The metric tensor defines geometry on curved surfaces.
It surface M with a yellow coordinate grid. Arrows at the point indicate the basis tangent vectors ∂/∂x¹ and ∂/∂x².
The g_ij components are their dot products:
g₁₁ = ∂/∂x¹ ⋅ ∂/∂x¹
g₁₂ = ∂/∂x¹ ⋅ ∂/∂x²
g₂₁ = ∂/∂x² ⋅ ∂/∂x¹
g₂₂ = ∂/∂x² ⋅ ∂/∂x²
These encode lengths and angles intrinsically. Used in real life for spacetime modeling in general relativity and surface rendering in 3D graphics and engineering.
e appears in mathematics through four primary characterizations.
It is the limit as n→∞ of (1 + 1/n)^n, linked to maximum compounded growth. It also equals ∑_{n=0}^∞ 1/n!.
The integral from 1 to a of dx/x defines ln(a), and ∫_1^e (1/t) dt = 1. The derivative of e^x equals e^x.
It is used to model continuous compound interest in finance and exponential growth in biology and physics.
In 1831 Michael Faraday demonstrated that moving a magnet near a coil of wire generates an electric current within the wire. He discovered that electricity.
Voronoi diagram (black points) computed by projecting vertically lower envelope of n 3D graphs of functions {(x,y_i(x))} with y_i(x)=D(x_i,x) (pink).
When distance D(x,x')=‖x-x'‖^2, graphs of y_i are paraboloids and Voronoi cell borders are linear