Foundations — Triple integrals in Cartesian, cylindrical, spherical coordinates

4.4.20 · D1 · Maths › Multivariable Calculus › Triple integrals in Cartesian, cylindrical, spherical coordi

Yeh page assume karta hai ki tum notation ke baare mein kuch nahi jaante. Hum har symbol build karenge, use ek picture se connect karenge, aur batayenge ki topic ko yeh kyun chahiye — ek aisi order mein jahan har idea apne pehle wale idea par tikaa ho.

0. "Region in space" kya hai? ($E$, aur 3D khud)

Kisi bhi integral se pehle, humein integrate karne ke liye ek jagah chahiye.

Figure dekho: point wahan baithta hai jahan teen dashed guide-lines milti hain. Har coordinate ek axis ke along ek shadow distance hai.

Topic ko yeh kyun chahiye: tum "har chunk pe add" tabhi kar sakte ho jab tum fence karo ki kaun se chunks — woh fence $E$ hai. Parent note ki puri mushkil yeh hai ki "$E$" ko sliders ke liye concrete number ranges mein badlo.

1. Function $f(x,y,z)$ — har point pe rehne wali ek value

Topic ko yeh kyun chahiye: parent ke do headline uses hain $f=1$ (volume deta hai) aur $f=\rho (x,y,z)$ = density (mass deta hai). Integral woh machine hai jo in readings ko sum karta hai.

2. Chopping aur summing — $\Delta$, $\sum$, $\lim$

Integral $E$ ko chote pieces mein kaatke aur add karke banta hai.

Figure mein, coarse chop (left) blob se upar/neeche jaata hai; fine chop (right) usse hug karta hai. Limit woh perfect fit hai jo tum kabhi draw nahi kar sakte lekin hamesha paate ho.

Topic ko yeh kyun chahiye: yahi topic hai. Baki sab ise evaluate karne ki machinery hai. Dekhein Center of mass and moments of inertia ki yeh sums physically kya compute karte hain.

3. $dV$, $dxdydz$ — ek tiny chunk ki shape

Topic ko yeh kyun chahiye: $dV$ parent ki Feynman story ka "ek sugar cube" hai. Cylindrical aur spherical coordinates ka poora drama yeh hai ki unke chunks straight boxes nahi hote, isliye unka $dV$ extra stretch factors ($r$, aur ${\rho }^2\sin \varphi$) uthata hai — jo humein §6 tak le jaata hai.

4. Nested integrals — $\int$ sign aur iterated limits

Topic ko yeh kyun chahiye: yeh nested limits set karna hi, parent ke shabdon mein, "wo single skill hai jo matter karta hai." Tum pehle se 2D version Double integrals & polar coordinates mein dekh chuke ho.

5. Trig jo tumhare paas hona chahiye — $\cos ,\sin ,\tan$, aur $\pi$

Cylindrical aur spherical coordinates angles se bane hain, isliye humein un do functions ki zaroorat hai jo ek angle ko coordinates mein badlein.

• $\theta =0$ par: point $(1,0)$, toh $\cos 0=1,\sin 0=0$.
• $\theta =\frac{\pi }{2}$ par (quarter turn, seedha upar): point $(0,1)$, toh $\cos \frac{\pi }{2}=0,\sin \frac{\pi }{2}=1$.
• $\theta =\pi$ par (half turn): $(-1,0)$.

Topic ko yeh kyun chahiye: substitutions $x=r\cos \theta$, $y=r\sin \theta$ (cylindrical) aur spherical formulas kuch nahi hain sirf "ek rotating arm ke horizontal/vertical shadows ke." Deeper geometry yahan hai: Spherical coordinates geometry.

6. Partial derivatives, determinants, aur Jacobian $J$

Yeh genuinely naya machine hai. Hum ise teen layers mein build karte hain.

6a. Partial derivative $\frac{\partial x}{\partial u}$

6b. Determinant — ek box ka signed volume

Ek $3\times 3$ array ke liye, $\det \left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)=a(ei-fh)-b(di-fg)+c(dh-eg).$

6c. Jacobian $J$ aur $|J|$

Topic ko yeh kyun chahiye: parent note mein mysterious extra $r$ (cylindrical) aur ${\rho }^2\sin \varphi$ (spherical) kuch nahi hain sirf un coordinates ke liye compute kiya gaya $|J|$. Ise master karo aur woh factors memorization nahi rahenge. Full treatment: Jacobian and change of variables. Yahi box-ka-volume idea Divergence theorem mein wapas aata hai.

7. Coordinate players — $r,\theta ,\rho ,\varphi$

Ab woh letters jo naye coordinates ko naam dete hain.

Prerequisite map

Ise upar ki taraf padho: points aur values chunk-and-sum idea ko feed karte hain; single integrals nesting ko feed karte hain; trig plus partials plus determinants Jacobian banate hain; milke yeh woh volume elements dete hain jo parent note use karta hai.

Equipment checklist

Right side cover karo aur parent note start karne se pehle har ek aloud jawab do.