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11.1 Introduction to Functions of Several Variables 11.2 Limits and Continuity

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Functions of Several Variables(多變量函數)

Hung-Yuan Fan (范洪源)

Department of Mathematics, National Taiwan Normal University, Taiwan

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本章預定授課範圍

11.4 Differentials and the Chain Rules11.5 Directional Derivatives and Gradients11.6 Tangent Planes and Normal Lines11.7 Extrema of Functions of Two Variables11.8 Lagrange Multipliers

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 2/160

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(多變量函數的介紹)

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Type I: Functions of Two Variables (雙自變量函數)

Let D⊆ R2. A function z = f(x, y) of variables x and y is a rule that assigns to each (x, y)∈ D a unique value f(x, y) ∈ R.

(1) D = dom(f) is the domain of f.

(2) range(f) ={z = f(x, y) ∈ R | (x, y) ∈ D} is the range of f.

(3) x and y are the independent variables (自變數), and z is the dependent variable (應變數) of f.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 4/160

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函數 z = f(x, y) 的示意圖

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 6/160

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 8/160

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Example 2 的示意圖

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 10/160

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Level Curves

Def. (等位線或等高線的定義)

Let f(x, y) be a real-valued function of x and y. For any c∈ R, the setC defined by

C = {(x, y) ∈ D | f(x, y) = c}

is called a level curve or contour curve of f.

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 12/160

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等高線的應用

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 14/160

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Type II: Functions of Three Variables (三自變量函數)

Let D⊆ R3. A function w = f(x, y, z) of x, y and z is a rule that assigns to each (x, y, z)∈ D a unique value f(x, y, z) ∈ R.

(1) D = dom(f) is the domain of f.

(2) range(f) ={w = f(x, y, z) ∈ R | (x, y, z) ∈ D} is the range of f.

(3) x, y, z are the independent variables, and w is the dependent variable of f.

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Level Surfaces

Def. (等位面的定義)

Let f(x, y, z) be a real-valued function of x, y and z. For any c∈ R, the setS defined by

S = {(x, y, z) ∈ D | f(x, y, z) = c}

is called a level surface (等位面) of f.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 16/160

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Example 6 的示意圖

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 18/160

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(極限與連續性)

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 20/160

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Open Disk 的示意圖 (承上頁)

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 22/160

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The Limit of f(x, y)

Def. (雙自變量函數的極限)

Let f(x, y) be a real-valued function defined on D\{(x0, y0)}, where D is an open region inR2. We say that f has a limit L as (x, y) approaches (x0, y0), denoted by

lim

(x,y)→(x0,y0)f(x, y) = L, if∀ ε > 0, ∃ δ > 0 s.t. (x, y) ∈ D and

0 <p

(x− x0)2+ (y− y0)2< δ =⇒ |f(x, y) − L| < ε.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 24/160

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雙自變量函數的極限 (承上頁)

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 26/160

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Limit Laws for f(x, y)

Thm (雙自變量函數的極限法則; 1/2)

Suppose that lim

(x,y)→(x0,y0)f(x, y) = L and lim

(x,y)→(x0,y0)g(x, y) = M.

(1) lim

(x,y)→(x0,y0)|x| = |x0| and lim

(x,y)→(x0,y0)|y| = |y0|.

(2) lim

(x,y)→(x0,y0)[f(x, y)± g(x, y)] = L ± M.

(3) lim

(x,y)→(x0,y0)[f(x, y)· g(x, y)] = L · M.

(4) lim f(x,y) = L with M̸= 0.

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Limit Laws for f(x, y)

Thm (雙自變量函數的極限法則; 2/2)

(5) lim

(x,y)→(x0,y0)[f(x, y)]n= Ln ∀ n ∈ N.

(6) lim

(x,y)→(x0,y0)

pn

f(x, y) =√n

L, provided that L≥ 0 when n is even.

(7) lim

(x,y)→(x0,y0)|f(x, y)| = 0 ⇐⇒ lim

(x,y)→(x0,y0)

f(x, y) = 0.

Example 2 (直接代入求極限)

lim

(x,y)→(1,2)

5x2y

x2+ y2 = 5(1)2(2)

(1)2+ (2)2 = 10 5 = 2.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 28/160

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Remark (極限存在的等價條件)

The following statements are equivalent.

(1) lim

(x,y)→(x0,y0)

f(x, y) = L.

(2) f(x, y)→ L as (x, y) → (x0, y0) along all paths passing (x0, y0).

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 30/160

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Nonexistence of a Limit

Thm (Two Paths Test; 雙路徑測試)

If the following limits satisfy

lim (x, y)→ (x0, y0)

alongC1

f(x, y) = L1 ̸= L2 = lim (x, y)→ (x0, y0)

alongC2

f(x, y)

fortwo different pathsC1 andC2 passing (x0, y0), then lim

(x,y)→(x0,y0)

f(x, y) @.

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 32/160

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Continuity of f(x, y)

Def. (雙自變量函數的連續性)

Let f(x, y) be a real-valued function defined on the open region D⊆ R2 with (x0, y0)∈ D.

(1) f is conti. at (x0, y0) if lim

(x,y)→(x0,y0)f(x, y) = f(x0, y0).

(2) f is conti. on D if it is conti. at every point (x0, y0)∈ D.

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Thm 11.1 (Properties of Continuity of f(x, y))

(1) If k∈ R and f, g are conti. at (x0, y0), then kf, f± g, fg and f/g with g(x0, y0)̸= 0 are conti. at (x0, y0), respectively.

(2) Continuity of Composite Functions:

If h is conti. at (x0, y0) and g is conti. at h(x0, y0), then (g◦ h)(x, y) ≡ g(h(x, y)) is conti. at (x0, y0).

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 34/160

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Section 11.3Partial Derivatives(偏導數、偏導函數)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 36/160

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Partial Derivatives of f(x, y)

Def. (雙自變量函數的偏微分; 1/2)

Let f(x, y) bee a real-valued function defined on an open region D⊆ R2.

(1) The first partial derivative of f w.r.t. x is fx(x, y)≡ lim

∆x→0

f(x + ∆x, y)− f(x, y)

∆x

= 將 y 視為常數, 僅對 x 微分

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Partial Derivatives of f(x, y)

Def. (雙自變量函數的偏微分; 2/2)

(2) The first partial derivative of f w.r.t. y is fy(x, y)≡ lim

∆y→0

f(x, y + ∆y)− f(x, y)

∆y

= 將 x 視為常數, 僅對 y 微分

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 38/160

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Notations

Let z = f(x, y) be a real-valued function of x and y.

1

zx= ∂z

∂x = ∂f

∂x = fx= Dxf = D1f.

2

zy= ∂z

∂y = ∂f

∂y = fy= Dyf = D2f.

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 40/160

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 42/160

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Partial Derivatives of f(x, y, z)

Def. (三自變量函數的偏微分; 1/3)

Let f(x, y, z) bee a real-valued function defined on an open region D⊆ R3.

(1) The first partial derivative of f w.r.t. x is fx(x, y, z)≡ lim

∆x→0

f(x + ∆x, y, z)− f(x, y, z)

∆x

= 將 y 和 z 視為常數, 僅對 x 微分

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Partial Derivatives of f(x, y, z)

Def. (三自變量函數的偏微分; 2/3)

(2) The first partial derivative of f w.r.t. y is fy(x, y, z)≡ lim

∆y→0

f(x, y + ∆y, z)− f(x, y, z)

∆y

= 將 x 和 z 視為常數, 僅對 y 微分

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 44/160

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Partial Derivatives of f(x, y, z)

Def. (三自變量函數的偏微分; 3/3)

(3) The first partial derivative of f w.r.t. z is fz(x, y, z)≡ lim

∆z→0

f(x, y, z + ∆z)− f(x, y, z)

∆z

= 將 x 和 y 視為常數, 僅對 z 微分

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 46/160

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Higher-Order Partial Derivatives (1/2)

Let z = f(x, y) be a real-valued function of x and y.

First Partial Derivatives: (一階偏導數) fx= ∂f

∂x, fy= ∂f

∂y. Second Partial Derivatives: (二階偏導數)

fxx= 2f

∂x2

∂x

∂f

∂x



, fxy= 2f

∂y∂x

∂y

∂f

∂x

 , fyy= 2f

∂f

, fyx= 2f

∂f .

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Higher-Order Partial Derivatives (2/2)

Third Partial Derivatives: (8 個三階偏導數) fxxx= 3f

∂x3

∂x

2f

∂x2



, fxxy= 3f

∂y∂x2

∂y

2f

∂x2

 , fxyx= 3f

∂x∂y∂x

∂x

 2f

∂y∂x



, fxyy= 3f

∂y2∂x

∂y

 2f

∂y∂x

 , fyyx= 3f

∂x∂y2

∂x

2f

∂y2



, fyyy= 3f

∂y3

∂y

2f

∂y2

 , fyxx= 3f

∂x2∂y

∂x

 2f

∂x∂y



, fyxy= 3f

∂y∂x∂y

∂y

 2f

∂x∂y

 .

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 48/160

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 50/160

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Thm 11.3 (Equality of Mixed Partial Derivatives)

If f(x, y) is a real-valued function s.t. fxy and fyx are conti. on an open regionR, then

fxy(x, y) = fyx(x, y) ∀ (x, y) ∈ R.

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Differentials and the Chain Rules(微分與連鎖律)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 52/160

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Def. (Total Differential 的定義)

Let z = f(x, y) be a real-valued function of x and y.

(1) The differentials of x and y are dx = ∆xand dy = ∆y, where

∆x and∆y are (small) increments of x and y.

(2) The total differential (全微分) of z is

dz = fx(x, y)dx + fy(x, y)dy.

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 54/160

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Differentiability of f(x, y) (1/2)

Recall (單變數函數微分的等價條件)

y = f(x) is diff. at x = x0.

⇐⇒f(x0) = lim

∆x→0

f(x0+ ∆x)− f(x0)

∆x ∃.

⇐⇒ lim

∆x→0

hf(x0+ ∆x)− f(x0)

∆x − f (x0) i

= 0.

⇐⇒ lim

∆x→0

∆y− f (x0)∆x

∆x = 0, with ∆y≡ f(x0+ ∆x)− f(x0).

⇐⇒∆y = f(x0)∆x + ε1∆x, with ε1 = ε1(x0, ∆x)→ 0 as ∆x → 0.

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Differentiability of f(x, y) (2/2)

Def. (雙自變量函數的可微分性)

Let z = f(x, y) be defined on an open region D with (x0, y0)∈ D.

Suppose that fx(x0, y0) and fy(x0, y0) both exist.

(1) f is diff. at (x0, y0) if ∃ two functions ε1 and ε2 s.t.

∆z≡ f(x0+ ∆x, y0+ ∆y)− f(x0, y0)

=fx(x0, y0)∆x + fy(x0, y0)∆y + ε1∆x + ε2∆y, with ε1→ 0 and ε2→ 0 as (∆x, ∆y) → (0, 0).

(2) f is diff. on D if it is diff. at every point (x0, y0)∈ D.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 56/160

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 58/160

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Thm 11.4 (可微分的充分條件)

Let f(x, y) be a real-valued function s.t. its first partial derivatives both exist on an open region D⊆ R2. Iffx and fy are conti. on D, then f isdiff. on D.

Example 2 Revisited

For f(x, y) = x2+ 3y, we see that fx= 2x and fy= 3 are both conti. on D = dom(f) =R2. Thus, it follows from Thm 11.4 that f must be diff. on D.

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Approximation by Differentials

If dx = ∆x and dy = ∆y aresuﬀiciently small, then

∆z = f(x0+ ∆x, y0+ ∆y)− f(x0, y0)

≈dz = fx(x0, y0)dx + fy(x0, y0)dy.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 60/160

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 62/160

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Thm 11.5 (可微分 =⇒ 連續)

Let f(x, y) be a real-valued function defined on an open region D⊆ R2 with (x0, y0)∈ D.

(1) If f is diff. at (x0, y0), then f is conti. at (x0, y0).

(2) If f is NOT conti. at (x0, y0), then f is NOT diff. at (x0, y0).

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 64/160

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 66/160

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Def. (可微分的等價定義)

Let z = f(x, y) be defined on an open region D with (x0, y0)∈ D.

Suppose that fx(x0, y0) and fy(x0, y0) both exist. We say that f is diff. at (x0, y0) if

lim

(∆x,∆y)→(0,0)

∆z− fx(x0, y0)∆x− fy(x0, y0)∆y p(∆x)2+ (∆y)2 = 0, where ∆z = f(x0+ ∆x, y0+ ∆y)− f(x0, y0).

Example (補充題; 連續; 可微分)

The function f(x, y) =p

|xy| is conti. at (0, 0), but it is NOT diff.

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Chain Rules (連鎖律)

Type I: One Parameter (Thm 11.6)

If w = f(x, y) is adiff. function of x and y, and x = g(t), y = h(t) arediff. functions of the parameter t, then w = w(t) is a diff.

function of t and

dw dt = ∂w

∂x dx dt +∂w

∂y dy dt.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 68/160

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Chain Rules (連鎖律)

Type II: Two Parameters (Thm 11.7)

If w = f(x, y) is adiff. function of x and y, and

x = g(s, t), y = h(s, t) arediff. functions of the parameters s and t, then w = w(s, t) is adiff. function of s and t, and

∂w

∂s = ∂w

∂x

∂x

∂s +∂w

∂y

∂y

∂s,

∂w

∂t = ∂w

∂x

∂x

∂t +∂w

∂y

∂y

∂t.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 70/160

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 72/160

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Chain Rules (連鎖律)

Type III: Multiple Parameters (多重參數)

If w = f(x1, x2,· · · , xn) is a diff. function of x1, x2,· · · , xn, and xj = xj(t1, t2,· · · , tm) is a diff. function oft1, t2,· · · , tm for each j = 1, 2, . . . , n, thenw = w(t1, t2,· · · , tm) is diff. and

∂w

∂t1 = ∂w

∂x1

∂x1

∂t1 + ∂w

∂x2

∂x2

∂t1 +· · · + ∂w

∂xn

∂xn

∂t1,

∂w

∂t2 = ∂w

∂x1

∂x1

∂t2 + ∂w

∂x2

∂x2

∂t2 +· · · + ∂w

∂xn

∂xn

∂t2, ...

∂w = ∂w ∂x1

+ ∂w ∂x2

+· · · + ∂w ∂xn

.

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 74/160

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Implicit Differentiation Revisited

Thm 11.8 (隱微分的公式)

Suppose that all first partial derivatives of F exist.

(1) If the equation F(x, y) = 0 defines y implicitly as a diff.

function of x, then dy

dx = −Fx

Fy with Fy̸= 0.

(2) If the equation F(x, y, z) = 0 defines z implicitly as a diff.

function of x and y, then

∂z

∂x = −Fx

Fz , ∂z

∂y = −Fy

Fz with Fz̸= 0.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 76/160

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 78/160

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Section 11.5

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 80/160

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Functions of Two Variables

Def. (雙自變量函數的方向導數)

Let f(x, y) be a real-valued function defined on D with (x0, y0)∈ D.

The directional derivative of f at (x0, y0) in the direction of aunit vector u = ai + bjis defined by

Duf(x0, y0)≡ lim

t→0

f(x0+ ta, y0+ tb)− f(x0, y0)

t .

(函數 f 在 (x0, y0) 處沿著單位向量 u 的方向導數)

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方向導數的示意圖 (1/2)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 82/160

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方向導數的示意圖 (2/2)

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Thm 11.9 (方向導數的計算公式)

Let f(x, y) bediff. on an open region D with (x0, y0)∈ D. If u = ai + bj is a unit vector inR2, then

Duf(x0, y0) = a· fx(x0, y0) + b· fy(x0, y0).

pf: Since f is diff. on D, g(t)

≡ f(x0+ ta, y0+ tb) is also diff. for suﬀiciently small t̸= 0. It follows from Chain Rule (Thm 11.6) that Duf(x0, y0) = lim

t→0

g(t)− g(0)

t = g (0)

= a· fx(x0+ ta, y0+ tb)

t=0+ b· fy(x0+ ta, y0+ tb)

t=0

= a· f(x0, y0) + b· fy(x0, y0).

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 84/160

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Example 1 的示意圖

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 86/160

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Def. (梯度向量的定義)

Let f(x, y) be a function of x and y s.t. fx and fy both exist. The gradient (vector) of f at (x, y) is

∇f(x, y) ≡ fx(x, y)i + fy(x, y)j∈ R2.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 88/160

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梯度向量的示意圖

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 11, Calculus B 90/160

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Thm 11.10 (方向導數的等價公式)

If f(x, y) is diff. on an open region D⊆ R2 and u = ai + bj is a unit vector inR2, then

Duf(x0, y0) =∇f(x0, y0)• u ∀ (x0, y0)∈ D.

pf: The proof follows immediately from Thm 11.9, since we have

Duf(x0, y0) = a·fx(x0, y0)+b·fy(x0, y0) =∇f(x0, y0)•u ∀ (x0, y0)∈ D.

11 　Oleg Benesch, Inventing the Way of the Samurai: Nationalism, Internationalism, and Bushidō in Modern Japan (Oxford: Oxford University Press, 2014), pp..

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