PRIDE CHEKERAdimensions. This is a crucial aspect when dealing with tensors.
Since we have looked at finding distance vectors in Euclidean
flat geometry: what about distanced between two points in a
curved space? Let say we a curve with two end points A and B.
Then we divide the distance between A and B into infinite
infinitesimally small distances. To go through with this, we need
to employ calculus credited to Newton and Leibniz, and from
here the journey starts into the world of manifolds of general
relativity. In high school we use calculus to find the slope
quotient ∆∆푦푥, (in differential terms 푑푦푑푥), surface areas enclosed by
curved functions, through integration, ∫푎푏푥 푑푦 and volumes
enclosed by certain functions rotated one revolution, 휋∫푎푏푥^2 푑푦.
Slope quotient (gradient) of a continuous curve at a certain point
is equal to the gradient of the tangent at that point. Here we use
differential calculus to find infinitesimally small distances, dr. In
Euclidean flat space we define a vector by 퐴^ = 퐴푥푒⃗⃗⃗푥 , but in this
case of small incremental distances, we define a differential
distance vector by dx = 푑푥푗푒⃗⃗푗. Thus dot product of two distance
vectors of the latter becomes 푑푠^2 = dx.dx= ⃗푒⃗⃗ (^) 푖.⃗푒⃗푗 푑푥푖푑푥푗. This
become: 푑푠^2 = 푔푖푗푑푥푖푑푥푗 = 푔휇푣푑푥휇푑푥푣. The latter equation is
fundamentally significant and can be simplified in Cartesian
geometry into Pythagoras theorem:
푑푠^2 = 푔 11 푑푥^1 푑푥^1 + 푔 12 푑푥^1 푑푥^2 + 푔 21 푑푥^2 푑푥^1 + 푔 22 푑푥^2 푑푥^2 =