet theorem to prove this
12. ### line of curvature is developable

Let β be a curve on M. Show that β is a line of curvature on M iff the surface defined by y(u,v)= β(u)+vU(u) is developable. Here U(u) denotes the unit normal to M along β
13. ### unknown Gaussian surfaces?

wrinkled sphere Gaussian surface? I invented this problem you don't have to do it, take 2 wrinkled spheres of opposite charge rhocharge2 and equal radius, volumerho1 of the spheres=1+\frac{1}{2}sin(m\pi) sin(n\phi), m=6, n=5 the spheres overlap in a region and the vector between the centers...
14. ### Find the curve, C, for a line integral

Solving line integrals isn't a problem for me, but, for this question, I'm having trouble finding the parametrized curve. I did draw the cone and plane, but I still can't figure out how to find the curve, r(t). Any help on the matter will be greatly appreciated, Giest
15. ### Leibniz Rule for Directional Derivatives

Show the following Leibniz Rule for directional derivatives holds. Given v ∈ T_p(M) we have v[fg]=v[f]g(p)+f(p)v[g]
16. ### inverse stereographic projection map

Consider the sphere S= {(x,y,z)} ∈ R^3 | x² + y²+ z²=1 }. Let N=(0,0,1) be the north pole of the sphere. The inverse stereographic projection map s is a homomorphism s:R^2 s →S-N defined by mapping the point (x,y) ∈ R^2 to the point on S that lies on the line connection (x,y,0) to N in R^3...
17. ### compact subsurfaces of bordered surfaces of infinite genus

Hello, I am a new user in this helpful forum and I start by posting a question that has been troubling me for quite some time now: I want to find a proof for the following proposition: "A bordered surface of infinite genus contains compact subsurfaces of arbitrarily large genus" A few...
18. ### Show that two surfaces intersect orthogonally at a point P

I was given this question to take home and practice but we have not been shown anything like this before and there in nothing in the text like this either. If anyone has any tips on how to start this question I would really appreciate it. Two surfaces are said to intersect orthogonally at a...
19. ### Question on Quadric Surfaces

Hey everyone, I am having a bit of difficulty understanding a problem. It says: The trace of the graph of z = f(x,y) = \ x^2 + 2y^2 on the plane z=3, is which of the following conic sections? At first glance, the equation of z = \ x^2 + y^2 is an elliptic paraboloid. However, if I...
20. ### Homeomorphism with surfaces

Let H=\{(x,y,z)\in\mathbb R^3|x^2+y^2=1+z^2\}, and C=\{(x,y,z)\in\mathbb R^3|x^2+y^2=1\}. Prove that H and C as subspaces of \mathbb R^3 are homeomorph. How to solve this analytically? I've seen a geometric solution but I don't see how to work it analytically. Thanks.