Math 375 - 11/28

Open Balls and Open Sets • Open Interval • Closed Interval • Interior Point ○ E⊆Rn is a subset ○ p∈E is an interior point if there is an r 0 ○ such that B_r (p)⊆E ○ where B_r (p) is the open disc of radius centered at p ○ B_r (p)={x∈Rn│‖x−p‖ r} • Koch s Snowflake • Open Sets ○ E⊆Rn is open if all x∈E are interior points in E • Example • Boundary Point ○ A point p∈Rn is a boundary point for E if for every r 0 ○ B_r (p) contains x,y with x∈E and y∉E Limits and Continuity • Limits ○ lim_(x→a)⁡f(x)=L⟺lim_(‖x−a‖→0)⁡‖f(x)−L‖=0 ○ If x→a, then f(x)→L • Properties ○ If f(x)→L∈Rm,g(x)→M∈Rm, when x→a, then ○ f(x)±g(x)→L±M ○ f(x)⋅g(x)→L⋅M ○ ‖f(x)‖→‖L‖ ○ f(x)/g(x) →L/M ○ (only when n=1, f(x),g(x)∈Rn) • Graph ○ Graph of f={(x,y,z)|z=f(x,y)} • Continuity ○ f:Rn→Rm is continuous at a∈Rn ○ if lim_(x→a)⁡f(x)=f(a) • Continuous Function Example ○ f(x_1,…,x_n )=x_k ○ f:Rn→R • Properties ○ If f,g is continuous ○ Then f±g, fg, f/g (g(a)≠0) are continuous • Example ○ f:R2→R ○ f(x,y)={■8(xy/(x^2+y2 )&(x,y)≠(0,0)@0&x=y=0)┤ ○ f is continuous at all point except (0,0) ○ Let (x,y)→(0,0) along a straight line with angle θ ○ x=rcos⁡θ, y=rsin⁡θ ○ f(x,y)=xy/(x^2+y2 )=(r^2 sin⁡θ cos⁡θ)/(r^2 cos^2⁡θ+r2 sin⁡θ )=cos⁡θ sin⁡θ ○ Note that f(x,y) does not depend on r ○ lim_█((x,y)→(0,0)@along line@with angle θ)⁡f(x,y)=sin⁡θ cos⁡θ ○ When θ=π/2⇒f=0, when θ=π/4⇒f=1/2⋯ ○ Therefore we get the counter plot near origin ○ And the graph near 0 Derivative • Directional Derivative ○ D_hf(x)=∇_hf(x)= f^′ (x;h⃗ )=df_x⋅h ○ =lim_(t→0)⁡〖(f(x+th⃗ )−f(x))/t〗 ○ =[d/dt f(x+th⃗ )]_(t=0) • Example ○ f:Rn→R ○ f(x)=‖x‖^2 ○ f^′ (x;h⃗ ) ○ =[d/dt f(x+th]_(t=0) ○ =[d/dt ‖x+th^2 ]_(t=0) ○ =[d/dt (h2 t^2+(2hx)t+x2 )]_(t=0) ○ =[2h2 t+2hx]_(t=0) ○ =2x⋅h • Partial Derivative • Total Derivative - • •