3103-Math 2500.sagews

Math 2500

Niles Johnson, University of Georgia



A function $f: \mathbb{R}^2 \to \mathbb{R}$ and its graph:

var('x,y')

f(x,y) = x^2 -y^2
fgraph = plot3d(f, (x,-3,3), (y,-3,3)) 
fgraph.show()

Slices of $f$ along $x = x_0$ and $y=y_0$:

x0 = 1
y0 = 0
xslice=parametric_plot((x0,y,f(x=x0)), (x,-3,3), (y,-3,3),color='red', thickness=4) 
yslice=parametric_plot((x,y0,f(y=y0)), (x,-3,3), (y,-3,3),color='red', thickness=4) 

fgraph+xslice+yslice

Contours of $f$:

contour_plot(f(x,y), (x,-3,3), (y,-3,3),colorbar=True,cmap=colormaps.Spectral)

c0 = 1
c1 = 2.5
c2 = 4
contour_plot(f(x,y), (x,-3,3), (y,-3,3),contours=[c0,c1,c2],fill=False,labels=True,cmap='flag')

Functions $\mathbb{R}^3 \to \mathbb{R}$

Level surfaces:

f(x,y,z) = z - x^2 - y^2
contours=[-1, 0, 1, 2]
fc=fast_callable(f,domain=float)

Alternate functions:

f(x,y,z)=sin(x)*cos(y)*z^2 
contours = [1, 2, 4]

f(x,y,z) = sin(x) - y*cos(z)
contours = [0,1,3]

fc=fast_callable(f,domain=float)

p=point([0,0,0]) 

for c in contours: 
    H = contours.index(c)*1.0/len(contours)
    p+=implicit_plot3d(fc, (x,-3,3), (y,-3,3), (z,-3,3), contour=c, color=hue(H), opacity=0.7) 

p.show()

Slice $y=0$: we plot the points $(x,\, z,\, f(x,0,z))$ as a parametric surface.

z0 = 1
parametric_plot((x,z,f(y=0)), (x,-2,2), (z,-2,2),color='red') + \
    parametric_plot((x,z0,f(y=0, z=z0)), (x,-2,2),color='cyan', thickness=5)

Slice $z=0$: we plot the points $(x,\, y,\, f(x,y,0))$ as a parametric surface.



y0 = 0
parametric_plot((x,y,f(z=0)), (x,-1,1), (y,-1,1),color='red') + \
    parametric_plot((x,y0,f(y=y0,z=0)), (x,-1,1),color='cyan', thickness=5)