Merge pull request #40 from coolbluealan/master

Fix typos.

Author: jcponce

README.md

@@ -37,7 +37,7 @@ Some of the topics covered here are basic arithmetic of complex numbers, complex
 Riemann surfaces, limits, derivatives, domain coloring, analytic landscapes and 
 some applications of conformal mappings.
 
-What distinguishes this online book from other traditional texts in the first instance is the use of interactive applets that allow you to explore properties of complex numbers geometrically and analyze complex functions by using different techniques to visualize them. For the design of applets I used the following open-source softwares: [GeoGebra](https://geogebra.org/), [p5.js](https://p5js.org/), [Cindy.js](https://cindyjs.org/) and [MathCell](http://mathcell.org/).
+What distinguishes this online book from other traditional texts in the first instance is the use of interactive applets that allow you to explore properties of complex numbers geometrically and analyze complex functions by using different techniques to visualize them. For the design of applets I used the following open-source software: [GeoGebra](https://geogebra.org/), [p5.js](https://p5js.org/), [Cindy.js](https://cindyjs.org/) and [MathCell](http://mathcell.org/).
 
 Although I advocate for the use of computers as an aid to geometric reasoning,
 I highly encourage you to practice your problem solving skills by solving
@@ -48,7 +48,7 @@ Think of the computer as a physicist would his laboratory. It may be used
 to check existing ideas about our world, or as a tool to discover new phenomena
 which then poses new ideas or challenges for their explanation. 
 Throughout the sections I have provided detailed instructions (in some cases)
-to explore concepts and relationships about complex numbers using specific softwares, 
+to explore concepts and relationships about complex numbers using specific software, 
 nevertheless you must still keep in mind that computer hardware and software 
 are ephemeral things in comparison with mathematical ideas, which are timeless.
 

content/analytic_landscapes.html

@@ -368,7 +368,7 @@ <h2>Dynamic exploration</h2>
           if(newN!=N,
             N = newN;
             //The following line is DIRTY, but it makes the application run smooth for high degrees. :-)
-            //Nethertheless, it might cause render errors for high degree surfaces. In fact, only a subset of the surface is rendered.
+            //Nevertheless, it might cause render errors for high degree surfaces. In fact, only a subset of the surface is rendered.
             //Adapt limit according to hardware.
             //values of kind 4*n-1 are good values, as it means to use vectors of length 4*n.
             N = min(N,11);
@@ -428,7 +428,7 @@ <h2>Dynamic exploration</h2>
           if(!intersect & id>0,
             s = gets(id); //s = floor(log_2(id))
 
-            //the intervals [a,b] are chossen such that (id in binary notation)
+            //the intervals [a,b] are chosen such that (id in binary notation)
             //id = 1   => [a,b]=[l,u]
             //id = 10  => [a,b]=[l,(u+l)/2]
             //id = 101 => [a,b]=[l,(u+3*l)/4]
@@ -855,4 +855,4 @@ <h2>Dynamic exploration</h2>
   <script async src="js/prism.js"></script>
 </body>
 
-</html>
+</html>

content/cauchy_goursat_theorem.html

@@ -194,7 +194,7 @@ <h2>Cauchy's Theorem</h2>
                 \end{eqnarray*}
                </div>
                <p>
-                Both intergrals, on the right side, are zero by the <a href="complex_differentiation.html">Cauchy-Riemann equations</a>. $\hspace{5pt} \blacksquare$
+                Both integrals, on the right side, are zero by the <a href="complex_differentiation.html">Cauchy-Riemann equations</a>. $\hspace{5pt} \blacksquare$
                </p>
 
                <p>Observe that once it has been established that the value 
@@ -211,7 +211,7 @@ <h2>Cauchy's Theorem</h2>
 
                 <p><strong>Example 1:</strong>
                   Consider the function $f(z)= \exp\left(z^3\right).$ If $C$ is 
-                  any simple closed countour, in either direction, then
+                  any simple closed contour, in either direction, then
                   \[
                   \int_C \exp\left(z^3\right)\,dz =0.
                   \]
@@ -436,7 +436,7 @@ <h2>Simply and multiply connected domains</h2>
               <p>
                 If $f$ is analytic in a multiply connected domain $D$ then we cannot conclude 
                 that $\int_C f(z)\,dz=0$  for every simple closed contour $C$ in $D.$
-                Suppose that $D$ is muliply connected with two "holes". 
+                Suppose that $D$ is multiply connected with two "holes". 
                 Let $C,$ $C_1$ and $C_2$ be simple 
                 closed contours such that each $C_k$ surrounds only one "hole" in the 
                 domain and are inside $C.$ See Figure 8.
@@ -737,7 +737,7 @@ <h2>Historical notes &amp; Proof</h2>
                   sides by line segments in the same way as shown in Figure 14 
                   and proceed to the equivalent of (\ref{sumcontours}) and (\ref{triangle01}). 
                   The integral of $f$ along one of these new triangular contous, let's call it 
-                  $\Delta_2$ then satisifies 
+                  $\Delta_2$ then satisfies 
                   \[
                    \left|\int_{\Delta_1} f(z)\,dz\right| \leq 4 \left|\int_{\Delta_2} f(z)\,dz\right| .
                   \]
@@ -806,7 +806,7 @@ <h2>Historical notes &amp; Proof</h2>
                   respectively. Then, if we keep in mind how the triangle $\Delta_1$ was constructed, 
                   it is a straightforward problem in similar triangles to show that 
                   $L_1$ is related to $L$ by $L_1=\dfrac{1}{2}L.$ Likewise, 
-                  if $L_2$ is the legth of $\Delta_2,$ then $L_2=\dfrac{1}{2}L_1 = \dfrac{1}{2^2}L.$ 
+                  if $L_2$ is the length of $\Delta_2,$ then $L_2=\dfrac{1}{2}L_1 = \dfrac{1}{2^2}L.$ 
                   In general we have that if $L_n$ is the length of $\Delta_n,$ then 
                   $L_n= \dfrac{1}{2^n}L.$ 
                 </p>
@@ -896,14 +896,14 @@ <h2>Historical notes &amp; Proof</h2>
               <figure>
                 <img src="../images/chp04/cauchy-theorem-final-proof.gif" alt="Approximation by polygonal path" title="Approximation by polygonal path" style="width:550px;">
                 <figcaption>
-                  The countour $C$ is approximated by a  polygonal contour $P.$
+                  The contour $C$ is approximated by a  polygonal contour $P.$
                 </figcaption>
               </figure>
 
               <p>
                 <em>Proof of Cauchy-Goursat Theorem.</em>
                 Consider a simple closed contour $C$ and $n$ points $z_1, z_2, \ldots, z_n$
-                on $C$ through wich a polygonal path $P$ has been constructed. 
+                on $C$ through which a polygonal path $P$ has been constructed. 
                 Then it can be shown that the difference 
                 \[
                 \abs{\int_C f(z)\,dz - \int_P f(z)\,dz} 
@@ -981,7 +981,7 @@ <h2>References</h2>
         </article>
     </main>
 
-   <!--GeoGebra embeding-->
+   <!--GeoGebra embedding-->
    <script src="https://www.geogebra.org/apps/deployggb.js"></script>
 
    <script>
@@ -1002,7 +1002,7 @@ <h2>References</h2>
            ggbApplet1.inject('ggb-element-1');
        });
    </script>
-   <!--GeoGebra embeding ends-->
+   <!--GeoGebra embedding ends-->
 
     <a href="cauchy_integral_formula.html" style="text-decoration: none;">
         <p class="nextPage">Cauchy Integral Formula <i class="fa-solid fa-angles-right"></i></p>
@@ -1089,4 +1089,4 @@ <h2>References</h2>
   <script async src="js/prism.js"></script>
 </body>
 
-</html>
+</html>

content/cauchy_integral_formula.html

@@ -184,7 +184,7 @@ <h1>Cauchy Integral Formula</h1>
         </div>
 
         <p>
-          Choose the radius $r$ of the circle $C_r$ smaller thatn the number
+          Choose the radius $r$ of the circle $C_r$ smaller than the number
           $\delta$ in the second of these inequalities. Since
           $\abs{z-z_0}=r\lt \delta$ when $z $ is on $C_r,$ it follows that
           the first of inequalities in (\ref{formula-03}) holds
@@ -287,7 +287,7 @@ <h2>An extension of the Cauchy Integral Formula</h2>
       <div class="practice">
         <p>
           <strong>Exercise 2:</strong>
-          Use the formal defition of derivative to verify that
+          Use the formal definition of derivative to verify that
           $f'(z)$ exists and the expression (\ref{integral-derivative})
           is in fact valid.
         </p>
@@ -443,7 +443,7 @@ <h2>Some consequences of the extension</h2>
       <p>
         As a consequence, when a function $ f (z) = u(x, y) + iv(x, y)$
         is analytic at a point $z = (x, y),$ the differentiability of
-        $f'$ ensures the continuity od $f'$ there. Then, since
+        $f'$ ensures the continuity of $f'$ there. Then, since
         \[
         f'(z) = u_x + iv_x = v_y - i u_y,
         \]
@@ -514,7 +514,7 @@ <h2>Some consequences of the extension</h2>
     </article>
   </main>
 
-  <!--GeoGebra embeding-->
+  <!--GeoGebra embedding-->
   <script src="https://www.geogebra.org/apps/deployggb.js"></script>
 
   <script>
@@ -545,7 +545,7 @@ <h2>Some consequences of the extension</h2>
       ggbApplet2.inject('ggb-element-2');
     });
   </script>
-  <!--GeoGebra embeding ends-->
+  <!--GeoGebra embedding ends-->
 
   <a href="fundamental_theorem_of_algebra.html" style="text-decoration: none;">
     <p class="nextPage">The Fundamental Theorem of Algebra<i class="fa-solid fa-angles-right"></i></p>
@@ -632,4 +632,4 @@ <h2>Some consequences of the extension</h2>
   <script async src="js/prism.js"></script>
 </body>
 
-</html>
+</html>

content/complex_functions.html

@@ -308,7 +308,7 @@ <h2>Explore the real and imaginary components</h2>
         </article>
     </main>
 
-    <!--GeoGebra embeding-->
+    <!--GeoGebra embedding-->
     <script src="https://www.geogebra.org/apps/deployggb.js"></script>
     <script>
         let params = {
@@ -326,7 +326,7 @@ <h2>Explore the real and imaginary components</h2>
             ggbApplet.inject('ggb-element');
         });
     </script>
-    <!--GeoGebra embeding ends-->
+    <!--GeoGebra embedding ends-->
 
     <a href="limits.html" style="text-decoration: none;">
         <p class="nextPage">Limits <i class="fa-solid fa-angles-right"></i></p>
@@ -413,4 +413,4 @@ <h2>Explore the real and imaginary components</h2>
   <script async src="js/prism.js"></script>
 </body>
 
-</html>
+</html>

content/complex_integration.html

@@ -375,7 +375,7 @@ <h2>Properties</h2>
                 </p>
                 <div class="scroll-wrapper">
                     \begin{eqnarray}
-                    \int_a^b k \, w(t)\, dt &amp;=&amp; k \int_a^b w(t)\,dt, \quad k\text{ is a complex contant,}\\
+                    \int_a^b k \, w(t)\, dt &amp;=&amp; k \int_a^b w(t)\,dt, \quad k\text{ is a complex constant,}\\
                     \int_a^b \big[ w(t) + s(t)\big] \,dt &amp;=&amp; \int_a^b w(t)\,dt + \int_a^b s(t) \,dt,\\
                     \int_a^b w(t) \,dt &amp;=&amp; \int_a^c w(t)\,dt + \int_c^b w(t) \,dt \quad c\in[a,b], \\
                     \int_a^b w(t) \,dt &amp;=&amp; -\int_b^a w(t)\,dt.
@@ -384,11 +384,11 @@ <h2>Properties</h2>
 
                 <p>
                     From definition (\ref{contour-integral}), and the properties just mentioned above,
-                    it also follows immediatly that
+                    it also follows immediately that
                 </p>
                 <div class="scroll-wrapper">
                     \begin{eqnarray}
-                    \int_{C} z_0 \, f(z) \, dz &amp;=&amp; z_0 \int_{C} f(z) \, dz, \quad z_0 \text{ is a complex contant,}\\
+                    \int_{C} z_0 \, f(z) \, dz &amp;=&amp; z_0 \int_{C} f(z) \, dz, \quad z_0 \text{ is a complex constant,}\\
                     \int_{C} \big[ f(z) + g(z) \big] \, dz &amp;=&amp; \int_{C} \, f(z) \, dz + \int_{C} g(z) \, dz \\
                     \end{eqnarray}
                 </div>
@@ -470,7 +470,7 @@ <h2>Properties</h2>
                     \[
                     z'(t) = x'(t) + iy'(t)
                     \]
-                    are continuos on the entire interval $[a,b].$
+                    are continuous on the entire interval $[a,b].$
                     Then the real-valued function
                     \[
                     \left|z'(t)\right| =\sqrt{\left[x'(t)\right]^2+ \left[y'(t)\right]^2}
@@ -483,8 +483,8 @@ <h2>Properties</h2>
                 </p>
 
                 <p>
-                    If $C$ is a contour of lenght $L$ and $f$ is a piecewise continuous
-                    function on $C,$ and $M$ is a nonnegative contant such that
+                    If $C$ is a contour of length $L$ and $f$ is a piecewise continuous
+                    function on $C,$ and $M$ is a nonnegative constant such that
                     $\left|f(z)\right|\leq M$ for all points $z$ on $C$ at which $f(z)$
                     is defined, then
                     \begin{eqnarray}\label{ML-inequality}
@@ -633,7 +633,7 @@ <h2>Antiderivatives</h2>
 
                 <div class="theorem" id="FTC">
                     Suppose that $f(z)$ is continuous on a domain $D.$
-                    If $f(z)$ has an antiderivative throughtout $D,$ then
+                    If $f(z)$ has an antiderivative throughout $D,$ then
                     for any contour $C$ lying entirely in $D$,
                     with initial fixed point $z_1$ and
                     final fixed point $z_2,$ we have that
@@ -760,7 +760,7 @@ <h2>Antiderivatives</h2>
                         Now, since $f$ is continuous, for any $\epsilon\gt 0 $ there 
                         exists a $\delta\gt 0$ such that $\abs{f(s)-f(z)}\lt \epsilon$
                         whenever $\abs{s-z}\lt \delta.$
-                        Thus, if we choose $z+ \Delta z$ clos enough to $z$ 
+                        Thus, if we choose $z+ \Delta z$ close enough to $z$ 
                         so that $\abs{\Delta z}\lt \delta,$
                         it follows from the ML-inequality that 
                     </p>
@@ -792,7 +792,7 @@ <h2>Antiderivatives</h2>
         </article>
     </main>
 
-    <!--GeoGebra embeding-->
+    <!--GeoGebra embedding-->
     <script src="https://www.geogebra.org/apps/deployggb.js"></script>
 
     <script>
@@ -823,7 +823,7 @@ <h2>Antiderivatives</h2>
             ggbApplet2.inject('ggb-element-2');
         });
     </script>
-    <!--GeoGebra embeding ends-->
+    <!--GeoGebra embedding ends-->
 
     <a href="integrals_of_functions_with_branch_cuts.html" style="text-decoration: none;">
         <p class="nextPage">Integrals of Functions with Branch Cuts <i class="fa-solid fa-angles-right"></i></p>
@@ -910,4 +910,4 @@ <h2>Antiderivatives</h2>
   <script async src="js/prism.js"></script>
 </body>
 
-</html>
+</html>

content/conformal_mapping.html

@@ -317,7 +317,7 @@ <h1>Conformal Mapping</h1>
 
       <div id="section2">
 
-        <h2>Analytics functions</h2>
+        <h2>Analytic functions</h2>
 
         <p>
           A remarkable geometrical property enjoyed by all complex
@@ -328,7 +328,7 @@ <h2>Analytics functions</h2>
 
         <div class="theorem">
           If $f$ is analytic in a domain $D$ containing $z_0,$ and if
-          $f'(z_0)\neq 0,$ then $w=f(z)$ is a conformal mapping at $z_0.$
+          $f'(z_0)\neq 0,$ then $w=f(z)$ is conformal at $z_0.$
         </div>
 
         <div class="proof">
@@ -383,7 +383,7 @@ <h2>Analytics functions</h2>
           <div class="scroll-wrapper">
             \begin{eqnarray*}
             \text{arg}\left(f'\left(z_0\right)\cdot z_2'\right)-\text{arg}\left(f'\left(z_0\right)\cdot z_1'\right) &amp;= &amp;
-            \text{arg}\left(f'\left(z_0\right) z_2'\right) +
+            \text{arg}\left(f'\left(z_0\right)\right) +
             \text{arg}\left(z_2'\right)-\left[\text{arg}\left(f'\left(z_0\right)\right)+ \text{arg}\left(z_1'\right)
             \right]\\
             &amp;=&amp; \text{arg}\left(z_2'\right) - \text{arg}\left(z_1'\right).
@@ -720,7 +720,7 @@ <h2>Local inverses</h2>
         <p>
           throughout $V.$
           Since the Cauchy-Riemann equations hold for $u$ and $v,$ they also
-          fold for $x$ and $y,$ that is
+          hold for $x$ and $y,$ that is
           \[
           x_u = y_v \quad \text{and}\quad x_v = - y_u.
           \]
@@ -851,7 +851,7 @@ <h2>The Riemann Mapping Theorem</h2>
     </article>
   </main>
 
-  <!--GeoGebra embeding-->
+  <!--GeoGebra embedding-->
   <script src="https://www.geogebra.org/apps/deployggb.js"></script>
 
   <script>
@@ -884,7 +884,7 @@ <h2>The Riemann Mapping Theorem</h2>
       ggbApplet2.inject('ggb-element-2');
     });
   </script>
-  <!--GeoGebra embeding ends-->
+  <!--GeoGebra embedding ends-->
 
   <a href="linear_fractional_transformations.html" style="text-decoration: none;">
     <p class="nextPage">Linear Fractional Transformations<i class="fa-solid fa-angles-right"></i></p>
@@ -971,4 +971,4 @@ <h2>The Riemann Mapping Theorem</h2>
   <script async src="js/prism.js"></script>
 </body>
 
-</html>
+</html>

content/continuity.html

@@ -343,7 +343,7 @@ <h1>Continuity</h1>
             </p>
 
             <div class="theorem" id="bounded-function">
-                If $f$ is continuous throughtout a region $R$ that is 
+                If $f$ is continuous throughout a region $R$ that is 
                 both bounded and closed, there there exists 
                 $M\gt 0$ such that $\abs{f(z)}\leq M$ for all points in $R,$ 
                 where the equality holds for at least one such $z.$
@@ -367,7 +367,7 @@ <h1>Continuity</h1>
             </p>
 
             <div class="theorem" id="contuity-properties">
-                Suppose that $f$ and $g$ are continous at $z_0,$ then the following
+                Suppose that $f$ and $g$ are continuous at $z_0,$ then the following
                 are continuous functions at $z_0:$
                 <ol class="roman-list">
                     <li>$c\cdot f,$ with $c$ a complex constant,</li>
@@ -474,4 +474,4 @@ <h1>Continuity</h1>
   <script async src="js/prism.js"></script>
 </body>
 
-</html>
+</html>