11.1 Introduction to Functions of Several Variables 11.2 Limits and Continuity

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Chapter 11 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

11.1 Introduction to Functions of Several Variables 11.2 Limits and Continuity

11.3 Partial Derivatives

11.4 Differentials and the Chain Rules 11.5 Directional Derivatives and Gradients 11.6 Tangent Planes and Normal Lines 11.7 Extrema of Functions of Two Variables 11.8 Lagrange Multipliers

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

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Section 11.1

Introduction to Functions of Several Variables

(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.

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函數 z = f(x, y) 的示意圖

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Example 2 的示意圖

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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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Example 3 的示意圖

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等高線的應用

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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.

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Example 6 的示意圖

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Section 11.2

Limits and Continuity

(極限與連續性)

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Open Disk 的示意圖 (承上頁)

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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.

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雙自變量函數的極限 (承上頁)

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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.

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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).

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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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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).

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

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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 f_x(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 f_y(x, y)≡ lim

∆y→0

f(x, y + ∆y)− f(x, y)

∆y

= 將 x 視為常數, 僅對 y 微分

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Notations

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

1

z_x= ∂z

∂x = ∂f

∂x = fx= Dxf = D₁f.

2

z_y= ∂z

∂y = ∂f

∂y = fy= Dyf = D₂f.

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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 f_x(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 f_y(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

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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 f_z(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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Higher-Order Partial Derivatives (1/2)

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

∂x, f_y= ∂f

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

f_xx= ∂²f

∂x² ≡ ∂

∂x

∂f

∂x

, f_xy= ∂²f

∂y∂x ≡ ∂

∂y

∂f

∂x

, f_yy= ∂²f

≡ ∂ ∂f

, f_yx= ∂²f

≡ ∂ ∂f .

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Higher-Order Partial Derivatives (2/2)

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

∂x³ ≡ ∂

∂x

∂²f

∂x²

, f_xxy= ∂³f

∂y∂x² ≡ ∂

∂y

∂²f

∂x²

, f_xyx= ∂³f

∂x∂y∂x ≡ ∂

∂x

∂²f

∂y∂x

, f_xyy= ∂³f

∂y²∂x ≡ ∂

∂y

∂²f

∂y∂x

, f_yyx= ∂³f

∂x∂y² ≡ ∂

∂x

∂²f

∂y²

, f_yyy= ∂³f

∂y³ ≡ ∂

∂y

∂²f

∂y²

, f_yxx= ∂³f

∂x²∂y ≡ ∂

∂x

∂²f

∂x∂y

, f_yxy= ∂³f

∂y∂x∂y ≡ ∂

∂y

∂²f

∂x∂y

.

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Thm 11.3 (Equality of Mixed Partial Derivatives)

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

f_xy(x, y) = fyx(x, y) ∀ (x, y) ∈ R.

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Section 11.4

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

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Def. (Total Differential 的定義)

(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 = f_x(x, y)dx + f_y(x, y)dy.

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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 f_x(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)

=f_x(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.

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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. Iff_x and f_y 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 = f_x(x0, y0)dx + fy(x0, y0)dy.

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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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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 f_x(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.

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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

∂s +∂w

∂y

∂s,

∂w

∂t = ∂w

∂x

∂t +∂w

∂y

∂t.

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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 x_j = 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

∂t₁,

∂w

∂t₂ = ∂w

∂x₁

∂t₂ + ∂w

∂x₂

∂t₂ +· · · + ∂w

∂x_n

∂xn

∂t₂, ...

∂w = ∂w ∂x1

+ ∂w ∂x2

+· · · + ∂w ∂xn

.

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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

F_y with F_y̸= 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

F_z , ∂z

∂y = −Fy

F_z with F_z̸= 0.

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Section 11.5

Directional Derivatives and Gradients (方向導數與梯度向量)

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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

D_uf(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)

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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

D_uf(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 D_uf(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).

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Example 1 的示意圖

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Gradient Vectors of f(x, y)

Def. (梯度向量的定義)

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

∇f(x, y) ≡ fx(x, y)i + f_y(x, y)j∈ R².

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梯度向量的示意圖

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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

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

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

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