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Why?

\begin{center}
\emph{
The soul cannot think without a picture.
}
--- Aristotle\index{Aristotle} (384-322 B.C.).
\end{center}

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Level curve problem (J01.3):

URL for gtqp
Scroll through the document & show them the math & the figure.

Claim that it's nice to be able to see what's going on.

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1D meshgrid

dx = 0.1
x = [0, dx, 10];
y = sin(x);
y = sin(x)  + cos(x); plot(x,y)
y = sin(x) .* cos(x); plot(x,y)

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y = sin(x);
plot(x, y, 'LineWidth', 2);
yp = gradient(y, dx);
hold on
plot(x, y, 'LineWidth', 1);

How would you numerically estimate a derivative?
* Forward difference:    [f(x+h) - f(x)  ] / h
* Backward differences:  [f(x)   - f(x-h)] / h
* Centered differences:  [f(x+h) - f(x-h)] / 2h

Matlab does:
* forward  differences on the left edge,
* backward differences on the right edge,
* centered differences in the middle.

x = [1:5]
y = x.^2
gradient(y,1)

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2D meshgrid

[x, y] = meshgrid(1:4, 5:9)
z = x+y
z = x.^2 .* y

[dzdx, dzdy] = gradient(z, 1, 1)

To fit on the screen:
[x,y]
z
[dzdx,dzdy]

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Slices
x(1,:) = 77
x(:,2) = 999

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Surfaces embedded in R^3

Show them the top of gdgproj.m
Run gdgproj hyp1
Show them hyp1setup.m

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Dx/Ds et al.


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quiver3 plots:  foot triple and vector triple