Foundations — Double integrals in polar coordinates — Jacobian r

4.4.19 · D1 · Maths › Multivariable Calculus › Double integrals in polar coordinates — Jacobian r

Is page par assume kiya gaya hai ki aapko parent note ki notation ke baare mein kuch bhi nahi pata. Hum har ek symbol build karenge — $x$, $y$, $r$, $\theta$, $\int$, $\iint$, $dA$, $\det$, $\partial$ — bilkul ground up se, har ek ko ek picture se anchor karke, aur har ek ko tab hi use karke jab woh earn ho chuka ho.

1. Plane aur uske do numbers $x$ aur $y$

Do number lines banao jo ek point par cross karein jise origin kehte hain. Horizontal waali $x$-axis hai, vertical waali $y$-axis.

Topic ko yeh kyun chahiye: poori problem hai "plane mein area measure karo." Area measure nahi kar sakte jab tak locations ka naam nahi de sakte, aur $(x,y)$ pehla naming scheme hai.

2. Circle equation $x^2+y^2=a^2$

Upar wale figure mein red circle dekho. Uska har point origin se same distance $a$ par hai. Pythagorean theorem se — right triangle mein, do chhoti sides ka square add ho ke badi side ka square banta hai — origin se $(x,y)$ tak ki distance $\sqrt{x^2+y^2}$ hai.

3. Angle $\theta$ aur radian

"Right aur up" ki jagah, ek point ko turn aur walk out se describe karo. Origin par khade ho, positive $x$-axis ki taraf muh karo, ek angle se rotate karo, phir aage chalo.

Lekin angles ko ek unit chahiye. Hum unhe degrees mein nahi, radians mein measure karte hain — aur yeh choice poore topic ki secret hero hai.

Aur dekho: Arc Length and Radian Measure.

4. Polar coordinates $(r,\theta )$

Ab "kitna door" aur "kis direction mein" ko combine karo.

Yahan $\cos \theta$ ("cosine") us triangle ki adjacent-over-hypotenuse hai aur $\sin \theta$ ("sine") opposite-over-hypotenuse hai. Hypotenuse $r$ ke saath, adjacent side $r\cos \theta$ aur opposite side $r\sin \theta$ hai — bilkul upar wale formulas ki tarah.

5. Integral sign $\int$ aur double integral $\iint$

Aur dekho: Area of Regions Bounded by Polar Curves.

6. Differentials $dr$ aur $d\theta$, aur polar tile

$r$ ko ek whisker $dr$ se slide karo aur $\theta$ ko ek whisker $d\theta$ se. Chaar grid lines (do circles, do rays) ek tiny curved box trap karti hain.

7. Partial derivatives $\partial$ aur determinant $\det$

Parent usi $r$ ka ek doosra, machine-jaisa proof deta hai Jacobian use karke. Yahan uske do ingredients hain.

Aur dekho: Jacobian Determinant aur Change of Variables Theorem.

8. Sab kuch kaise fit hota hai

Top se bottom tak padho: coordinates aur radians tile ko feed karte hain; tile $dA=rdrd\theta$ deta hai; determinant ise confirm karta hai; aur integral ka sum-idea ise master formula mein wrap karta hai.

Equipment checklist

Apne aap ko test karo — sirf tab reveal karo jab tum zor se jawab de chuke ho.

Connections

• Double integrals in polar coordinates — Jacobian r — woh parent jiske liye yeh page aapko prepare karta hai.
• Arc Length and Radian Measure — arc side $rd\theta$ ka source.
• Jacobian Determinant — woh determinant tool jo $r$ ko reconfirm karta hai.
• Change of Variables Theorem — woh general law jiska polar ek special case hai.
• Area of Regions Bounded by Polar Curves — jahan $\iint 1dA$ area ban jaata hai.
• Gaussian Integral — foundations set hone ke baad ka killer payoff.
• Triple Integrals in Cylindrical and Spherical Coordinates — same idea, ek dimension upar.