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+====== Math 445 lecture 20: 3D graphics ======
+Some examples of plotting 2d functions $f(x,y)$ in Matlab.
+----
+==== 1d graphs ====
+How would you make a graph of $f(x) = (x-1)^2$ on $-3 \leq x \leq 3$ in Matlab? You would
+  - Create vector ''x'' of gridpoints with ''linspace''
+  - Evaluate $f(x)$ on gridpoints using dot operations on ''x''
+  - Use ''plot(x,f)'' to draw the graph
+For example,
+<file matlab plot1d.m>
+% Plot 1d graph of f(x) = (x-1)^2 on -3 < x < 3
+x = linspace(-3, 3, 100); % create vector x of gridpoints
+f = (x-1).^2;             % evaluate f(x) on gridpoints
+plot(x,f)                 % plot  the graph
+xlabel('x')
+ylabel('f(x) = (x-1)^2')
+</file>
+produces
+{{ :gibson:teaching:spring-2016:math445:onedplot.png?direct&400 |}}
+----
+==== 2d graphs with meshgrid ====
+Graphing a 2d function $f(x,y)$ is very similar. For example, to make a graph of the function
+$f(x,y) = 2*(x-1)^2 + (y-1)^2$ on $-4 \leq x \leq 4, -4 \leq y \leq 4$
+  - Create vectors ''x'' and ''y'' of gridpoints on $x$ and $y$ axes with ''linspace''
+  - Create matrices ''X'' and ''Y'' of gridpoints on 2d mesh with ''meshgrid''
+  - Evaluate $f(x,y)$ on mesh of gridpoints using dot operations on ''X,Y''
+  - Use ''pcolor(x,y,f)'' to draw the graph (or ''contour'', ''contourf'', ''surf'', ''surfc'')
+For example,
+<file matlab plot2d.m>
+% Plot f(x,y) = 2*(x-1)^2 + (y-1)^2 on -4 < x < 4, -4 < y < 4
+x = linspace(-4,4,30);     % create 1d gridpoints on x and y axes
+y = linspace(-4,4,30);
+[X,Y] = meshgrid(x,y);     % create 2d mesh of gridpoints over x,y
+F = 2*(X-1).^2 + (Y-1).^2; % evaluate the function f(x,y) over mesh
+pcolor(x,y,F);             % or contour(X,Y,F), surf(x,y,F), etc.
+colorbar()                 % plot a colorbar
+xlabel('x');
+ylabel('y');
+title('f(x,y) = 2(x-1)^2 + (y-1)^2')
+shading interp   % or faceted, flat
+</file>
+{{ :gibson:teaching:spring-2016:math445:lecture:twodplot.png?direct&400 |}}
+----
+==== quiver plots and the gradient===
+The function $f(x,y) = x e^{-x^2 - y^2}$ has gradient
+\begin{eqnarray*}
+\nabla f = \left( \begin{array}{c} \partial f/\partial x \\  \partial f/\partial y \end{array} \right) = \left( \begin{array}{c} (1-x^2) e^{-x^2 - y^2} \\ -2xy e^{-x^2 - y^2} \end{array} \right)
+\end{eqnarray*}
+Here's the Matlab code to make a contour plot of $f(x,y)$ with its gradient superimposed as a quiver plot.
+<file matlab plotgradient.m>
+% Plot f(x,y) = x e^(-x^2 - y^2) with contours and its gradient as arrows.
+x = linspace(-2, 2, 30);
+y = linspace(-2, 2, 30);
+[X,Y] = meshgrid(x,y);
+F = X .* exp(-X.^2 - Y.^2);  % evaluate f(x,y) on mesh
+contour(x,y,F);              % draw contour plot of f
+% let dfdy = y component of grad F = df/dy
+dfdx = (1-2*X.^2) .* exp(-X.^2 - Y.^2); % evaluate dfdx on mesh
+dfdy = -2*X.*Y .* exp(-X.^2 - Y.^2);    % evaluate dfdy on mesh
+hold on
+quiver(x,y,dfdx,dfdy);       % draw quiver plot
+xlabel('x')
+ylabel('y')
+title('f(x,y) = x e^{-x^2 - y^2} with \nabla f arrow plot')
+colorbar()
+</file>
+{{ :gibson:teaching:spring-2016:math445:lecture:quiver.png?direct&500 |}}

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