channelflow.org

====== Differences ====== This shows you the differences between two versions of the page.

--- gibson:teaching:fall-2013:math445:lecture11 [2013/10/09 19:17]
gibson
+++ gibson:teaching:fall-2013:math445:lecture11 [2013/10/09 19:27] (current)
gibson
@@ Line 3: / Line 3: @@
 Below are two sample codes for making contour plots of a
 function of two variables. In the first sample code, the
-function f is expressed as a function of two separate scalar
+function $f(x,y) = x^2 + y^2$ is expressed as a function
-variables x and y.
+of two separate scalar variables x and y.
 <code>
@@ Line 34: / Line 34: @@
 </code>
-The second script expresses f as a function of a vector variable x.
+The second script suggests a good initial guess for zeros of
+the function
+<latex>
+f\left(\begin{array}{c} x_1 \\ x_2 \end{array}}\right) =
+\left(\begin{array}{l} x_1^2 + x_2^2 - 7 \\ x_1^{-1} - x_2 \end{array} \right)
+</latex>
+i.e. points $x$ for which $f(x) = 0$. The script plots contour lines near $f_1=0$ and $f_2=0$.
+The intersection of these curves are points where both components of $f$ are near zero, and
+so serve as good guesses for a Newton search.
 <code>