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Author: Juan Carlos Bustamante
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typeset_mode(True)

Exercice n. 28, section 1.1

var('x,y') f(x,y)=sqrt(16-x^2 - 16 * y ^2) contour_plot(f,(x,-4,4),(y,-1,1), colorbar=True)
(x\displaystyle x, y\displaystyle y)

Exercice n. 9, section 1.2

var('x,y') f(x,y) = y^4/(x^4+3*y^4) contour_plot(f,(x,-1,1),(y,-1,1), colorbar=True)
(x\displaystyle x, y\displaystyle y)

Exercice n. 39, section 1.2

var('x,y') f(x,y) = x*y /(x^2 +x*y+y^2) contour_plot(f,(x,-1,1),(y,-1,1), colorbar = True)
(x\displaystyle x, y\displaystyle y)
%md ## Exercice n. 94. section 2.1 Il faut définir

Exercice n. 94. section 2.1

def f(x,y) : if (x!=0)&(y!=0) : return (x^3*y-x*y^3)/(x**2+y**2) else : return 0
f(0,0)
0\displaystyle 0
cmsel = [colormaps['autumn'](i) for i in sxrange(0,1,0.05)] S = plot3d(f,(x,-1,1),(y,-1,1), adaptive=True, color = cmsel) S.show()
3D rendering not yet implemented
diff(f,x).factor() diff(f,y).factor()
(x**2+y**2contour_plot?
/ext/sage/sage-8.0/local/lib/python2.7/site-packages/urllib3/contrib/pyopenssl.py:46: DeprecationWarning: OpenSSL.rand is deprecated - you should use os.urandom instead import OpenSSL.SSL
File: /ext/sage/sage-8.0/local/lib/python2.7/site-packages/sage/misc/decorators.py Signature : contour_plot(f, xrange, yrange, axes=False, linestyles=None, region=None, labels=False, plot_points=100, linewidths=None, colorbar=False, contours=None, aspect_ratio=1, legend_label=None, frame=True, fill=True, label_inline=None, label_fmt='%1.2f', label_fontsize=9, label_colors='blue', label_inline_spacing=3, colorbar_spacing=None, colorbar_orientation='vertical', colorbar_format=None, **kwds) Docstring : "contour_plot" takes a function of two variables, f(x,y) and plots contour lines of the function over the specified "xrange" and "yrange" as demonstrated below. "contour_plot(f, (xmin,xmax), (ymin,ymax), ...)" INPUT: * "f" -- a function of two variables * "(xmin,xmax)" -- 2-tuple, the range of "x" values OR 3-tuple "(x,xmin,xmax)" * "(ymin,ymax)" -- 2-tuple, the range of "y" values OR 3-tuple "(y,ymin,ymax)" The following inputs must all be passed in as named parameters: * "plot_points" -- integer (default: 100); number of points to plot in each direction of the grid. For old computers, 25 is fine, but should not be used to verify specific intersection points. * "fill" -- bool (default: "True"), whether to color in the area between contour lines * "cmap" -- a colormap (default: "'gray'"), the name of a predefined colormap, a list of colors or an instance of a matplotlib Colormap. Type: "import matplotlib.cm; matplotlib.cm.datad.keys()" for available colormap names. * "contours" -- integer or list of numbers (default: "None"): If a list of numbers is given, then this specifies the contour levels to use. If an integer is given, then this many contour lines are used, but the exact levels are determined automatically. If "None" is passed (or the option is not given), then the number of contour lines is determined automatically, and is usually about 5. * "linewidths" -- integer or list of integer (default: None), if a single integer all levels will be of the width given, otherwise the levels will be plotted with the width in the order given. If the list is shorter than the number of contours, then the widths will be repeated cyclically. * "linestyles" -- string or list of strings (default: None), the style of the lines to be plotted, one of: ""solid"", ""dashed"", ""dashdot"", ""dotted"", respectively ""-"", ""--"", ""-."", "":"". If the list is shorter than the number of contours, then the styles will be repeated cyclically. * "labels" -- boolean (default: False) Show level labels or not. The following options are to adjust the style and placement of labels, they have no effect if no labels are shown. * "label_fontsize" -- integer (default: 9), the font size of the labels. * "label_colors" -- string or sequence of colors (default: None) If a string, gives the name of a single color with which to draw all labels. If a sequence, gives the colors of the labels. A color is a string giving the name of one or a 3-tuple of floats. * "label_inline" -- boolean (default: False if fill is True, otherwise True), controls whether the underlying contour is removed or not. * "label_inline_spacing" -- integer (default: 3), When inline, this is the amount of contour that is removed from each side, in pixels. * "label_fmt" -- a format string (default: "%1.2f"), this is used to get the label text from the level. This can also be a dictionary with the contour levels as keys and corresponding text string labels as values. It can also be any callable which returns a string when called with a numeric contour level. * "colorbar" -- boolean (default: False) Show a colorbar or not. The following options are to adjust the style and placement of colorbars. They have no effect if a colorbar is not shown. * "colorbar_orientation" -- string (default: 'vertical'), controls placement of the colorbar, can be either 'vertical' or 'horizontal' * "colorbar_format" -- a format string, this is used to format the colorbar labels. * "colorbar_spacing" -- string (default: 'proportional'). If 'proportional', make the contour divisions proportional to values. If 'uniform', space the colorbar divisions uniformly, without regard for numeric values. * "legend_label" -- the label for this item in the legend * "region" - (default: None) If region is given, it must be a function of two variables. Only segments of the surface where region(x,y) returns a number >0 will be included in the plot. EXAMPLES: Here we plot a simple function of two variables. Note that since the input function is an expression, we need to explicitly declare the variables in 3-tuples for the range: sage: x,y = var('x,y') sage: contour_plot(cos(x^2 + y^2), (x,-4,4), (y,-4,4)) Graphics object consisting of 1 graphics primitive Here we change the ranges and add some options: sage: x,y = var('x,y') sage: contour_plot((x^2) * cos(x*y), (x,-10,5), (y,-5,5), fill=False, plot_points=150) Graphics object consisting of 1 graphics primitive An even more complicated plot: sage: x,y = var('x,y') sage: contour_plot(sin(x^2+y^2) * cos(x) * sin(y), (x,-4,4), (y,-4,4), plot_points=150) Graphics object consisting of 1 graphics primitive Some elliptic curves, but with symbolic endpoints. In the first example, the plot is rotated 90 degrees because we switch the variables x, y: sage: x,y = var('x,y') sage: contour_plot(y^2 + 1 - x^3 - x, (y,-pi,pi), (x,-pi,pi)) Graphics object consisting of 1 graphics primitive sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi)) Graphics object consisting of 1 graphics primitive We can play with the contour levels: sage: x,y = var('x,y') sage: f(x,y) = x^2 + y^2 sage: contour_plot(f, (-2,2), (-2,2)) Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (-2,2), (-2,2), contours=2, cmap=[(1,0,0), (0,1,0), (0,0,1)]) Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (-2,2), (-2,2), ....: contours=(0.1,1.0,1.2,1.4), cmap='hsv') Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (-2,2), (-2,2), contours=(1.0,), fill=False) Graphics object consisting of 1 graphics primitive sage: contour_plot(x - y^2, (x,-5,5), (y,-3,3), contours=[-4,0,1]) Graphics object consisting of 1 graphics primitive We can change the style of the lines: sage: contour_plot(f, (-2,2), (-2,2), fill=False, linewidths=10) Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (-2,2), (-2,2), fill=False, linestyles='dashdot') Graphics object consisting of 1 graphics primitive sage: P = contour_plot(x^2 - y^2, (x,-3,3), (y,-3,3), ....: contours=[0,1,2,3,4], linewidths=[1,5], ....: linestyles=['solid','dashed'], fill=False) sage: P Graphics object consisting of 1 graphics primitive sage: P = contour_plot(x^2 - y^2, (x,-3,3), (y,-3,3), ....: contours=[0,1,2,3,4], linewidths=[1,5], ....: linestyles=['solid','dashed']) sage: P Graphics object consisting of 1 graphics primitive sage: P = contour_plot(x^2 - y^2, (x,-3,3), (y,-3,3), ....: contours=[0,1,2,3,4], linewidths=[1,5], ....: linestyles=['-',':']) sage: P Graphics object consisting of 1 graphics primitive We can add labels and play with them: sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), ....: fill=False, cmap='hsv', labels=True) Graphics object consisting of 1 graphics primitive sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), ....: fill=False, cmap='hsv', ....: labels=True, label_fmt="%1.0f", ....: label_colors='black') sage: P Graphics object consisting of 1 graphics primitive sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), ....: fill=False, cmap='hsv', labels=True, ....: contours=[-4,0,4], ....: label_fmt={-4:"low", 0:"medium", 4: "hi"}, ....: label_colors='black') sage: P Graphics object consisting of 1 graphics primitive sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), ....: fill=False, cmap='hsv', labels=True, ....: contours=[-4,0,4], label_fmt=lambda x: "$z=%s$"%x, ....: label_colors='black', label_inline=True, ....: label_fontsize=12) sage: P Graphics object consisting of 1 graphics primitive sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), ....: fill=False, cmap='hsv', labels=True, ....: label_fontsize=18) sage: P Graphics object consisting of 1 graphics primitive sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), ....: fill=False, cmap='hsv', labels=True, ....: label_inline_spacing=1) sage: P Graphics object consisting of 1 graphics primitive sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), ....: fill=False, cmap='hsv', labels=True, ....: label_inline=False) sage: P Graphics object consisting of 1 graphics primitive We can change the color of the labels if so desired: sage: contour_plot(f, (-2,2), (-2,2), labels=True, label_colors='red') Graphics object consisting of 1 graphics primitive We can add a colorbar as well: sage: f(x, y)=x^2-y^2 sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True) Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True, colorbar_orientation='horizontal') Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4], colorbar=True) Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4], ....: colorbar=True, colorbar_spacing='uniform') Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[0,2,3,6], ....: colorbar=True, colorbar_format='%.3f') Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (x,-3,3), (y,-3,3), labels=True, ....: label_colors='red', contours=[0,2,3,6], ....: colorbar=True) Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (x,-3,3), (y,-3,3), cmap='winter', ....: contours=20, fill=False, colorbar=True) Graphics object consisting of 1 graphics primitive This should plot concentric circles centered at the origin: sage: x,y = var('x,y') sage: contour_plot(x^2 + y^2-2,(x,-1,1), (y,-1,1)) Graphics object consisting of 1 graphics primitive Extra options will get passed on to show(), as long as they are valid: sage: f(x,y) = cos(x) + sin(y) sage: contour_plot(f, (0,pi), (0,pi), axes=True) Graphics object consisting of 1 graphics primitive sage: contour_plot(f, (0,pi), (0,pi)).show(axes=True) # These are equivalent One can also plot over a reduced region: sage: contour_plot(x**2 - y**2, (x,-2,2), (y,-2,2), region=x - y, plot_points=300) Graphics object consisting of 1 graphics primitive Note that with "fill=False" and grayscale contours, there is the possibility of confusion between the contours and the axes, so use "fill=False" together with "axes=True" with caution: sage: contour_plot(f, (-pi,pi), (-pi,pi), fill=False, axes=True) Graphics object consisting of 1 graphics primitive