Foundations — Properties — linearity, first - second shift theorems, scaling

4.6.27 · D1 · Maths › Ordinary Differential Equations › Properties — linearity, first - second shift theorems, scali

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Yeh page kyun exist karta hai

Parent note mein ${\int }_0^{\infty }$, $e^{-st}$, $\mathcal{L}\{\cdot \}$, $F(s)$, $u(t-a)$, $\tau$, $dt$, aur "substitution" aise throw kiye jaate hain jaise tum inhe pehle se jaante ho. Yahan hum har ek ko earn karte hain, ek aisi order mein jahan har symbol sirf unhi symbols ka use karta hai jo pehle define ho chuke hain. Agar parent page ki koi bhi line foreign language jaisi lagi, toh yeh tumhara dictionary hai — pictures ke saath.

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0. "Function of time" $f(t)$ kya hota hai?

Topic ko yeh kyun chahiye: har Laplace rule "given a function $f(t)$…" se shuru hota hai. Agar tum $f(t)$ ko ek curve ki tarah picture nahi kar sakte, toh poora chapter symbols hai jo space mein float kar rahe hain.

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1. Exponential $e^t$ aur uske cousins $e^{at}$, $e^{-st}$

Topic ko yeh kyun chahiye: first-shift theorem literally "kya hota hai jab tum apni function ko $e^{at}$ se multiply karte ho" hai, aur poora transform fader $e^{-st}$ par built hai. Exponential nahi toh Laplace nahi.

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2. Variable $s$ aur transform output $F(s)$

Topic ko yeh kyun chahiye: har property "$F(s)\to$ kuch" likhi hoti hai. First shift $F(s-a)$ deta hai, second $e^{-as}F(s)$ deta hai. Tumhe pata hona chahiye ki $F$ ek function hai jisme tum shifted inputs plug kar sakte ho, bilkul waise jaise $s-a$ ko $f(\text{box})={\text{box}}^2$ mein plug karne se $(s-a{)}^2$ milta hai.

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3. Integral sign ${\int }_0^{\infty }\dots dt$

Yeh parent page ka sabse scary-looking symbol hai. Hum ise teen simple steps mein build karte hain.

Topic ko yeh kyun chahiye: definition $\mathcal{L}\{f\}={\int }_0^{\infty }{e}^{-st}f(t)dt$ ek integral hai, aur har property is integral ko manipulate karke prove ki jaati hai. Linearity kaam karta hai kyunki area-of-a-sum = sum-of-areas.

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4. Sab kuch jodna: operator $\mathcal{L}\{\cdot \}$

Ab parent page ka har symbol defined hai: $\mathcal{L}$, $f(t)$, $e^{-st}$, ${\int }_0^{\infty }$, $dt$, $F(s)$, aur $s$. Yeh integral kab converge hone ki guarantee hai, iske liye Laplace Transform — Definition and Existence note dekho, aur jin building blocks par properties act karti hain unke liye Laplace Transforms of Standard Functions (1, e^at, sin, cos, t^n) dekho.

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5. Do aur symbols jo properties mein kaam aate hain

5a. Unit step $u(t-a)$

Topic ko yeh kyun chahiye: second shift theorem $\mathcal{L}\{f(t-a)u(t-a)\}=e^{-as}F(s)$ is switch par built hai. Switch ki poori details Unit Step and Dirac Delta Functions mein hain.

5b. Dummy variable $\tau$ aur "substitution"

Topic ko yeh kyun chahiye: second shift aur scaling derivations substitutions hain. Agar tum trust nahi karte ki naam badalne se integral ki value nahi badlti, toh woh proofs magic jaisi lagti hain.

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Prerequisite map

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Equipment checklist

Recall Self-test: kya tum colon se pehle har ek answer de sakte ho?

$f(t)$ ka kya matlab hai, ek picture ke roop mein? ::: Ek curve; $t$ ke har time ke liye tum graph se height $f(t)$ read karte ho. Curve $e^t$ mein kya special hai? ::: Har point par uski steepness uski apni height ke barabar hai; yeh $(0,1)$ se guzarti hai aur tezi se tezi se grow karti hai. Growth vs decay: $e^{at}$ kab decay karta hai? ::: Jab $a<0$ (aur grow karta hai jab $a>0$). Ek exponential algebra rule ::: $e^p\cdot e^q=e^{p+q}$ — multiply karne se exponents add hote hain. ${\int }_0^{\infty }g(t)dt$ kya compute karta hai? ::: $t=0$ se $\infty$ tak $g(t)$ ke neeche ka total signed area, thin strips se sum kiya hua. $dt$ kya hai? ::: Ek area-strip ki infinitely thin width, jo $t$ ko summed variable mark karti hai. Laplace integral ko $e^{-st}$ factor ki zaroorat kyun hai? ::: Yeh far future ko fade karta hai taaki infinite-area sum finite rahe (converge kare). Lower-case $f$ vs capital $F$? ::: $f(t)$ time-world mein rehta hai; $F(s)$ s-world mein uska transform hai. $\mathcal{L}\{f(t)\}$ kya hai — ek number ya ek machine? ::: Ek machine (operator): yeh time-function ko s-function $F(s)$ mein turn karta hai. $u(t-a)$ ek function ke saath kya karta hai? ::: Use $t=a$ se pehle erase karta hai (value $0$), $t=a$ se unchanged pass karta hai (value $1$). Shift/scaling proofs mein $\tau$ kyun use hota hai? ::: Yeh time variable ka ek dummy rename hai taaki integral $F(s)$ ki definition se match kare; naam badalne se value nahi badlti, lekin $dt\to d\tau$ factor ko track karna zaroori hai.

Connections

• Laplace Transform — Definition and Existence
• Laplace Transforms of Standard Functions (1, e^at, sin, cos, t^n)
• Inverse Laplace Transform and Partial Fractions
• Unit Step and Dirac Delta Functions
• Solving ODEs with Laplace Transforms
• Convolution Theorem