第26讲 西罗定理（选修） Jan 10, 2018 Shawn Abstract Algebra No comments yet 第27讲 西罗定理的应用（选修） Jan 10, 2018 Shawn Abstract Algebra No comments yet 第28讲 可解群 Jan 10, 2018 Shawn Abstract Algebra No comments yet Math 375 – 12/13 Dec 13, 2017 Shawn Math 375 No comments yet Math 375 – 12/12 Dec 12, 2017 Shawn Math 375 No comments yet 1 … 43 44 45 46 47 … 87
![Question 1 • Find a basis in which the matrix (■8(3&0@3&−2)) becomes diagonalized • Let A=(■8(3&0@3&−2)) • det(A−λI)=|■8(3−λ&0@3&−2−λ)|=λ^2−λ−6=0 • ⇒λ_1=3, λ_2=−2 • ⇒Λ=(■8(3&0@0&−2)) • When λ_1=3 ○ A−λI=(■8(0&0@3&−5)) ○ ⇒v_1=k(5,3), k∈R • When λ_2=−2 ○ A−λI=(■8(5&0@3&0)) ○ ⇒v_2=k(0,1), k∈R • The basis is (5,3), (0,1) Exercise 8.17 Question 8 • Find a Cartesian equation for the tangent plane • to the surface xyz=a^3 at a general point (x_0,y_0,z_0 ). ○ ∇f=(█(yz@xz@xy)) ○ H={(x,y,z)∈R3│∇f(x_0,y_0,z_0 )⋅(█(x−x_0@y−y_0@z−z_0 ))=0} ○ ={(x,y,z)∈R3│y_0 z_0 (x−x_0 )+x_0 z_0 (y−y_0 )+x_0 y_0 (z−z_0 )=0} ○ ={(x,y,z)∈R3│xy_0 z_0+x_0 yz_0+x_0 y_0 z=3x_0 y_0 z_0=3a^3 } Question 2 • Let f:R2→R smooth • Let g(x,y)=f(u,v)=f(sin〖(x)y,x^y 〗 ) • Find g_x,g_y in terms of f_u,f_v ○ Let h(x,y)=[█(u(x,y)@v(x,y) )]=[█(sin(x)y@x^y )], Then T_g=T_f∘T_ℎ ○ T_f=[■8(∂f/∂u&∂f/∂v)]=[■8(f_u&f_v )] ○ T_ℎ=[■8(∂u/∂x&∂u/∂y@∂v/∂x&∂v/∂y)]=[■8(cos(x)y&sin(x)@y⋅x^(y−1)&x^y (y ln(x)+1) )] ○ T_g=T_f∘T_ℎ=[■8(f_u&f_v )][■8(cos(x)y&sin(x)@y⋅x^(y−1)&x^y (y ln(x)+1) )]=[█(f_u cos(x)y+f_v y⋅x^(y−1)@f_u sin(x)+f_v x^y (y ln(x)+1) )]^T ○ ⇒{█(g_x=f_u cos(x)y+f_v y⋅x^(y−1)@g_y=f_u sin(x)+f_v x^y (y ln(x)+1) )┤](https://shawnzhong.com/wp-content/uploads/2017/12/img_5a31df1529981.png)
![Derivative of Vector Fields • Map ○ f:Rn→Rm ○ f(x_1,…,x_n )=[█(f_1 (x_1,…,x_n )@⋮@f_m (x_1,…,x_n ) )] • Example 1 ○ n=1, m=2 ○ f:R→Rn ○ f:t↦[█(f_1 (t)@f_2 (t) )]=[█(x(t)@y(t) )] ○ Also called parametric curve • Example 2 ○ f:Rn→R, where n≥2 ○ f is a function of n variables f(x_1,…,x_n ) • Example 3 ○ Rn→Rm, where n≥2, m≥2 ○ f:R2→R2 ○ f:(u,v)⟼(x,y) ○ Polar Coordinates § f(r,θ)=[█(x(r,θ)@y(r,θ) )]=[█(r cosθ@r sinθ )] • Example 4 ○ f(u,v)=[■8(2&3@1&−1@1&1)][█(u@v)]=[█(2u+3v@u−v@u+v)] • Definition ○ A map f:Rn→Rm is differentiable at a∈Rn if ○ f(a+y)=f(a)+T_a (y)+E(a,y)‖y‖ ○ Where T_a:Rn→Rm is a linear transformation, and lim_(y→0)E(a,y)=0 • Theorem ○ f differentiable at a⇒f continuous at a • Theorem ○ If f(x)=[█(f_1 (x_1,…,x_n )@⋮@f_m (x_1,…,x_n ) )], and f_1,…,f_m have continuous partial derivatives ○ Then f is differentiable and T_a (y)=[■8((∂f_1)/(∂x_1 )&⋯&(∂f_1)/(∂x_n )@⋮&⋱&⋮@(∂f_m)/(∂x_1 )&⋯&(∂f_m)/(∂x_n ))][█(y_1@⋮@y_n )] • Example ○ f(r,θ)=[█(r cosθ@r sinθ )] ○ The linear transformation T_a has the matrix § mat(T_a )=[■8((∂f_1)/∂r&(∂f_1)/∂θ@(∂f_2)/∂r&(∂f_2)/∂θ)]=[■8(∂x/∂r&∂x/∂θ@∂y/∂r&∂y/∂θ)]=[■8(cosθ&−r sinθ@sinθ&r cosθ )] ○ At a=(√2,π/4) § mat(T_a )=[■8(√2∕2&−1@√2∕2&1)] § T_a [█(1@0)]=[█(√2∕2@√2∕2)] § T_a [█(0@1)]=[█(−1@1)] ○ In general § mat(T_((r,θ) ) )=[■8(cosθ&−r sinθ@sinθ&r cosθ )] § T_a [█(ε@0)]=ε[█(cosθ@sinθ )] § T_a [█(0@ε)]=εr[█(−sinθ@cosθ )] • Common notations ○ T_a (v)=df_a⋅v=Df(a)⋅v=f^′ (a)⋅v • Jacobian Matrix ○ [■8((∂f_1)/(∂x_1 )&⋯&(∂f_1)/(∂x_n )@⋮&⋱&⋮@(∂f_m)/(∂x_1 )&⋯&(∂f_m)/(∂x_n ))] is called Jacobian Matrix • Chain Rule ○ Given f:Rn→Rm, g:Rm→Rk ○ Consider h(x)=g(f(x))=g∘f(x) ○ {█(f is differentiable at x@g is differentiable at f(x) )┤⇒h=g∘f is differentiable at x ○ And d(g∘f)_x=dg_f(x) ⋅df_x ○ Another notation: (g∘f)^′ (x)=g^′ (f(x))⋅f^′ (x) ○ In components § x=[█(x_1@⋮@x_n )]∈Rn § u=[█(u_1@⋮@u_m )]=[█(f_1 (x)@⋮@f_m (x) )]=[█(f_1 (x_1,…,x_n )@⋮@f_m (x_1,…,x_n ) )]∈Rm § v=[█(v_1@⋮@v_k )]=[█(g_1 (u)@⋮@g_k (u) )]=[█(g_1 (u_1,…,u_m )@⋮@g_k (u_1,…,u_m ) )]=[█(g_1 (f_1 (x_1,…,x_n ),…,f_m (x_1,…,x_n ))@⋮@g_k (f_1 (x_1,…,x_n ),…,f_m (x_1,…,x_n )) )]∈Rn § mat[(g∘f)^′ ] § =[■8((∂v_1)/x_1 &⋯&(∂v_1)/x_n @⋮&⋱&⋮@(∂v_k)/x_1 &⋯&(∂v_k)/x_n )] § =[■8(∂/(∂x_1 ) g_1 (f_1 (x_1,…,x_n ),…,f_m (x_1,…,x_n ))&⋯&∂/(∂x_n ) g_1 (f_1 (x_1,…,x_n ),…,f_m (x_1,…,x_n ))@⋮&⋱&⋮@∂/(∂x_1 ) g_k (f_1 (x_1,…,x_n ),…,f_m (x_1,…,x_n ))&⋯&∂/(∂x_n ) g_k (f_1 (x_1,…,x_n ),…,f_m (x_1,…,x_n )) )] § =[■8((∂g_1)/(∂u_1 )⋅(∂f_1)/(∂x_1 )+…+(∂g_1)/(∂u_m )⋅(∂f_m)/(∂x_1 )&⋯&(∂g_1)/(∂u_1 )⋅(∂f_1)/(∂x_n )+…+(∂g_1)/(∂u_m )⋅(∂f_m)/(∂x_n )@⋮&⋱&⋮@(∂g_k)/(∂u_1 )⋅(∂f_1)/(∂x_1 )+…+(∂g_k)/(∂u_m )⋅(∂f_m)/(∂x_1 )&⋯&(∂g_k)/(∂u_1 )⋅(∂f_1)/(∂x_n )+…+(∂g_k)/(∂u_m )⋅(∂f_m)/(∂x_n ))] § =[■8((∂g_1)/(∂u_1 )&⋯&(∂g_1)/(∂u_m )@⋮&⋱&⋮@(∂g_k)/(∂u_1 )&⋯&(∂g_k)/(∂u_m ))][■8((∂f_1)/(∂x_1 )&⋯&(∂f_1)/(∂x_n )@⋮&⋱&⋮@(∂f_m)/(∂x_1 )&⋯&(∂f_m)/(∂x_n ))] § =mat(g^′ (f(x)))mat(f^′ (x))](https://shawnzhong.com/wp-content/uploads/2017/12/img_5a308cf6ccaea.png)
