MA102: Multivariable Calculus Rupam Barman and Shreemayee Bora - - PowerPoint PPT Presentation

Directional derivatives Implications of differentiability

MA102: Multivariable Calculus

Rupam Barman and Shreemayee Bora Department of Mathematics IIT Guwahati

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Directional derivatives of f : Rn → R

Let f : Rn → R and X0 ∈ Rn. Also let U ∈ Rn with U = 1. Then the limit, when exists, DUf (X0) := lim

t→0

f (X0 + tU) − f (X0) t = d dt f (X0 + tU)|t=0, = rate of change of f at X0 in the direction of U, is called directional derivative of f at X0 in the direction of U.

❼ DUf (X0), also denoted by ∂f

∂U (X0), is the rate of change

• f f at X0 in the direction U.
• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Examples

• 1. Consider f : R2 → R given by f (x, y) :=
• |xy|. Then

∂1f (0, 0) = 0 = ∂2f (0, 0) and f is continuous at (0, 0). However, DUf (0, 0) does NOT exist for u1u2 = 0.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Examples

• 1. Consider f : R2 → R given by f (x, y) :=
• |xy|. Then

∂1f (0, 0) = 0 = ∂2f (0, 0) and f is continuous at (0, 0). However, DUf (0, 0) does NOT exist for u1u2 = 0.

• 2. Consider f : R2 → R given by f (0, 0) = 0 and

f (x, y) := x2y x4 + y 2 if (x, y) = (0, 0). Then f is NOT continuous at (0, 0), ∂1f (0, 0) = 0 = ∂2f (0, 0) and DUf (0, 0) exits for all U. Further, DUf (0, 0) = u2

1/u2 for

u1u2 = 0.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Examples

• 1. Consider f : R2 → R given by f (x, y) :=
• |xy|. Then

∂1f (0, 0) = 0 = ∂2f (0, 0) and f is continuous at (0, 0). However, DUf (0, 0) does NOT exist for u1u2 = 0.

• 2. Consider f : R2 → R given by f (0, 0) = 0 and

f (x, y) := x2y x4 + y 2 if (x, y) = (0, 0). Then f is NOT continuous at (0, 0), ∂1f (0, 0) = 0 = ∂2f (0, 0) and DUf (0, 0) exits for all U. Further, DUf (0, 0) = u2

1/u2 for

u1u2 = 0. Moral: Partial derivatives ⇒ Directional derivative ⇒ Continuity ⇒ Directional derivative.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Properties of directional derivatives

Let f : Rn → R and X0 ∈ Rn. Also let U ∈ Rn with U = 1. Then

❼ Sum, product and chain rule similar to those of ∂if (X0)

hold for DUf (X0).

❼ If DUf (X0) exists for all nonzero U ∈ Rn then f is said to

have directional derivatives in all directions.

❼ Obviously ∂if (X0) = Deif (X0). Hence DUf (X0) exists in

all directions U ⇒ ∂if (X0) exist for i = 1, 2, . . . , n.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differential Calculus for f : Rn → R

Question: Let f : Rn → R. What does it mean to say that f is differentiable?

❼ ❼ ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differential Calculus for f : Rn → R

Question: Let f : Rn → R. What does it mean to say that f is differentiable? Task: Define differentiability of f at X0 ∈ Rn and determine the derivative Df (X0).

❼ ❼ ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differential Calculus for f : Rn → R

Question: Let f : Rn → R. What does it mean to say that f is differentiable? Task: Define differentiability of f at X0 ∈ Rn and determine the derivative Df (X0). Wish List:

❼ f is differentiable at X0 ⇒ f is continuous at X0. ❼ ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differential Calculus for f : Rn → R

Question: Let f : Rn → R. What does it mean to say that f is differentiable? Task: Define differentiability of f at X0 ∈ Rn and determine the derivative Df (X0). Wish List:

❼ f is differentiable at X0 ⇒ f is continuous at X0. ❼ Sum, product and chain rules hold for Df (X0). ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differential Calculus for f : Rn → R

Question: Let f : Rn → R. What does it mean to say that f is differentiable? Task: Define differentiability of f at X0 ∈ Rn and determine the derivative Df (X0). Wish List:

❼ f is differentiable at X0 ⇒ f is continuous at X0. ❼ Sum, product and chain rules hold for Df (X0). ❼ Mean Value Theorem and Taylor’s Theorem hold for f .

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : (c, d) ⊂ R → R

• 1. f is differentiable at a ∈ (c, d) if there exists α ∈ R such

that α = lim

h→0

f (a + h) − f (a) h .

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : (c, d) ⊂ R → R

• 1. f is differentiable at a ∈ (c, d) if there exists α ∈ R such

that α = lim

h→0

f (a + h) − f (a) h . In other words, f is differentiable at a if there exists ε = ε(h) and a constant α satisfying f (a + h) − f (a) = h · α + h · ε such that ε → 0 as h → 0.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

Differentiability of f : R2 → R: Let D be an open subset of R2. Definition 1: A function f : D → R is differentiable at a point (a, b) ∈ D if there exist (α1, α2) ∈ R2 and ε1 = ε1(h, k), ε2 = ε2(h, k) such that f (a + h, b + k) − f (a, b) = h · α1 + k · α2 + hε1 + kε2, where ε1, ε2 → 0 as (h, k) → (0, 0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

Differentiability of f : R2 → R: Let D be an open subset of R2. Definition 1: A function f : D → R is differentiable at a point (a, b) ∈ D if there exist (α1, α2) ∈ R2 and ε1 = ε1(h, k), ε2 = ε2(h, k) such that f (a + h, b + k) − f (a, b) = h · α1 + k · α2 + hε1 + kε2, where ε1, ε2 → 0 as (h, k) → (0, 0). We call the pair (α1, α2) the total derivative of f at (a, b).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

Fact: If (α1, α2) is the total derivative of f at (a, b), then letting (h, k) approach (0, 0) along the x-axis and y-axis, we have α1 = fx(a, b) and α2 = fy(a, b), respectively.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

Fact: If (α1, α2) is the total derivative of f at (a, b), then letting (h, k) approach (0, 0) along the x-axis and y-axis, we have α1 = fx(a, b) and α2 = fy(a, b), respectively. Example 1: The following function is NOT differentiable at (0, 0). f (x, y) =

• x sin 1

x + y sin 1 y ,

xy = 0 xy = 0. Solution: |f (x, y)| ≤ |x| + |y| ≤ 2

• x2 + y 2 implies that f is

continuous at (0, 0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

We have

❼

fx(0, 0) = lim

h→0

f (h, 0) − f (0, 0) h = 0.

❼

fy(0, k) = lim

k→0

f (0, k) − f (0, 0) k = 0. If f is differentiable at (0, 0), then we can deduce that sin 1

h → 0 as h → 0, which is a contradiction.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

Example 2: The function f defined by f (x, y) =

• |xy| is

NOT differentiable at the origin. Solution: If f is differentiable at (0, 0), then there exist ε1, ε2 such that f (h, k) = ε1h + ε2k.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

Example 2: The function f defined by f (x, y) =

• |xy| is

NOT differentiable at the origin. Solution: If f is differentiable at (0, 0), then there exist ε1, ε2 such that f (h, k) = ε1h + ε2k. Taking h = k, we get |h| h = ε1 + ε2.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

Example 2: The function f defined by f (x, y) =

• |xy| is

NOT differentiable at the origin. Solution: If f is differentiable at (0, 0), then there exist ε1, ε2 such that f (h, k) = ε1h + ε2k. Taking h = k, we get |h| h = ε1 + ε2. This implies that (ε1 + ε2) → 0 as h → 0 along the line h = k.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Another definition of differentiability of f : R2 → R

Recall that f : (c, d) → R is differentiable at a ∈ (c, d) if there exists α ∈ R such that lim

h→0

|f (a + h) − f (a) − αh| |h| = 0.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Another definition of differentiability of f : R2 → R

Recall that f : (c, d) → R is differentiable at a ∈ (c, d) if there exists α ∈ R such that lim

h→0

|f (a + h) − f (a) − αh| |h| = 0. Definition 2: Let D be open in R2. Then f : D → R is differentiable at a point (a, b) ∈ D if lim

(h,k)→(0,0)

|f (a + h, b + k) − f (a, b) − fx(a, b)h − fy(a, b)k| √ h2 + k2 = 0.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Another definition of differentiability of f : R2 → R

We use the following notations

• 1. ∆f = f (a + h, b + k) − f (a, b), the total variation of f
• 2. df = hfx(a, b) + kfy(a, b), the total differential of f .
• 3. ρ =

√ h2 + k2 Then Definition 2 takes the form: f is differentiable at (a, b) if lim

ρ→0

∆f − df ρ = 0. Theorem Definition 1 and Definition 2 of differentiability of f : D ⊆ R2 → R are equivalent.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

Example 3: Let f : R2 → R be defined by f (x, y) =

• x2y2

x2+y2,

(x, y) ≡ (0, 0) 0, x = y = 0. Then, f is differentiable at (0, 0). Solution: We have fx(0, 0) = 0, fy(0, 0) = 0. By taking h = ρ cos θ, k = ρ sin θ, we obtain ∆f − df ρ = h2k2 ρ3 = ρ4 cos2 θ sin2 θ ρ3 = ρ cos2 θ sin2 θ. This implies that

• ∆f −df

ρ

• ≤ ρ → 0 as ρ → 0. Hence, f is

differentiable at (0, 0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : R2 → R

Example 4: Let f : R2 → R be defined by f (x, y) =

• x2y

x2+y2,

(x, y) ≡ 0 0, x = y = 0 . Then, f is NOT differentiable at (0, 0). Solution: We have fx(0, 0) = 0, fy(0, 0) = 0. By taking h = ρ cos θ, k = ρ sin θ, we obtain ∆f − df ρ = h2k ρ3 = ρ3 cos2 θ sin θ ρ3 = cos2 θ sin θ. The limit does not exist as ρ → 0. Therefore, f is NOT differentiable at (0, 0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Sufficient condition for differentiability

We now prove a sufficient condition for differentiability of f : D ⊂ R2 → R. Theorem Suppose that one of the partial derivatives fx and fy exists at (a, b) and the other is continuous at (a, b). Then, f is differentiable at (a, b). Proof: Board. Remark: Continuity of fx or fy at (a, b) is not necessary for differentiability of f at (a, b).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Sufficient condition for differentiability

Example 5: Let f : R2 → R be defined by f (x, y) =

• x3 sin 1

x2 + y 3 sin 1 y2

xy = 0 xy = 0. Then f is differentiable at (0, 0), but fx and fy are not continuous at (0, 0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Sufficient condition for differentiability

Example 5: Let f : R2 → R be defined by f (x, y) =

• x3 sin 1

x2 + y 3 sin 1 y2

xy = 0 xy = 0. Then f is differentiable at (0, 0), but fx and fy are not continuous at (0, 0). Solution: We have fx(x, y) =

• 3x2 sin 1

x2 − 2 cos 1 x2

xy = 0 x = y = 0 and fy(x, y) =

• 3y 2 sin 1

y2 − 2 cos 1 y2

xy = 0 x = y = 0.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Sufficient condition for differentiability

Clearly, partial derivatives fx and fy are not continuous at (0, 0). However, f (0 + h, 0 + k) − f (0, 0) = f (h, k) = h3 sin 1 h2 + k3 sin 1 k2 Then, f (h, k) = 0 + 0 + ε1h + ε2k, where ε1 = h2 sin 1

h2 and ε2 = k2 sin 1 k2.

It is easy to check that ε1, ε2 → 0 as (h, k) → (0, 0). Hence, f is differentiable at (0, 0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Sufficient condition for differentiability

Remark: There are functions for which directional derivatives exist in any direction, but the function is not differentiable. Example 6: The function f (x, y) =

• y

|y|

• x2 + y 2

y = 0 y = 0 is not differentiable, but all the directional derivatives exist.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : A ⊂ Rn → R

A function f : A → R is said to be differentiable at X0 ∈ A if there exists α = (α1, . . . , αn) ∈ Rn such that lim

H→0

|f (X0 + H) − f (X0) − α • H| H = 0. If α = (α1, α2, . . . , αn) and H = (h1, h2, . . . , hn), then α • H = α, H =

n

• i=1

αi · hi. Fact: If f is differentiable at X0, then α1 = ∂1f (X0), . . ., αn = ∂nf (X0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : A ⊂ Rn → R

Differentiability and linear maps f is differentiable at a ∈ (c, d) if there exists a linear map L : R → R such that lim

h→0

|f (a + h) − f (a) − L(h)| |h| = 0. Let A ⊂ Rn be open. Then f : A ⊂ Rn → R is differentiable at X0 ∈ A if there exists a linear map L : Rn → R such that lim

H→0

|f (X0 + H) − f (X0) − L(H)| H = 0. (∗) The linear map L is called the derivative of f at X0 and is denoted by Df (X0), that is, L = Df (X0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : A ⊂ Rn → R

Fact: If L : Rn → R is linear, then L(X) = P • X = X, P for some P ∈ Rn. Theorem: If f : A ⊂ Rn → R is differentiable at X0 ∈ A then partial derivatives ∂1f (X0), . . . , ∂nf (X0) exist and the derivative Df (X0) : Rn → R is given by Df (X0)(X) = ∇f (X0) • X = X, ∇f (X0), where ∇f (X0) := (∂1f (X0), . . . , ∂nf (X0)) is the gradient of f at X0.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : A ⊂ Rn → R

Fact: If L : Rn → R is linear, then L(X) = P • X = X, P for some P ∈ Rn. Theorem: If f : A ⊂ Rn → R is differentiable at X0 ∈ A then partial derivatives ∂1f (X0), . . . , ∂nf (X0) exist and the derivative Df (X0) : Rn → R is given by Df (X0)(X) = ∇f (X0) • X = X, ∇f (X0), where ∇f (X0) := (∂1f (X0), . . . , ∂nf (X0)) is the gradient of f at X0. Proof. Considering H := tei for t ∈ R in (∗) and letting t → 0, it ready follows that ∂if (X0) exists for all i = 1, 2, . . . , n.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Differentiability of f : A ⊂ Rn → R

Example: Consider f : R2 → R given by f (0, 0) = 0 and f (x, y) := xy x2 − y 2 x2 + y 2 if (x, y) = (0, 0). Then

❼ f is continuous at (0, 0) and ∇f (0, 0) = (0, 0). ❼

|f (h, k) − f (0, 0) − ∇f (0, 0) • (h, k)| (h, k) ≤ |hk| (h, k) → 0. Hence, f is differentiable at (0, 0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Affine approximation

Let X0 ∈ Rn. Define the error function E : Rn \ {0} → R by E(H) := f (X0 + H) − f (X0) − ∇f (X0) • H H .

❼ ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Affine approximation

Let X0 ∈ Rn. Define the error function E : Rn \ {0} → R by E(H) := f (X0 + H) − f (X0) − ∇f (X0) • H H .

❼ Then f is differentiable at X0 if and only if

f (X0 + H) = f (X0) + ∇f (X0) • H + E(H)H and E(H) → 0 as H → 0.

❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Affine approximation

Let X0 ∈ Rn. Define the error function E : Rn \ {0} → R by E(H) := f (X0 + H) − f (X0) − ∇f (X0) • H H .

❼ Then f is differentiable at X0 if and only if

f (X0 + H) = f (X0) + ∇f (X0) • H + E(H)H and E(H) → 0 as H → 0.

❼ The affine function y = f (X0) + ∇f (X0) • H approximates

f (X0 + H) for small H ⇐ ⇒ f is differentiable at X0.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Geometric interpretation

Let f : Rn → R be differentiable at X0 ∈ Rn. Then y = f (X0) + ∇f (X0) • X represents

❼ ❼ ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Geometric interpretation

Let f : Rn → R be differentiable at X0 ∈ Rn. Then y = f (X0) + ∇f (X0) • X represents

❼ For n = 1 : a line y = f (a) + f ′(a)x passing through

(0, f (a)) ∈ R2 that approximates f (a + x).

❼ ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Geometric interpretation

Let f : Rn → R be differentiable at X0 ∈ Rn. Then y = f (X0) + ∇f (X0) • X represents

❼ For n = 1 : a line y = f (a) + f ′(a)x passing through

(0, f (a)) ∈ R2 that approximates f (a + x).

❼ For n = 2 : a plane z = f (a, b) + fx(a, b)x + fy(a, b)y

passing through (0, 0, f (a, b)) ∈ R3 that approximates f (a + x, b + y).

❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Geometric interpretation

Let f : Rn → R be differentiable at X0 ∈ Rn. Then y = f (X0) + ∇f (X0) • X represents

❼ For n = 1 : a line y = f (a) + f ′(a)x passing through

(0, f (a)) ∈ R2 that approximates f (a + x).

❼ For n = 2 : a plane z = f (a, b) + fx(a, b)x + fy(a, b)y

passing through (0, 0, f (a, b)) ∈ R3 that approximates f (a + x, b + y).

❼ For n ≥ 3 : a hyperplane

y = f (X0) + ∂1f (X0)x1 + · · · + ∂nf (X0)xn passing through (0, 0, . . . , 0, f (X0)) ∈ Rn+1 that approximates f (X0 + X).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Implications of differentiability

Theorem: Let f : Rn → R and X0 ∈ Rn.

❼ If f is differentiable at X0 then f is continuous at X0. ❼ If f is differentiable at X0 then directional derivatives exist

for all U ∈ Rn and DUf (X0) = ∇f (X0) • U.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Implications of differentiability

Theorem: Let f : Rn → R and X0 ∈ Rn.

❼ If f is differentiable at X0 then f is continuous at X0. ❼ If f is differentiable at X0 then directional derivatives exist

for all U ∈ Rn and DUf (X0) = ∇f (X0) • U. Proof: Continuity follows from f (X0 + H) = f (X0) + ∇f (X0) • H + E(H)H and the fact that E(H) → 0 as H → 0. Second part: Put H = tU. Note that H = tU = |t|.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Example

Consider f : R2 → R given by f (0, 0) = 0 and f (x, y) := x2y x4 + y 2 if (x, y) = (0, 0). Then

❼ ❼ ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Example

Consider f : R2 → R given by f (0, 0) = 0 and f (x, y) := x2y x4 + y 2 if (x, y) = (0, 0). Then

❼ f is NOT continuous at (0, 0) ⇒ f is not differentiable at

(0, 0).

❼ ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Example

Consider f : R2 → R given by f (0, 0) = 0 and f (x, y) := x2y x4 + y 2 if (x, y) = (0, 0). Then

❼ f is NOT continuous at (0, 0) ⇒ f is not differentiable at

(0, 0).

❼ DUf (0, 0) exists for all U ∈ R2 and ∇f (0, 0) = (0, 0). ❼

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Example

Consider f : R2 → R given by f (0, 0) = 0 and f (x, y) := x2y x4 + y 2 if (x, y) = (0, 0). Then

❼ f is NOT continuous at (0, 0) ⇒ f is not differentiable at

(0, 0).

❼ DUf (0, 0) exists for all U ∈ R2 and ∇f (0, 0) = (0, 0). ❼ For U = (u1, u2) such that u1u2 = 0, we have

DUf (0, 0) = u2

1/u2 = ∇f (0, 0) • U.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Example

Consider f : R2 → R given by f (0, 0) = 0 and f (x, y) := x2y x4 + y 2 if (x, y) = (0, 0). Then

❼ f is NOT continuous at (0, 0) ⇒ f is not differentiable at

(0, 0).

❼ DUf (0, 0) exists for all U ∈ R2 and ∇f (0, 0) = (0, 0). ❼ For U = (u1, u2) such that u1u2 = 0, we have

DUf (0, 0) = u2

1/u2 = ∇f (0, 0) • U.

Moral: The equality DUf (X0) = ∇f (X0) • U may not hold if f is NOT differentiable at X0.

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

A geometric interpretation of gradient

Let D ⊆ R2 and let (a, b) be an interior point of D. Let f : D → R be differentiable at (a, b) and suppose ∇f (a, b) = (0, 0). DUf (a, b) = ∇f (a, b) • U = ∇f (a, b) cos θ, where θ is the angle between ∇f (a, b) and U.

• 1. DUf (a, b) is maximum when cos θ = 1. Thus, near (a, b),

U = ∇f (a, b)/∇f (a, b) is the direction in which f increases most rapidly.

• 2. DUf (a, b) is minimum when cos θ = −1. Thus, near

(a, b), U = −∇f (a, b)/∇f (a, b) is the direction in which f decreases most rapidly.

• 3. DUf (a, b) = 0 when cos θ = 0. Thus, near (a, b),

U = ±(fy(a, b), −fx(a, b))/∇f (a, b) are the directions

• f no change in f .
• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Properties of derivative

Fact: Let f , g : Rn → R be differentiable at X0 ∈ Rn. Then

❼ D(f + αg)(X0) = Df (X0) + αDg(X0). ❼ D(fg)(X0) = Df (X0)g(X0) + f (X0)Dg(X0).

• R. Barman & S. Bora

MA-102 (2017)

Directional derivatives Implications of differentiability

Properties of derivative

Fact: Let f , g : Rn → R be differentiable at X0 ∈ Rn. Then

❼ D(f + αg)(X0) = Df (X0) + αDg(X0). ❼ D(fg)(X0) = Df (X0)g(X0) + f (X0)Dg(X0).

Theorem: Let f : Rn → R and X0 ∈ Rn. If ∂if (X0) exists for i = 1, 2, . . . , n, and are continuous on B(X0, ε) for some ε > 0, then f is differentiable at X0.

• R. Barman & S. Bora

MA-102 (2017)