WHAT hai problem? Ek vector v ek real geometric arrow hai — use coordinates ki koi parwah nahi. Lekin use describe karne wale numbers sirf ek chosen basis {e1,e2} ke relative mein exist karte hain. Agar basis vectors orthonormal nahi hain (alag lengths, 90° par nahi), to "kitna e1, v mein hai" — isme genuinely ambiguity hai.
WHY ambiguity? Do reasonable rules hain:
Parallelogram rule — axes ke saath slide karo tip tak pahunchne ke liye. Ye coefficients vicontravariant components hain, upper index ke saath likhe jaate hain:
v=v1e1+v2e2=viei
Orthogonal-projection rule — har axis par perpendicular daalo aur dot product lo. Ye covariant components vi (lower index) dete hain:
vi=v⋅ei
Covariant components ki definition se shuru karo aur contravariant expansion plug in karo. Ye step kyun? Kyunki covariant components define hote hain ei ke saath dot karke, aur hum pehle se jaante hain v ko ej's ke sum ke roop mein — to substitute karo.
vi=v⋅ei=(vjej)⋅ei=vj(ej⋅ei)
Quantity ei⋅ej naturally appear hoti hai. Ise naam do:
WHY hai gij identity jab orthonormal ho? Tab ei⋅ej=δij, to vi=vi. Poora distinction collapse ho jaata hai. Ye tumhara sanity check hai.
WHICH object covariant components ko multiply karta hai? Ek doosra basis {ei} hai, dual basis, jo define hota hai:
ei⋅ej=δji
yaani e1 perpendicular hai e2 ke, aur scale kiya hua hai taaki e1⋅e1=1. Tab wahi vector ke do clean expansions hote hain:
v=viei=viei
To covariant components simply contravariant components hain dual basis ke respect mein. Sundar symmetry hai.
Socho tum ek vector ko normal square paper ki jagah stretchy, slanted graph paper par draw kar rahe ho. Apne arrow ko describe karne ke liye tum ya to keh sakte ho "slanted lines ke saath itna door chalo" (contravariant) ya "arrow ka shadow seedha har line par kitna padta hai" (covariant). Normal square paper par dono answers same hain. Slanted paper par alag hain — to hum numbers ke do sets rakhte hain, aur ek chhoti si table jise metric kehte hain jo unke beech translate karti hai.
