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Cross and dot products of vector fields under fundamental correspondence

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What happens to the cross product and the dot product of two vector fields under the fundamental correspondence?

$$ \newcommand{\lra}[1]{\!\!\!\!\!\!\!\xleftarrow{}\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ra}[1]{\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{rlllllllllll} & A & \ra{\lambda_0} & \varphi^0 \\ {\rm grad} & \da{} & & \da{d} & \\ & V & \ra{\lambda_1} & \varphi^1 \\ {\rm curl} & \ua{} & & \da{d} \\ & W & \ra{\lambda_2} & \varphi^2 \\ {\rm div} & \da{} & & \da{d} \\ & B & \ra{\lambda_3} & \varphi^3 \end{array} $$

Theorem: Suppose

  • $1$-form $\alpha$ corresponds to vector field $v$, $\alpha = \lambda_1(v)$, and
  • $1$-form $\beta$ corresponds to vector field $w$, $\beta = \lambda_1(w)$,

under the fundamental correspondence. Then:

  • (a) The $2$-form $\alpha \wedge \beta$ corresponds to vector field $v \times w$, $\alpha \wedge \beta = \lambda_2(v \times w)$.
  • (b) The $3$-form $\alpha \wedge \beta^*$ corresponds to scalar function $v \cdot w$, $\alpha \wedge \beta^* = \lambda_3(v \cdot w)$.

Proof. Let $$\alpha = A_1dx + A_2dy + A_3dz,$$ $$\beta = B_1dx + B_2dy + B_3dz,$$ $$v = <v_1,v_2,v_3>,$$ $$w = <w_1,w_2,w_3>.$$ Compute...

  • (a)

$$\begin{array}{} v \times w &= \left| \begin{array}{} v_2 & v_3 \\ w_2 & w_3 \end{array} \right| i - \left| \begin{array}{} v_1 & v_3 \\ w_1 & w_3 \end{array} \right| j + \left| \begin{array}{} v_1 & v_2 \\ w_1 & w_2 \end{array} \right| k \\ & =(v_2w_3 - v_3w_2)i - (v_1w_3 - v_3w_1)j + (v_1w_2 - v_2w_1)k \end{array}$$

$$\begin{array}{} \alpha \wedge \beta &= (A_1dx + A_2dy + A_3dz) \wedge (B_1dx+B_2dy+B_3dz) \\ &= A_1B_2 dx \hspace{1pt} dy + A_1B_3dx \hspace{1pt} dz+A_2B_1dy \hspace{1pt} dx+A_2B_3dy \hspace{1pt} dz+A_3B_1dz \hspace{1pt} dx+A_3B_2dz \hspace{1pt} dy \\ &= (A_1B_2-A_2B_1)dx \hspace{1pt} dy+(A_1B_3-A_3B_1)dx \hspace{1pt} dz + (A_2B_3-A_3B_2)dy \hspace{1pt} dz \\ &= (A_2B_3-A_3B_2)dy \hspace{1pt} dz - (A_1B_3-A_3B_1)dz \hspace{1pt} dx+(A_1B_2-A_2B_1)dx \hspace{1pt} dy \end{array}$$

  • (b)

$$v \cdot w = v_1w_1+v_2w_2 + v_3w_3$$

$$\alpha = A_1dx + A_2dy + A_3 dz$$

$$\beta^* = B_1 dy \hspace{1pt} dz + B_2 dz \hspace{1pt} dx + B_3dx \hspace{1pt} dy$$

$$\begin{array}{} \alpha \wedge \beta^* &= (A_1dx + A_2dy + A_3dz)\wedge (B_1 dy \hspace{1pt} dz + B_2dz \hspace{1pt} dx + B_3 dx \hspace{1pt} dy) \\ &= (A_1B_1 + A_2B_2 + A_3B_3)dx \hspace{1pt} dy \hspace{1pt} dz \end{array}$$

$\blacksquare$