added to week 35

doc/pub/week35/html/._week35-bs000.html

@@ -404,7 +404,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Aug 28, 2022</h4>
+<h4>Aug 29, 2022</h4>
 </center> <!-- date -->
 <br>
 

doc/pub/week35/html/._week35-bs069.html

@@ -964,6 +964,31 @@ $$
 
 <p>and finally find the second derivative of this function with respect to the vector \( \boldsymbol{s} \).</p>
 
+<p><b>Hint</b>: In these exercises it is always useful to write out with summation indices the various quantities.
+As an example, consider the function
+</p>
+
+$$
+f(\boldsymbol{x}) =\boldsymbol{A}\boldsymbol{x},
+$$
+
+<p>which reads for a specific component \( f_i \) (we define the matrix \( \boldsymbol{A} \) to have dimension \( n\times n \) and the vector $\boldsymbol{x} to have length \( n \))</p>
+
+$$
+f_i =\sum_{j=0}^{n-1}a_{ij}x_j, 
+$$
+
+<p>which leads to</p>
+$$
+\frac{\partial f_i}{\partial x_j}= a_{ij},
+$$
+
+<p>and written out in terms of the vector \( \boldsymbol{x} \) we have</p>
+$$
+\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}= \boldsymbol{A}.
+$$
+
+
 <!-- --- end exercise --- -->
 <p>
 <!-- navigation buttons at the bottom of the page -->

doc/pub/week35/html/week35-bs.html

@@ -404,7 +404,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Aug 28, 2022</h4>
+<h4>Aug 29, 2022</h4>
 </center> <!-- date -->
 <br>
 

doc/pub/week35/html/week35-reveal.html

@@ -184,7 +184,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Aug 28, 2022</h4>
+<h4>Aug 29, 2022</h4>
 </center> <!-- date -->
 <br>
 
@@ -4080,6 +4080,39 @@ $$
 
 <p>and finally find the second derivative of this function with respect to the vector \( \boldsymbol{s} \).</p>
 
+<p><b>Hint</b>: In these exercises it is always useful to write out with summation indices the various quantities.
+As an example, consider the function
+</p>
+
+<p>&nbsp;<br>
+$$
+f(\boldsymbol{x}) =\boldsymbol{A}\boldsymbol{x},
+$$
+<p>&nbsp;<br>
+
+<p>which reads for a specific component \( f_i \) (we define the matrix \( \boldsymbol{A} \) to have dimension \( n\times n \) and the vector $\boldsymbol{x} to have length \( n \))</p>
+
+<p>&nbsp;<br>
+$$
+f_i =\sum_{j=0}^{n-1}a_{ij}x_j, 
+$$
+<p>&nbsp;<br>
+
+<p>which leads to</p>
+<p>&nbsp;<br>
+$$
+\frac{\partial f_i}{\partial x_j}= a_{ij},
+$$
+<p>&nbsp;<br>
+
+<p>and written out in terms of the vector \( \boldsymbol{x} \) we have</p>
+<p>&nbsp;<br>
+$$
+\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}= \boldsymbol{A}.
+$$
+<p>&nbsp;<br>
+
+
 <!-- --- end exercise --- -->
 </section>
 

doc/pub/week35/html/week35-solarized.html

@@ -330,7 +330,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Aug 28, 2022</h4>
+<h4>Aug 29, 2022</h4>
 </center> <!-- date -->
 <br>
 
@@ -3893,6 +3893,31 @@ $$
 
 <p>and finally find the second derivative of this function with respect to the vector \( \boldsymbol{s} \).</p>
 
+<p><b>Hint</b>: In these exercises it is always useful to write out with summation indices the various quantities.
+As an example, consider the function
+</p>
+
+$$
+f(\boldsymbol{x}) =\boldsymbol{A}\boldsymbol{x},
+$$
+
+<p>which reads for a specific component \( f_i \) (we define the matrix \( \boldsymbol{A} \) to have dimension \( n\times n \) and the vector $\boldsymbol{x} to have length \( n \))</p>
+
+$$
+f_i =\sum_{j=0}^{n-1}a_{ij}x_j, 
+$$
+
+<p>which leads to</p>
+$$
+\frac{\partial f_i}{\partial x_j}= a_{ij},
+$$
+
+<p>and written out in terms of the vector \( \boldsymbol{x} \) we have</p>
+$$
+\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}= \boldsymbol{A}.
+$$
+
+
 <!-- --- end exercise --- -->
 <!-- ------------------- end of main content --------------- -->
 <center style="font-size:80%">

doc/pub/week35/html/week35.html

@@ -407,7 +407,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Aug 28, 2022</h4>
+<h4>Aug 29, 2022</h4>
 </center> <!-- date -->
 <br>
 
@@ -3970,6 +3970,31 @@ $$
 
 <p>and finally find the second derivative of this function with respect to the vector \( \boldsymbol{s} \).</p>
 
+<p><b>Hint</b>: In these exercises it is always useful to write out with summation indices the various quantities.
+As an example, consider the function
+</p>
+
+$$
+f(\boldsymbol{x}) =\boldsymbol{A}\boldsymbol{x},
+$$
+
+<p>which reads for a specific component \( f_i \) (we define the matrix \( \boldsymbol{A} \) to have dimension \( n\times n \) and the vector $\boldsymbol{x} to have length \( n \))</p>
+
+$$
+f_i =\sum_{j=0}^{n-1}a_{ij}x_j, 
+$$
+
+<p>which leads to</p>
+$$
+\frac{\partial f_i}{\partial x_j}= a_{ij},
+$$
+
+<p>and written out in terms of the vector \( \boldsymbol{x} \) we have</p>
+$$
+\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}= \boldsymbol{A}.
+$$
+
+
 <!-- --- end exercise --- -->
 <!-- ------------------- end of main content --------------- -->
 <center style="font-size:80%">

doc/src/week35/week35.do.txt

@@ -2921,3 +2921,31 @@ and
 \]
 !et
 and finally find the second derivative of this function with respect to the vector $\bm{s}$.
+
+_Hint_: In these exercises it is always useful to write out with summation indices the various quantities.
+As an example, consider the function
+
+!bt
+\[
+f(\bm{x}) =\bm{A}\bm{x},
+\]
+!et
+which reads for a specific component $f_i$ (we define the matrix $\bm{A}$ to have dimension $n\times n$ and the vector $\bm{x} to have length $n$)
+
+!bt
+\[
+f_i =\sum_{j=0}^{n-1}a_{ij}x_j, 
+\]
+!et
+which leads to
+!bt
+\[
+\frac{\partial f_i}{\partial x_j}= a_{ij},
+\]
+!et
+and written out in terms of the vector $\bm{x}$ we have
+!bt
+\[
+\frac{\partial f(\bm{x})}{\partial \bm{x}}= \bm{A}.
+\]
+!et