tex

1 files changed, 10 insertions, 10 deletions

diff --git a/ClassJune26.page b/ClassJune26.page
index 9bb4ded..8b14dc1 100644
--- a/ClassJune26.page
+++ b/ClassJune26.page
@@ -70,20 +70,20 @@ projections onto the real and imaginary axis. If we use column vector
 notation, we have
 $z = \left(\begin{array}{c}
 a\\
-b\end{array}\right)\\
+b\end{array}\right)
  = a\left(\begin{array}{c}
 1\\
 0\end{array}\right)+b\left(\begin{array}{c}
 0\\
-1\end{array}\right)\\
- = a\cdot1+b\cdot i\\
+1\end{array}\right)
+ = a\cdot1+b\cdot i
  = a+bi$
 Put another way, we are using $1$ and $i$ as basis vectors. For
-example, 
+example,  
 $\left(\begin{array}{c}
 3\\
-2\end{array}\right)=3(\rightarrow)+2(\uparrow)=3+2i$
-and 
+2\end{array}\right)=3(\rightarrow)+2(\uparrow)=3+2i$  
+and  
 $\overset{2}{\nwarrow}=\left(\begin{array}{c}
 -\sqrt{2}\\
 \sqrt{2}\end{array}\right)=-\sqrt{2}(\rightarrow)+\sqrt{2}(\uparrow)=-\sqrt{2}+\sqrt{2}i$
@@ -93,8 +93,8 @@ Polar coordinates represent each complex number by its magnitude and argument, i
 $i=\uparrow=(1,\pi/2)$ and $\overset{2}{\nwarrow}=(2,3\pi/4)$.
 
 The familiar angle addition formula from trigonometry now follows
-from the distributive law applied to unit vectors.
-$\cos(\theta+\phi)+i\sin(\theta+\phi)=(\cos\theta+i\sin\theta)(\cos\phi+i\sin\phi)=\cos\theta\cos\phi-\sin\theta\sin\phi+i(\cos\theta\sin\phi+\sin\theta\cos\phi)$
+from the distributive law applied to unit vectors.  
+    $\cos(\theta+\phi)+i\sin(\theta+\phi)=(\cos\theta+i\sin\theta)(\cos\phi+i\sin\phi)=\cos\theta\cos\phi-\sin\theta\sin\phi+i(\cos\theta\sin\phi+\sin\theta\cos\phi)$
 
 
 In the workshop session, you'll use a similar trick to compute triple
@@ -139,8 +139,8 @@ As a sanity check, note that $i$ corresponds to $\left(\begin{array}{cc}
 0 & -1\end{array}\right)\sim-1$.
 
 Another natural transformation of the complex plane is given by squaring,
-sending $z\rightsquigarrow z^{2}$. This squares the length of each
-vector, and doubles its angle. PICTURE. What does $z\rightsquigarrow z^{n}$
+sending $z\mapsto z^{2}$. This squares the length of each
+vector, and doubles its angle. PICTURE. What does $z\mapsto z^{n}$
 look like? PICTURE.
 
 Another key example is the exponential map. Recall the power series