
\section{Exterior Derivative}

\begin{enumerate}[1.]


\item 
{\it Let $\Psi=x_1^2\, dx_1\wedge dx_3 -x_3 \,dx_2 \wedge dx_4$ 
in $\rr^4$. 

\medskip

(a) Compute $\Psi \wedge \Phi$, if
$\Phi= x_3^5  \, dx_2 \wedge dx_4$.
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

{\it (b) Find $d\Psi$. Simplify your answer.}


\bigskip


(a)

\begin{align*}
\Psi \wedge \Phi &= x_1^2 x_3^5 \, dx_1\wedge dx_3 \wedge dx_2 \wedge dx_4
\\
&= - x_1^2 x_3^5\,  dx_1\wedge dx_2 \wedge dx_3 \wedge dx_4.
\end{align*}




\medskip

(b)
\begin{align*}
d \Psi &= 2 x_1 \, dx_1\wedge dx_1\wedge dx_3 -dx_3 \wedge dx_2 \wedge dx_4
\\
&= dx_2 \wedge dx_3 \wedge dx_4
\end{align*}






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\item
{\it Calculate the exterior derivative of the differential forms:}

\medskip

(a) $\Phi=xydy + x^2z^2dz$

\medskip

\begin{align*}
d\Phi &= d(xydy + x^2z^2dz)\\
&= d(xy) \wedge dy + d(x^2 z^2) \wedge dz\\
&=(y dx + x dy) \wedge dy + (2xz^2 dx + 2x^2 z dz) \wedge dz\\
&= y dx \wedge dy + 2xz^2 dx \wedge dz.
\end{align*}

\medskip

(b) $\Phi= (x_1+x_2^2) dx_1\wedge dx_2+ x_2\cos(x_3) dx_2\wedge dx_4$

\medskip


\begin{align*}
d\Phi &= d\left((x_1+x_2^2) dx_1\wedge dx_2+ x_2\cos(x_3) dx_2\wedge dx_4\right)\\
&= d(x_1 + x_2^2) \wedge dx_1\wedge dx_2 + d(x_2\cos(x_3)) \wedge dx_2\wedge dx_4\\
&= (dx_1 + 2x_2 dx_2) \wedge dx_1\wedge dx_2 + (\cos(x_3)dx_2 - x_2\sin(x_3)dx_3) \wedge dx_2\wedge dx_4\\
&= - x_2\sin(x_3)dx_3 \wedge dx_2\wedge dx_4\\
&= x_2\sin(x_3)dx_2 \wedge dx_3 \wedge dx_4.
\end{align*}





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\item
{\it Calculate the exterior derivative of the differential forms:}

\medskip

(a) $\Phi= 5x_2 \, dx_1\wedge dx_3+ (3x_3 - x_1)  \, dx_3\wedge dx_4$

\begin{align*}
		d \Phi
		=& \; 5\, dx_2 \wedge dx_1 \wedge dx_3 - dx_1 \wedge dx_3 \wedge dx_4 \\
		=& \; -5\, dx_1 \wedge dx_2 \wedge dx_3 - dx_1 \wedge dx_3 \wedge dx_4
	\end{align*}

\medskip

(b) $\Psi = x_1 x_2^2  \,dx_1 \wedge dx_3 \wedge dx_4 
 + x_3^2 x_4  \, dx_2 \wedge dx_3 \wedge dx_4$. 
 
\begin{align*}
		d \Psi
		=& \; 2x_1x_2 \, dx_2 \wedge dx_1 \wedge dx_3 \wedge dx_4 \\
		=&  -2x_1x_2 \, dx_1 \wedge dx_2 \wedge dx_3 \wedge dx_4
	\end{align*}
	
	\medskip
	
	
 (c) $\Phi = e^{x_1-x_3}  dx_1 \wedge dx_2 \wedge dx_4 
 + e^{4x_2-5x_3}  dx_1 \wedge dx_3 \wedge dx_4$. 
 
 \medskip
 
\begin{align*}
d\Phi
&= -e^{x_1-x_3} dx_3 \wedge dx_1 \wedge dx_2 \wedge dx_4 + 4e^{4x_2-5x_3} dx_2 \wedge dx_1 \wedge dx_3 \wedge dx_4\\
&= \left(-e^{x_1-x_3} -4e^{4x_2-5x_3}\right) dx_1 \wedge dx_2 \wedge dx_3 \wedge dx_4.
\end{align*}

 
 



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\item
{\it Find the exterior derivative of the 3-form
$x_1x_2^2 x_3^4 \, dx_1\wedge dx_3 \wedge dx_4$ in $\rr^4$.
Give your answer in the form $g(x_1,x_2,x_3,x_4)\,  
dx_1\wedge dx_2 \wedge dx_3 \wedge dx_4$.}


\medskip


\begin{align*}
d \Psi &= 2 x_1 x_2 x_3^4 \, dx_2 \wedge dx_1\wedge dx_3 \wedge dx_4
\\
&= - 2 x_1 x_2 x_3^4 \, 
dx_1\wedge dx_2 \wedge dx_3 \wedge dx_4
\end{align*}





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\item {\it 
 Let $\Psi=xy dx + z^2 dy$ and $\Phi= (2y+z)dx\wedge dy + (xy+z^2) dy \wedge dz$.
 
 \medskip
 
(a)  Find $\Psi \wedge \Phi$. Simplify your answer.

\medskip

(b) Find $d\Phi$ and $d\Psi$.}

\bigskip

(a) 
\begin{align*}
\Psi \wedge \Phi &= xy(2y+z) dx \wedge dx\wedge dy
+ xy(xy+z^2) dx \wedge dy\wedge dz
\\
&+ z^2 (2y+z) dy \wedge dx\wedge dy
+ z^2 (xy+z^2) dy \wedge dy\wedge dz
\\
&= xy(xy+z^2) dx \wedge dy\wedge dz.
\end{align*}

\medskip

(b)
\begin{align*}
d \Phi &=  dz \wedge dx\wedge dy
+ y dx \wedge dy\wedge dz
\\
&= (1+y) dx \wedge dy\wedge dz
\end{align*}
\begin{align*}
d \Psi &=  x dy\wedge dx
+ 2z  dz\wedge dy
\\
&= -x dx\wedge dy
- 2z  dy\wedge dz
\end{align*}



\end{enumerate}

