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 : R n → R Let f : R n → R and X 0 ∈ R n . Also let U ∈ R n with � U � = 1 . Then the limit, when exists, f ( X 0 + tU ) − f ( X 0 ) = d D U f ( X 0 ) := lim d t f ( X 0 + tU ) | t =0 , t t → 0 = rate of change of f at X 0 in the direction of U , is called directional derivative of f at X 0 in the direction of U . ❼ D U f ( X 0 ) , also denoted by ∂ f ∂ U ( X 0 ) , is the rate of change of f at X 0 in the direction U . R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Examples 1. Consider f : R 2 → R given by f ( x , y ) := � | xy | . Then ∂ 1 f (0 , 0) = 0 = ∂ 2 f (0 , 0) and f is continuous at (0 , 0). However, D U f (0 , 0) does NOT exist for u 1 u 2 � = 0 . R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Examples 1. Consider f : R 2 → R given by f ( x , y ) := � | xy | . Then ∂ 1 f (0 , 0) = 0 = ∂ 2 f (0 , 0) and f is continuous at (0 , 0). However, D U f (0 , 0) does NOT exist for u 1 u 2 � = 0 . 2. Consider f : R 2 → R given by f (0 , 0) = 0 and x 2 y f ( x , y ) := x 4 + y 2 if ( x , y ) � = (0 , 0) . Then f is NOT continuous at (0 , 0) , ∂ 1 f (0 , 0) = 0 = ∂ 2 f (0 , 0) and D U f (0 , 0) exits for all U . Further, D U f (0 , 0) = u 2 1 / u 2 for u 1 u 2 � = 0 . R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Examples 1. Consider f : R 2 → R given by f ( x , y ) := � | xy | . Then ∂ 1 f (0 , 0) = 0 = ∂ 2 f (0 , 0) and f is continuous at (0 , 0). However, D U f (0 , 0) does NOT exist for u 1 u 2 � = 0 . 2. Consider f : R 2 → R given by f (0 , 0) = 0 and x 2 y f ( x , y ) := x 4 + y 2 if ( x , y ) � = (0 , 0) . Then f is NOT continuous at (0 , 0) , ∂ 1 f (0 , 0) = 0 = ∂ 2 f (0 , 0) and D U f (0 , 0) exits for all U . Further, D U f (0 , 0) = u 2 1 / u 2 for u 1 u 2 � = 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 : R n → R and X 0 ∈ R n . Also let U ∈ R n with � U � = 1 . Then ❼ Sum, product and chain rule similar to those of ∂ i f ( X 0 ) hold for D U f ( X 0 ) . ❼ If D U f ( X 0 ) exists for all nonzero U ∈ R n then f is said to have directional derivatives in all directions. ❼ Obviously ∂ i f ( X 0 ) = D e i f ( X 0 ) . Hence D U f ( X 0 ) exists in all directions U ⇒ ∂ i f ( X 0 ) exist for i = 1 , 2 , . . . , n . R. Barman & S. Bora MA-102 (2017)

❼ ❼ ❼ Directional derivatives Implications of differentiability Differential Calculus for f : R n → R Question: Let f : R n → R . What does it mean to say that f is differentiable? R. Barman & S. Bora MA-102 (2017)

❼ ❼ ❼ Directional derivatives Implications of differentiability Differential Calculus for f : R n → R Question: Let f : R n → R . What does it mean to say that f is differentiable? Task: Define differentiability of f at X 0 ∈ R n and determine the derivative D f ( X 0 ) . R. Barman & S. Bora MA-102 (2017)

❼ ❼ Directional derivatives Implications of differentiability Differential Calculus for f : R n → R Question: Let f : R n → R . What does it mean to say that f is differentiable? Task: Define differentiability of f at X 0 ∈ R n and determine the derivative D f ( X 0 ) . Wish List: ❼ f is differentiable at X 0 ⇒ f is continuous at X 0 . R. Barman & S. Bora MA-102 (2017)

❼ Directional derivatives Implications of differentiability Differential Calculus for f : R n → R Question: Let f : R n → R . What does it mean to say that f is differentiable? Task: Define differentiability of f at X 0 ∈ R n and determine the derivative D f ( X 0 ) . Wish List: ❼ f is differentiable at X 0 ⇒ f is continuous at X 0 . ❼ Sum, product and chain rules hold for D f ( X 0 ) . R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Differential Calculus for f : R n → R Question: Let f : R n → R . What does it mean to say that f is differentiable? Task: Define differentiability of f at X 0 ∈ R n and determine the derivative D f ( X 0 ) . Wish List: ❼ f is differentiable at X 0 ⇒ f is continuous at X 0 . ❼ Sum, product and chain rules hold for D f ( X 0 ) . 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 f ( a + h ) − f ( a ) α = lim . h h → 0 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 f ( a + h ) − f ( a ) α = lim . h h → 0 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 : R 2 → R Differentiability of f : R 2 → R : Let D be an open subset of R 2 . Definition 1: A function f : D → R is differentiable at a point ( a , b ) ∈ D if there exist ( α 1 , α 2 ) ∈ R 2 and ε 1 = ε 1 ( h , k ) , ε 2 = ε 2 ( h , k ) such that f ( a + h , b + k ) − f ( a , b ) = h · α 1 + k · α 2 + h ε 1 + k ε 2 , where ε 1 , ε 2 → 0 as ( h , k ) → (0 , 0). R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Differentiability of f : R 2 → R Differentiability of f : R 2 → R : Let D be an open subset of R 2 . Definition 1: A function f : D → R is differentiable at a point ( a , b ) ∈ D if there exist ( α 1 , α 2 ) ∈ R 2 and ε 1 = ε 1 ( h , k ) , ε 2 = ε 2 ( h , k ) such that f ( a + h , b + k ) − f ( a , b ) = h · α 1 + k · α 2 + h ε 1 + k ε 2 , where ε 1 , ε 2 → 0 as ( h , k ) → (0 , 0). We call the pair ( α 1 , α 2 ) the total derivative of f at ( a , b ). R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Differentiability of f : R 2 → R Fact: If ( α 1 , α 2 ) 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 α 1 = f x ( a , b ) and α 2 = f y ( a , b ), respectively. R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Differentiability of f : R 2 → R Fact: If ( α 1 , α 2 ) 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 α 1 = f x ( a , b ) and α 2 = f y ( a , b ), respectively. Example 1: The following function is NOT differentiable at (0 , 0). � x sin 1 x + y sin 1 y , xy � = 0 f ( x , y ) = 0 xy = 0 . x 2 + y 2 implies that f is � Solution: | f ( x , y ) | ≤ | x | + | y | ≤ 2 continuous at (0 , 0). R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Differentiability of f : R 2 → R We have ❼ f ( h , 0) − f (0 , 0) f x (0 , 0) = lim = 0 . h h → 0 ❼ f (0 , k ) − f (0 , 0) f y (0 , k ) = lim = 0 . k 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 : R 2 → 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 ε 1 , ε 2 such that f ( h , k ) = ε 1 h + ε 2 k . R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Differentiability of f : R 2 → 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 ε 1 , ε 2 such that f ( h , k ) = ε 1 h + ε 2 k . Taking h = k , we get | h | h = ε 1 + ε 2 . R. Barman & S. Bora MA-102 (2017)

Directional derivatives Implications of differentiability Differentiability of f : R 2 → 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 ε 1 , ε 2 such that f ( h , k ) = ε 1 h + ε 2 k . Taking h = k , we get | h | h = ε 1 + ε 2 . This implies that ( ε 1 + ε 2 ) �→ 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 : R 2 → R Recall that f : ( c , d ) → R is differentiable at a ∈ ( c , d ) if there exists α ∈ R such that | f ( a + h ) − f ( a ) − α h | lim = 0 . | h | h → 0 R. Barman & S. Bora MA-102 (2017)