A question about covariant derivative (find $D_{v_p}W$)

Question

the quest. on my book is $(y_1,y_2,y_3)\quad R^3\quad coordinate \quad system$ . if $W=y_1y_2^2\frac{\partial}{\partial y_1}+(y_3-y_2^2)\frac{\partial}{\partial y_2}+3y_1\frac{\partial}{\partial y_3}$

$p=(-1,2,0)$ and $v_p=(3,0,1)_p$ then find $D_{v_p}W$

I did:

Let $a:I\mapsto R^3$ be conjoint curve(not sure if it's a right call) to $v_p$, then covariant derivative is $D_{v_p}W=(Woa)'(0)$. this $a$ curve can be $a(t)=(3t,t^22,t)$ because $a'(0)=v_p$

so $(Woa)(t)=(y_1y_2^2)(a(t))\frac{\partial}{\partial y_1}+(y_3-y_2^2)(a(t))\frac{\partial}{\partial y_2}+(3y_1)(a(t))\frac{\partial}{\partial y_3}$

$=3tt^4\frac{\partial}{\partial y_1}+(t-t^4)\frac{\partial}{\partial y_2}+9t\frac{\partial}{\partial y_3}$

$(Woa)'(t)=15t^5\frac{\partial}{\partial y_1}+(1-4t^3)\frac{\partial}{\partial y_2}+9\frac{\partial}{\partial y_3}$

$(Woa)'(0)=15.0\frac{\partial}{\partial y_1}+(1-0)\frac{\partial}{\partial y_2}+9\frac{\partial}{\partial y_3}=(0,1,9)_p$

But when I try $D_{v_p}W=\sum_{j=1}^n v_p[W_j]\frac{\partial}{\partial y_j}(p) $ formula I get

$D_{v_p}W=(v_p[W_1,v_p[W_2],v_p[W_3])_p$

$v_p[W_1]=\frac{\partial W_1}{\partial y_1}|_pv_1+\frac{\partial W_1}{\partial y_2}|_pv_2+\frac{\partial W_1}{\partial y_3}|_pv_3=12$

$v_p[W_2]=0$

$v_p[W_3]=9]$

$D_{v_p}W=(12,0,9)_{(-1,2,0)}$

which is different result. Which one is correct? where did I do wrong?

Answer 1

There are several places where you go wrong:

• Your curve $a$ needs to satisfy $a(0) = p$ (as well as $a'(0) = p$)
• Also, no need to take anything more than affine expressions of $t$. Here you can take $a(t) = (3t-1, 2, t)$
• I think the first component in $(W \circ a)'(t)$ in your computation should be $15 t^4$
• I think you made a mistake in the computation of what you denote by $\nu_p [W_2]$: you should find $1$.

NB:

• Instead of "conjoint", just say that $a(t)$ is tangent to $\nu_p$ (at $t=0$)
• This method (both actually) for computing the covariant derivative of a vector field only works because you're working here with the flat covariant derivative on $\mathbb{R}^3$.