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Differential Geometry

Curves, surfaces, curvature, manifolds: the language of general relativity.

经验教训

Advanced Curves, arc length and curvature Parametrised curves, the unit tangent, curvature as the turning rate. integrate sqrt(1 + 4x^2) dx from 0 to 1 Advanced Surfaces and Gaussian curvature Principal curvatures, Gauss's Theorema Egregium, geodesics. x^2 - y^2

Calculus on curved things. Curvature measures how a surface bends, geodesics are its straight lines, and manifolds let the same ideas run in any dimension. Einstein's equations are a statement in this language.

更多经验教训

Curves

Differential geometry of curvesCurvatureArc lengthFrenet–Serret formulasTorsion of a curveOsculating circleParametric equation

Surfaces

Differential geometry of surfacesGaussian curvatureMean curvatureTheorema EgregiumFirst fundamental formSecond fundamental formGeodesicGauss–Bonnet theoremMinimal surfaceDevelopable surface

Manifolds

ManifoldDifferentiable manifoldTangent spaceVector fieldDifferential formRiemannian manifoldMetric tensorChristoffel symbolsCovariant derivativeRiemann curvature tensorParallel transportLie groupGeneral relativity

其中一些经验教训从开放文本中改编: Wikipedia. 每页课目都列出其来源和执照。

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人们对人们提出的问题

What is curvature?

For a curve, how fast its direction turns per unit length; for a surface, Gauss's combination of the two principal bendings, and Gauss's theorem says it can be measured from inside the surface without leaving it.

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输入它或将其输入求解器页面:每个步骤、图片、隐藏的答案直到您询问。 解决 →

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