Foundations — Covariant and contravariant components

4.10.8 · D1 · Maths › Advanced Topics (Elite Level) › Covariant and contravariant components

Yeh page kuch bhi assume nahi karta. Parent note Covariant and contravariant components padhne se pehle, tumhe neeche diye gaye har symbol mein fluent hona chahiye. Hum unhe ek-ek karke build karte hain, har ek pichle par, har ek ek picture se anchored.

1. Ek vector — woh arrow jo pehle aata hai

Picture. Neeche diye gaye black arrow ko dekho. Woh ek blank sheet par baitha hai. Abhi tak koi numbers nahi hain — numbers tabhi aate hain jab hum measure karne ke liye reference arrows choose karte hain.

Yeh topic kyun iske saath shuru hota hai. Covariant/contravariant ka poora drama yahi hai ki arrow fixed hai lekin uske numbers grid ke saath change ho jaate hain. Agar tumhe pehle yeh yakeen nahi hai ki arrow coordinates se alag exist karta hai, toh do-readings wali baat ka koi matlab nahi banega.

2. Basis — woh reference arrows jo tum choose karte ho

Picture. Neeche, ${\mathbf{e}}_1$ aur ${\mathbf{e}}_2$ lavender aur coral mein draw kiye gaye hain. Dhyan do ki yeh $90°$ par nahi hain aur ek hi length ke nahi hain. Yeh ek skew basis hai — deliberately awkward, taaki do-readings effect dikhaayi de.

Aage link: alag basis choose karna exactly wahi hai jo Change of basis and transformation laws study karta hai.

3. Ek arrow ko number se multiply karna, aur arrows ko add karna

Yeh topic kyun iska use karta hai. Har component statement in do moves se bani hai. Jab hum $\mathbf{v}=v^1{\mathbf{e}}_1+v^2{\mathbf{e}}_2$ likhte hain, hum keh rahe hain: ${\mathbf{e}}_1$ ko number $v^1$ se scale karo, ${\mathbf{e}}_2$ ko $v^2$ se scale karo, phir dono scaled arrows ko add karo taaki exactly $\mathbf{v}$ ki tip par land ho.

4. Upper aur lower indices — staircase notation

Yahi notation logon ko darr deti hai. Yeh bas chhote label ki jagah hai — itna hi.

Picture. Ek staircase socho: lower index neeche ki step par baitha hai, upper index upar ki step par. Parent note ka mnemonic — "CO cellar mein neeche jaata hai, contra upar jaata hai" — exactly yahi staircase hai.

Poori index machinery Tensors and index notation mein hai.

5. Einstein summation convention — gayab hota $\sum$

Ek upper aur ek lower kyun? "Ek upar, ek neeche" ki pairing ek signal hai ki result ek genuine coordinate-free number hai (hum §8 mein dekhenge kyun). Agar tum kabhi wahi letter do baar dono upar ya dono neeche dekho, toh kuch galat hai.

6. Dot product — hum length aur angle kaise measure karte hain

Is topic mein har cheez geometric — lengths, angles, projections, metric — ek operation se bani hai: dot product.

Yeh exact formula kyun, aur $\cos$ kyun? Hum ek aisa gadget chahte hain jo (a) length ke saath bade, aur (b) bataye ki do arrows kitne aligned hain. Cosine natural "alignment dial" hai:

• Arrows same direction mein point karte hain → $\theta =0°$, $\cos 0°=1$ → sabse bada value.
• Arrows right angles par → $\theta =90°$, $\cos 90°=0$ → dot product zero hai. (Aise hum perpendicularity detect karte hain!)
• Arrows opposite mein point karte hain → $\theta =180°$, $\cos 180°=-1$ → sabse negative.

Picture. Figure mein, coral arrow ka shadow seedha lavender arrow par girta hai jiska length $|\mathbf{b}|\cos \theta$ hai. Dot product woh shadow length times $|\mathbf{a}|$ hai. Yeh "axis par shadow" wala idea parent note mein ek covariant component ki definition literally hai.

Dot product ek poore subject ka seed hai, Inner product spaces.

7. Arrow ke numbers padhne ke do tarike — poora point

Ab jab hamare paas arrows, ek basis, scaling, aur dot product hain, do readings naturally dikhti hain.

Picture. Neeche, wahi $\mathbf{v}$ dono tareekon se padha gaya hai. Dashed parallel lines $v^1,v^2$ deti hain (tirchi grid ke saath chalo). Dotted perpendicular drops $v_1,v_2$ dete hain (har axis par shadows). Is skew grid par yeh differ karte hain — square paper par yeh ek doosre ke upar aa jaate.

Topic ko dono ki zaroorat kyun hai. Na hi koi reading "sahi wali" hai — yeh dono ek hi arrow ke do faces hain. Parent note ka kaam dono ko carry karna aur metric naam ke dot products ke chhote table se unke beech translate karna hai.

8. Metric $g_{ij}$ aur Kronecker delta ${\delta }_{ij}$ — translator

Yeh topic kyun iska use karta hai. Metric dono readings ke beech dictionary hai: $v_i=g_{ij}{v}^j$ (stairs neeche) aur $v^i=g^{ij}{v}_j$ (stairs upar, inverse table $g^{ij}$ use karke). Jab basis orthonormal hoti hai, har dot product ek Kronecker delta hota hai, isliye $g_{ij}={\delta }_{ij}$, metric kuch-nahi-karta table ban jaata hai, aur dono readings identical ho jaati hain — isliye square-paper school maths ne kabhi yeh kuch mention nahi kiya.

Deeper study: Metric tensor, aur General relativity — raising and lowering indices mein iski role. Perpendicular-reading ka apna alag reference arrows ka set bhi hai, Dual (reciprocal) basis.

Yeh pieces topic ko kahan feed karte hain

Woh applications jahan skew grid unavoidable hai: Curvilinear coordinates (polar, spherical).

Equipment checklist

Right side cover karo aur parent note kholne se pehle har ek ka jawab do.