Write $\omega$ in terms of the standard basis $dx^i ∧dx^j$ at each point.

Question

At each point $p ∈ \mathbb R^3$, define a bilinear function $\omega_p$ on $T_p(\mathbb R^3)$ by

$$\omega_p(\textbf{a},\textbf{b})=\omega_p(\begin{bmatrix} a^1 \\ a^2 \\ a^3 \\ \end{bmatrix}, \begin{bmatrix} b^1 \\ b^2 \\ b^3 \\ \end{bmatrix})= p^3\det \begin{bmatrix} a^1&a^2 \\ b^1 &b^2 \\ \end{bmatrix} $$ , for tangent vectors $\textbf{a},\textbf{b} ∈ T_p(\mathbb R^3)$, where $p^3$ is the third component of $p = (p^1, p^2, p^3)$. Since $\omega_p$ is an alternating bilinear function on $T_p(\mathbb R^3)$, $\omega$ is a $2-$form on $\mathbb R^3$. Write $\omega$ in terms of the standard basis $dx^i ∧dx^j$ at each point.

I know that $\omega=a_{12}dx^1 ∧dx^2+a_{13}dx^1 ∧dx^3+a_{23}dx^2 ∧dx^3,$ $\textbf{a}=a^1\frac{\partial}{\partial x^1}+a^2\frac{\partial}{\partial x^2}+a^3\frac{\partial}{\partial x^3},\textbf{b}=b^1\frac{\partial}{\partial x^1}+b^2\frac{\partial}{\partial x^2}+b^3\frac{\partial}{\partial x^3}.$

$\omega(\textbf{a},\textbf{b})=\omega(a^1\frac{\partial}{\partial x^1}+a^2\frac{\partial}{\partial x^2}+a^3\frac{\partial}{\partial x^3},b^1\frac{\partial}{\partial x^1}+b^2\frac{\partial}{\partial x^2}+b^3\frac{\partial}{\partial x^3})$ $$=a_{12}dx^1 ∧dx^2+a_{13}dx^1 ∧dx^3+a_{23}dx^2 ∧dx^3 (a^1\frac{\partial}{\partial x^1}+a^2\frac{\partial}{\partial x^2}+a^3\frac{\partial}{\partial x^3},b^1\frac{\partial}{\partial x^1}+b^2\frac{\partial}{\partial x^2}+b^3\frac{\partial}{\partial x^3})=a_{12}a^1b^2+a_{13}a^1b^3+a_{23}a^2b^3.$$

How does this answer related to $$p^3\det \begin{bmatrix} a^1&a^2 \\ b^1 &b^2 \\ \end{bmatrix}. $$ Please help me.

Answer 1

The general formula for $\omega_p=a_{12}dx_1\wedge dx_2+a_{13}dx_1\wedge dx_3+a_{23}dx_2\wedge dx_3$ is $$\omega_p(a,b)=a_{12}\det \begin{pmatrix}a^1&a^2\\b^1&b^2\end{pmatrix}+a_{13}\det\begin{pmatrix}a^1&a^3\\b^1&b^3\end{pmatrix}+a_{23}\det\begin{pmatrix}a^2&a^3\\b^2&b^3\end{pmatrix}.$$

Choose a special $\omega_p=p^3dx^1\wedge dx^2$,

$\omega_p(a,b)=\omega_p(a^1\frac{\partial}{\partial x^1}+a^2\frac{\partial}{\partial x^2}+a^3\frac{\partial}{\partial x^3},b^1\frac{\partial}{\partial x^1}+b^2\frac{\partial}{\partial x^2}+b^3\frac{\partial}{\partial x^3})=p^3\det \begin{pmatrix}a^1&a^2\\b^1&b^2\end{pmatrix}$