Math 375 - 12/12

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 cos⁡θ@r sin⁡θ )] • 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 cos⁡θ@r sin⁡θ )] ○ 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(cos⁡θ&−r sin⁡θ@sin⁡θ&r cos⁡θ )] ○ 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(cos⁡θ&−r sin⁡θ@sin⁡θ&r cos⁡θ )] § T_a [█(ε@0)]=ε[█(cos⁡θ@sin⁡θ )] § T_a [█(0@ε)]=εr[█(−sin⁡θ@cos⁡θ )] • 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))