
\section{Introduction to Differential Forms}

\begin{enumerate}[1.]
\item
{\it  Let ${\bf u}= (1,-1,0,5)$, ${\bf v}=(3,2,6,-1)$, ${\bf w}=(5,-2,3,1)$. Evaluate
$dx_1\wedge dx_3 \wedge dx_4({\bf u}, {\bf v}, {\bf w})$.}


$$dx_1\wedge dx_3 \wedge dx_4({\bf u, v, w}) =
		\left|\begin{matrix}
			1 & 3 & 5 \\
			0 & 6 & 3 \\
			5 & -1 & 1
		\end{matrix} \right|
		= 6+45-150+3 = -96$$





\bigskip




\item
{\it  Let ${\bf u}= (2,1,4,-1)$, ${\bf v}=(1,3,5,0)$, ${\bf w}=(2,7,-2,3)$. Evaluate:}

\medskip

(a) 
\begin{align*}
dx_1\wedge dx_2 \wedge dx_4({\bf u}, {\bf v}, {\bf w}) &=
\begin{vmatrix}
2 & 1 & 2\\
1 & 3 & 7\\
-1 & 0 & 3
\end{vmatrix}
\\
&= (-1)
\begin{vmatrix}
1 & 2\\
3 & 7
\end{vmatrix}
+3
\begin{vmatrix}
2 & 1\\
1 & 3
\end{vmatrix}
\\
&= (-1)(1) + 3(5)\\
&= 14,
\end{align*}
where we expanded the $3\times 3$ determinant along the third row.

\medskip

(b) 
\begin{align*}
dx_2\wedge dx_3({\bf u}, {\bf v}) -2 dx_1 \wedge dx_4({\bf u}, {\bf w}) &=
\begin{vmatrix}
1 & 3\\
4 & 5
\end{vmatrix}
-2
\begin{vmatrix}
2 & 2\\
-1 & 3
\end{vmatrix}
\\
&= -7 -2(8)\\
&= -23.
\end{align*}

\bigskip

\item
{\it  Calculate and simplify the wedge product $\phi\wedge \psi$. Write the $dx_i$'s in the answers in increasing subscript order.}

\medskip

(a) $\phi=dx_1\wedge dx_3\wedge dx_4,$
$\psi= a_1 dx_1+ a_2 dx_2 + a_3 dx_3$.

\medskip


\begin{align*}
		\phi \wedge \psi =& \; (dx_1\wedge dx_3\wedge dx_4)\wedge (a_1dx_1+a_2dx_2+a_3dx_3)\\
		=& \; a_1(dx_1\wedge dx_3 \wedge dx_4\wedge dx_1) + a_2 (dx_1 \wedge dx_3\wedge dx_4 \wedge dx_2) 
		\\
		&+ a_3(dx_1\wedge dx_3\wedge dx_4\wedge dx_3)\\
		=&\; a_1(-1)^2(dx_1 \wedge dx_1 \wedge dx_3\wedge dx_4) + a_2(-1)^2(dx_1\wedge dx_2\wedge dx_3\wedge dx_4) \\
		&+ a_3(-1) (dx_1 \wedge dx_3 \wedge dx_3 \wedge dx_4)\\
		=& \; a_2\, (dx_1 \wedge dx_2 \wedge dx_3 \wedge dx_4)
	\end{align*}
	
	\bigskip	


(b) $\phi= dx_2\wedge dx_3 +  dx_1\wedge dx_4,$
$\psi= 4 dx_1\wedge dx_2 - dx_1
\wedge dx_3 + 5 dx_2\wedge dx_3$.

\medskip

\begin{align*}
		\phi \wedge \psi =& \; (dx_2 \wedge dx_3 + dx_1 \wedge dx_4) \wedge (4dx_1 \wedge dx_2 - dx_1 \wedge dx_3 + 5dx_2 \wedge dx_3)\\
		=& \; 4(dx_2\wedge dx_3 \wedge dx_1 \wedge dx_2) - (dx_2\wedge dx_3 \wedge dx_1 \wedge dx_3) \\
		&+ 5(dx_2 \wedge dx_3 \wedge dx_2 \wedge dx_3) + 4(dx_1 \wedge dx_4 \wedge dx_1 \wedge dx_2) 
		\\
		& - (dx_1 \wedge dx_4 \wedge dx_1 \wedge dx_3) + 5 (dx_1 \wedge dx_4 \wedge dx_2 \wedge dx_3) \\
		=& \; 5(-1)^2(dx_1 \wedge dx_2 \wedge dx_3 \wedge dx_4) \\
		=& \; 5\, dx_1 \wedge dx_2 \wedge dx_3 \wedge dx_4
	\end{align*}
	
	

\bigskip
	


\item
{\it Calculate and simplify the wedge product $\phi \wedge \psi$. Write the $dx_i$'s in the answers in increasing subscript order.}

\medskip

(a) $\phi=3 dx_2\wedge dx_3 - 7 dx_2 \wedge dx_4,$
$\psi= 2 dx_1+ 3 dx_2 - dx_4$

\begin{align*}
\phi \wedge \psi & = (3 dx_2\wedge dx_3 - 7 dx_2 \wedge dx_4) \wedge (2 dx_1+ 3 dx_2 - dx_4)\\
&= 6 dx_2 \wedge dx_3 \wedge dx_1 - 3 dx_2 \wedge dx_3 \wedge dx_4 - 14 dx_2 \wedge dx_4 \wedge dx_1\\
&= 6 dx_1 \wedge dx_2 \wedge dx_3 - 14 dx_1 \wedge dx_2 \wedge dx_4 - 3 dx_2 \wedge dx_3 \wedge dx_4,
\end{align*}
where in the second line the three missing terms vanish because they each have two appearances of some $dx_i$.

\medskip

(b) $\phi= dx_2\wedge dx_3\wedge dx_4,$
$\psi= 3 dx_1 - 5 dx_2$

\begin{align*}
\phi \wedge \psi &= (dx_2\wedge dx_3\wedge dx_4) \wedge (3 dx_1 - 5 dx_2)\\
&= 3 dx_2\wedge dx_3\wedge dx_4 \wedge dx_1\\
&= -3 dx_1 \wedge dx_2\wedge dx_3\wedge dx_4.
\end{align*}

\medskip

(c) $\phi=2 dx_1\wedge dx_3$, $\psi= - 2 dx_1 \wedge dx_4 + 3 dx_2 \wedge dx_4$.

\begin{align*}
\phi \wedge \psi &= (2 dx_1\wedge dx_3) \wedge (- 2 dx_1 \wedge dx_4 + 3 dx_2 \wedge dx_4)\\
&= 6 dx_1\wedge dx_3 \wedge dx_2\wedge dx_4\\
&= -6 dx_1\wedge dx_2 \wedge dx_3\wedge dx_4.
\end{align*}



\bigskip

	
 
 
 
\item
{\it Compute the exterior product 
$\Phi_1 \wedge \Phi_2$, if
$\Phi_1 = x_1  \, dx_1 \wedge dx_2 \wedge dx_4$ and
$\Phi_2 = x_4 dx_1 + x_2 dx_3 + x_3 dx_4$ in $\rr^4$.
Give your answer in the form $f(x_1,x_2,x_3,x_4) 
dx_1\wedge dx_2 \wedge dx_3 \wedge dx_4$.}


\medskip


\begin{align*}
\Psi \wedge \Phi &= 
x_1 x_2 \,  dx_1\wedge dx_2 \wedge dx_4 \wedge dx_3
\\
&= - x_1 x_2 \, dx_1\wedge dx_2 \wedge dx_3 \wedge dx_4.
\end{align*}




\end{enumerate}

