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Differential Geometry

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.

Izifundo

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

Symbols used here

f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\frac{\partial f}{\partial x}
partial derivative
Derivative with respect to x, holding the other variables fixed.
\nabla f
gradient (nabla, del)
Vector of partial derivatives; points uphill.
\kappa,\ K
curvature, Gaussian curvature
How fast a curve turns; the product of a surface's principal curvatures.

Questions people ask

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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