SymPy — symbolic algebra, calculus, ODE solving: Foundations

5.4.20 · D1 · Coding › Scientific Computing (Python) › SymPy — symbolic algebra, calculus, ODE solving

Parent note ki ek bhi line padhne se pehle, tumhe kuch chote-chote ideas apne karne honge. Yeh page unme se har ek ko bilkul zero se build karta hai — koi bhi notation use hone se pehle draw aur explain ki jaati hai. Hum jaanbujhkar dheere chalte hain.

1. Number vs. symbol — sabse gehra farak

Picture. Ek number line imagine karo. Ek number ek exact jagah par laga hua pin hai. Ek symbol ek slider hai jo kahin bhi baith sakta hai — tum ise slider ke roop mein isliye rakhte ho taaki tum saari positions ke baare mein ek saath soch sako.

Figure — SymPy — symbolic algebra, calculus, ODE solving

YEH TOPIC ISKO KYUN CHAHTA HAI. NumPy (dekho NumPy — numerical arrays) har cheez ko turant ek number par pin kar deta hai. SymPy mana kar deta hai: woh slider ko rakhta hai, taaki woh koi bhi value choose hone se pehle factor, differentiate aur solve kar sake. Yahi puri wajah hai ki SymPy exist karta hai.

2. Operation — woh verb jo symbols ko jodta hai

Picture. $+$ ko ek aisi machine samjho jiske upar do funnels hain aur neeche ek spout. Do cheezein funnels mein daalo, ek cheez neeche aati hai. Funnels mein numbers, symbols, ya kisi doosri machine ka output — yeh stacking hi trees ko possible banata hai.

3. Expression tree — ek idea ki shape

Tree padhna. Sabse bahar ki machine $+$ hai. Uske do funnels $x^{2}$ (ek power machine), $2 x$ (ek multiply machine) aur $1$ (ek plain number) se feed ho rahe hain. Ise top-down draw karo aur tumhe ek branching diagram milta hai — isliye "tree" kehte hain.

YEH TOPIC ISKO KYUN CHAHTA HAI. Har ek SymPy command is tree ko rewrite karne ka ek rule hai:

expand multiplications ko andar dhakelta hai (brackets kholta hai),
factor common structure ko bahar kheenchta hai,
diff tree par walk karta hai aur har node par derivative rule apply karta hai.

Agar tree picture karo, toh yeh sab magic lagni band ho jaati hai.

4. Equation, aur "zero set karna"

Picture. Expression $a - b$ ko ek curve ki tarah plot karo. Jo jagahen curve horizontal axis ko cross karti hai (height zero) wahi exactly solutions hain. Solve karna = crossings dhundhna.

5. Teen calculus symbols jo tumhe milenge

Tumhe yahan calculus master karne ki zaroorat nahi — parent note usse derive karta hai. Tumhe bas teen symbols pehchanne hain taaki woh tumhe surprise na karein.

YEH TEEN KYUN. Parent ka Section 3 derivative ko limit se recover karta hai, aur ise integral ke saath pair karta hai. Taylor and Maclaurin Series section phir baar baar derivatives use karta hai. Filhaal symbols pehchaanna kaafi hai.

6. Function object $y (x)$ aur arbitrary constant $C_{1}$

YEH TOPIC ISKO KYUN CHAHTA HAI. dsolve return karta hai Eq(y(x), C1*sin(x) + C2*cos(x)) — poori family. Initial conditions (ics) dials ghuma ke ek single curve deti hain. Agar tumhe nahi pata tha ki $y (x)$ ek shape hai aur $C_{1}$ ek dial, toh woh output noise lagta.

7. Exact vs. decimal — aakhri gate

sqrt(2) ko ek sealed envelope samjho: SymPy ise band rakhke liye le jaata hai, label par algebra karta rehta hai. .evalf() woh kaam hai jab tum envelope kholte ho (rounded) contents padhne ke liye. Actual plotting numbers ki taraf bridge hai Lambdify — bridging SymPy to NumPy for plotting.

Prerequisite map

Equipment checklist

Khud test karo — right side cover karo, answer bolkar bolo, phir reveal karo.

Number aur symbol mein farak

Number ek settled value hai (line par ek pin); symbol ek unknown placeholder hai (ek slider) jo isliye rakha jaata hai taaki value choose karne se pehle reason kar sako.

SymPy ek value ki jagah expression tree kyun store karta hai

Kyunki unknown symbols ke saath abhi compute karne ko kuch nahi hota, toh woh structure rakhta hai — jo ise baad mein factor, expand aur differentiate karne deta hai.

SymPy code mein "x squared" kaise likhte hain

x**2 — kabhi x^2 nahi, kyunki Python mein ^ ka matlab xor hota hai.

Har equation jis universal form mein rewrite hoti hai

expression = 0, kyunki

a = b ⟺ a - b = 0

, toh solve karna = expression kahan zero cross karta hai woh dhundhna.

(x+1)**2 == x**2+2*x+1 kyun False hai

== tree structure compare karta hai, mathematical value nahi; simplify(a-b)==0 ya a.equals(b) use karo.

lim_{h \to 0}

ka matlab words mein

Woh value jis par tum pahunchte ho jab

h

zero ki taraf simat ta hai bina kabhi pahunche.

Derivative

f^{'} (x)

kya measure karta hai, ek picture ke roop mein

Point

x

par curve ki steepness/slope.

Definite integral

\int_{a}^{b} f d x

kya measure karta hai, ek picture ke roop mein

a

b

tak curve aur horizontal axis ke beech signed area.

y (x)

x

se kaise alag hai, aur ise kaise banate hain

x

ek unknown number hai;

y (x)

ek unknown function hai (ek poori shape), Function('y') se banaya jaata hai.

dsolve ke answer mein

C_{1}

kya hai

Ek dial jo solutions ki poori family mein se ek curve select karta hai; ics ise set karta hai.

Decimal aakhir kab aata hai

Tabhi jab tum .evalf() ya N(...) se maangite ho; tab tak sab kuch exact rehta hai.