update week 35

doc/src/week35/week35.do.txt

@@ -539,7 +539,32 @@ $1\times n$ matrix.
 !split
 ===== Derivatives, example 1 =====
 
-Let now $\bm{y}=\bm{A}\bm{x}$, where $\bm{A}$ is an $m\times n$ matrix and the matrix does not depend on $\bm{x}$. If we write out the vector $\bm{y}$
+Let now $\bm{y}=\bm{A}\bm{x}$, where $\bm{A}$ is an $m\times n$ matrix and the matrix does not depend on $\bm{x}$. If we write out the vector $\bm{y}$ compoment by component we have
+
+!bt
+\[
+y_i = \sum_{j=0}^{n-1}a_{ij}x_j,
+\]
+!et
+with $\all i=0,1,2,\dots,m-1$. The individual matrix elements of $\bm{A}$ are given by the symbol $a_{ij}$.
+It follows that the partial derivatives of $y_i$ with respect to $x_k$ 
+!bt
+\[
+\frac{\partial y_i }{\partial x_k}= a_{ik} \all i=0,1,2,\dots,m-1.
+\]
+!et
+
+From this we have, using the definition of the Jacobian
+
+!bt
+\[
+\frac{\partial \bm{y} }{\partial \bm{x}}= \bm{A}.
+\]
+!et
+
+
+
+
 
 
 !bt