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Differential Geometry of Curves and Surfaces from Riemannian Geometry

My personal preference was Lee for the more direct presentation along the lines youâ€™re asking, and Spivak for bedtime reading. The others are essentially treatments of surfaces using differential ...
• 57.3k

Let C be the path in $\Bbb R^2$ from from $(0,0)$ to $(2,1)$ defined by the equation $x^4-6xy^3-4y^2=0$. Find $\int_C (10x^4-2xy^3)dx -3x^2y^2dy.$

Your second idea is the right one (but I don't think your statement is correct): You should recognise that a parametrisation of this curve (although probably not impossible) will be very difficult. ...
• 1,129
1 vote

Exercise about contact between a curve and a surface in $\mathbb{R}^3$

It is not easy to understand this fact easily for a general $f$. It is, instead, really easier to understand the fact for a function of the form $$g(x,y,z) = z - \varphi(x,y).$$ Such a (smooth) ...
1 vote
Accepted

Confused on the result of Sequence of Geometric Transformations

You are correct if the transformations are applied in the order stated. If you translate up 4, the asymptote becomes $y=4$. Dilating vertically by factor 3 moves it to $y=12$. The reflection then ...
• 2,671
1 vote
Accepted

Lemniscate of Bernoulli using Watt's linkage

I'll use your numbers, with $a=1$ and consequently $AB=2$. Set up then a cartesian frame such that $A=(-1,0)$, $B=(1,0)$ and:  C=(-1+\sqrt2\cos\theta,\sqrt2\sin\theta), \quad D=(1+\sqrt2\cos\phi,\...
• 51.2k
1 vote
Accepted

Curvature of an Embedded Curve

Given an oriented curve in the plane, one can define a signed curvature for it. For an oriented closed curve, the integral of the signed curvature will give $2\pi$ (up to sign). On a general surface, ...
• 43.4k

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