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Questions tagged [surfaces]

For questions about two-dimensional manifolds.

Formally, a surface is a two-dimensional topological manifold. Some examples of surfaces are the plane, the cylinder, the sphere, and the graph of a real-valued function of two variables.

More generally, the term "hypersurface" can be used to denote an $(n-1)$-dimensional submanifold of an $n$-dimensional manifold.

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Find the equation of the plane through a point which is perpendicular to a curve

Find the equation of the plane through the point $(1, -1, 2)$ which is perpendicular to the curve of intersection of the two surfaces $x^2 + y^2 - z = 0$ and $2x^2 + 3y^2 + z^2 - 9 = 0$. And would you be able to explain the steps? thanks!
Ryan
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Elongation of edges of Moebius strip

Find length of edges of Moebius strip formed by cutting along length and re-joining from a circular cylinder segment: $( a \cos \theta, a \sin \theta, z), ( \theta, 0, 2 \pi ),( z,0,b) $ after half rotation. Since the Moebius strip has negative…
Narasimham
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How is the gradient of a curve or function its normal

I see in several books that the gradient of a function (curve), f(x,y) is defined as the normal to the curve without the proof. Can someone prove mathematically that this is true?
Rishab
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Equation of Elliptical Orbit of an Ellipse

I was wondering how to formulate the equation of an ellipse which is rotated about an elliptical orbit. Here's the example: Say I have ellipse in the x-y plane: $$ \frac{x^2}{a^2}+\frac{(y-y_0)^2}{b^2}=1$$ where $a>b$. Suppose I want to rotate this…
RGG
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