Math 375 - 12/5

Example • f:R2→R • f(x,y)={■8(xy/(x^2+y2 )&(x,y)≠(0,0) @0&x=y=0)┤ • Counterplot • Graph • Partial Derivative ○ ∂f/∂x,∂f/∂y exist at all points in R2 including the origin ○ When (x,y)≠(0,0), we have x^2+y2≠0 § So xy/(x^2+y2 ) is differentiable as function of x ○ When (x,y)=(0,0) § ∂f/∂x=lim_(h0)⁡〖(f(x+hy)−f(x,y))/h=lim_(h0)⁡〖(⏞(f(h0) )┴0−⏞(f(0,0) )┴0)/h=lim_(h0)⁡〖0/h=0 § Similarly ∂f/∂y=0 ○ Note § Both ∂f/∂x,∂f/∂y exist, but f is not differentiable everywhere § There we need a different definition for differentiable Differentiable • Definition ○ f:Rn→R is a differentiable at a∈Rn if ○ f(a+v)=⏟(f(a)+T_a (v) )┬(linear approximation)+⏟(‖v‖E(a,v) )┬(error term) ○ Where § T_a=Rn→R is a linear map § lim_(v→0)⁡E(a,v)=0 ○ Alternative formulation § lim_(v→0)⁡〖(f(a+v)−f(a)−T_a (v))/‖v‖ 〗=0 § Where E(a,v)=(f(a+v)−f(a)−T_a (v))/‖v‖ ○ Find T_a (v) § T_a (v)=T_a (v_1 e_1+…+v_n e_n )=v_1 ⏞(T_a (e_1 ) )┴(c_1 )+…+v_n ⏞(T_a (e_n ) )┴(c_n ) § For c_1=T_a (e_1 ), choose v=he_1 § lim_(h0)⁡〖(f(a+h�_1 )−f(a)−T_a (h�_1 ))/h=0 § ⇒lim_(h0)⁡((f(a_1+ha_2,…,a_n )−f(a_1,a_2,…,a_n ))/h(T_a (h�_1 ))/h=0 § ⇒∂f/(∂x_1 ) (a_1,…,a_n )−lim_(h0)⁡〖(c_1 h/h=0 § ⇒∂f/(∂x_1 ) (a_1,…,a_n )−c_1=0 § ⇒∂f/(∂x_1 ) (a_1,…,a_n )=c_1 § Similarly c_k=∂f/(∂x_k ) (a) Total Derivative • Definition ○ If f:Rn→R is differentiable at a, then ○ T_a (v)=v_1 ∂f/(∂x_1 ) (a)+…+v_n ∂f/(∂x_n ) ○ Here, the linear map T_a:Rn→R is called the total derivative of f at a • Alternative notations ○ f^′ (a) ○ df_a ○ Df_a ○ Df(a) • Theorem ○ If f is differentiable at a then f is continuous at a • Proof ○ We want to show lim_(h0)⁡f(a+v)=f(a) § lim_(v→0)⁡〖f(a+v)−f(a)〗 § =lim_(v→0)⁡(f(a+v)−f(a)−T_a (v)+T_a (v)) § =lim_(v→0)⁡(‖v‖ (f(a+v)−f(a)−T_a (v))/‖v‖ +T_a (v)) § =lim_(v→0)⁡(‖v‖)⋅lim_(v→0)⁡((f(a+v)−f(a)−T_a (v))/‖v‖ )+lim_(v→0)⁡(T_a (v)) § =0⋅0+0 § =0 ○ Therefore lim_(h0)⁡f(a+v)=f(a) ○ Note § ‖T_a (v)‖=‖c_1 v_1+…+c_n v_n ‖=‖c⋅v‖≤‖c‖⋅‖v‖ § So ‖T_a (v)‖≤‖c‖⋅‖v‖ § If v→0, then T_a (v)→0 • Properties of differentiable functions ○ Differentiable ⇒ Continuous ○ Differentiable ⇒ Partial derivative exist • Example ○ For f(x,y)={■8(xy/(x^2+y2 )&(x,y)≠(0,0) @0&x=y=0)┤ ○ Partial derivatives exist at (0,0), but not continuous at (0,0) ○ Therefore f(x,y) is NOT DIFFERENTIABLE • Therorem ○ If ∂f/(∂x_1 ),…,∂f/(∂x_n ) exist and are continuous at a ○ Then f is differentiable at a, and the total derivative is given below ○ T_a (v)=v_1 ∂f/(∂x_1 ) (a)+…+v_n ∂f/(∂x_n ) (a) Continuity • Definition ○ f is contunuous at x=a if ○ lim_(x→0)⁡f(x)=f(a) ○ lim_(‖x−a‖→0)⁡‖f(x)−f(a)‖=0 • Example ○ For f(x,y)=xy/(x^2+y2 ), (x,y)≠(0,0) ○ ∂f/∂x,∂f/∂y exist and continuous when (x,y)≠(0,0) ○ ⇒f Fréchet differentiable everywhere except at (0,0) Chain Rule • Definition ○ g(t)=f(x_1 (t),…,x_n (t)) ○ dg/dt=df(x_1 (t),…,x_n (t))/dt=∂f/(∂x_1 ) (dx_1)/dt+…+∂f/(∂x_n ) (dx_n)/dt • Proof ○ Let a=x(t)=(x_1 (t),…,x_n (t)) ○ Let v=x(t+h−x(t), then lim_(h0)⁡v=0 ○ lim_(h0)⁡〖(f(x(t+h)−f(x(t)))/h ○ (To be continued)