Tangent planes and linear approximations to surfaces

4.4.5 · Maths › Multivariable Calculus

WHAT are we approximating?

Hamare paas ek surface $z = f (x, y)$ hai. Ek point $(a, b)$ ke paas hum ek aisi plane chahte hain jo:

Surface ke point $(a, b, f (a, b))$ se guzre.
Us point par har direction mein surface jaisi hi slope rakhe.

HOW to derive it (from first principles)

Step 1 — $(a, b, z_{0})$ se guzarne wali ek general plane. Koi bhi non-vertical plane is tarah likhi ja sakti hai: $z = z_{0} + A (x - a) + B (y - b) .$ Yeh form kyun? Yeh guarantee karta hai ki plane $(a, b, z_{0})$ se guzregi, aur $A, B$ $x$ - aur $y$ -directions mein slopes hain.

Step 2 — Height match karo. $(a, b)$ par hume $z = f (a, b)$ chahiye, isliye $z_{0} = f (a, b)$ .

Step 3 — $x$ -slope match karo. $y = b$ hold karo. Tab $z = z_{0} + A (x - a)$ , slope $A$ . Surface ki slope yahan $f_{x} (a, b)$ hai. Isliye $A = f_{x} (a, b)$ .

Step 4 — $y$ -slope match karo. Usi tarah $x = a$ hold karo: $B = f_{y} (a, b)$ .

Figure — Tangent planes and linear approximations to surfaces

WHY does this work? (Differentiability)

Error $E (x, y) = f (x, y) - L (x, y)$ ko $(a, b)$ tak ki distance se zyada tezi se shrink karna chahiye: $lim_{(x, y) \to (a, b)} \frac{f ( x , y ) - L ( x , y )}{( x - a ) ^{2} + ( y - b ) ^{2}} = 0.$ Yahi differentiable ka precise matlab hai. Sirf partials ka exist karna kaafi nahi — unhe continuous bhi hona chahiye (ek sufficient condition) taaki plane sach mein surface ko "hug" kare.

Gradient / normal-vector view

$F (x, y, z) = f (x, y) - z = 0$ ko ek level surface define karo. Tab $\nabla F = (f_{x}, f_{y}, - 1)$ surface ke liye normal hai. Tangent plane woh sab kuch hai jo is normal ke perpendicular hai: $f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b) - 1 \cdot (z - f (a, b)) = 0,$ jo rearrange hoke same formula deta hai. Normal vector hai $n = ⟨ f_{x}, f_{y}, - 1 ⟩$ .

Worked Examples

Common Mistakes

Recall Feynman: explain to a 12-year-old

Ek pahadi chocolate-coated surface socho. Agar ek tiny flat sticker ek jagah bilkul sahi rakh do, toh sticker chocolate ko perfectly touch karta hai aur usi taraf jhukta hai jis tarah pahaadi jhukti hai. Woh sticker hi tangent plane hai. Kisi nearby point ki height guess karne ke liye, tum bumpy chocolate ki jagah flat sticker par chalo — pass ke points nearly sahi answer dete hain, door ke points galat ho jaate hain. Sticker ka left-right jhukav $f_{x}$ hai aur front-back $f_{y}$ .

Connections

Partial derivatives — slopes $f_{x}, f_{y}$ provide karte hain.
The gradient vector — normal $⟨ f_{x}, f_{y}, - 1 ⟩$ deta hai.
Differentiability of multivariable functions — woh condition jo plane ko chahiye.
Tangent line and linear approximation (single variable) — 1D parent idea.
The chain rule (multivariable) — linear-approximation engine par bana hai.
Directional derivatives — kisi bhi direction mein slope, tangent plane mein hi rehti hai.

#flashcards/maths

z = f (x, y)

ka tangent plane equation

(a, b)

par kya hota hai?

z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)

Tangent plane ke liye

f

differentiable kyun honi chahiye (sirf partials kaafi kyun nahi)?

Partials sirf axis directions test karte hain; differentiability require karti hai ki error

f - L

, distance to

(a, b)

se zyada tezi se vanish ho, taaki plane har direction mein fit ho. :::

Surface

z = f (x, y)

ka normal vector kya hai?

n = ⟨ f_{x}, f_{y}, - 1 ⟩

Linear approximation

L (x, y)

kya hai?

L (x, y) = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)

Differential

d z

kya hai aur yeh kya estimate karta hai?

d z = f_{x} d x + f_{y} d y

; yeh chhote steps ke liye true change

Δ z = f (x, y) - f (a, b)

estimate karta hai.

Ek point par differentiability ki sufficient condition kya hai?

f_{x}

aur

f_{y}

exist karein aur point ke paas continuous hon.

Jaisi door jaate ho linear approximation kitna accurate rehti hai?

Error roughly

(a, b)

se distance ke square jaisi badhti hai; sirf paas mein reliable hai.

A = x y

ka area error

d x, d y

errors ke saath estimate karo.

d A = y d x + x d y