{{{id=6|
# Here are some examples for the following two procedures:
    
# f(x,y) = -(x^3+y^3-3*x*y)       on [-1,2] x [-1,2]
# f(x,y) = -(x^2+y^2)             on [-1,1] x [-1,1]
# f(x,y) = -(x^2+y^2-2*x*y+1)     on [-1,1] x [-1,1]
# f(x,y) = x*exp(-x^2-y^2)        on [-1,1] x [-1,1]
# f(x,y) = six(x)*cos(y)          on [-pi,pi] x [-pi,pi]
# f(x,y) = six(x)*sin(y)          on [-pi,pi] x [-pi,pi]
# f(x,y) = cos(x)*cos(y)          on [-pi,pi] x [-pi,pi]
# f(x,y) = sin(x)*sin(y)*sin(x+y) on [0,pi] x [0,pi]
# f(x,y) = sin(x)+sin(y)+sin(x+y) on [0,pi] x [0,pi]
///
}}}

{{{id=1|
var('x,y')
@interact
def example(f=x*exp(-x^2-y^2), Range_x='(-1,1)', Range_y='(-1,1)', View = selector(['Filled 2d Contour Plot', '2d Plot with Contour Lines', '3d Plot'], nrows=1, label="View")):
    
    xmin, xmax = sage_eval(Range_x); ymin, ymax = sage_eval(Range_y)
    
    if View == '3d Plot':
        P = plot3d(f, (x, xmin, xmax), (y, ymin, ymax))
        
        ff = fast_float(f.diff(x), 'x', 'y')
        gg = fast_float(f.diff(y), 'x', 'y')
        hh = fast_float(f, 'x', 'y')
    
        rects = Graphics()
    
        step = 20

        xborder = (xmax-xmin)/(2*step)
        yborder = (ymax-ymin)/(2*step)

        for i in range(step):
            for j in range(step):
 
                xx=xmin+i*(xmax-xmin)/(step-1)
                yy=ymin+j*(ymax-ymin)/(step-1)
            
                vx=ff(xx,yy)
                vy=gg(xx,yy)
            
                norm=sqrt(vx^2/xborder^2+vy^2/yborder^2)
 
                if norm != 0:
                    vx=vx/norm
                    vy=vy/norm
 
                rects += line3d([(xx,yy,hh(xx,yy)), (xx+vx,yy+vy,hh(xx+vx,yy+vy))], arrow_head=True, rgbcolor=(0,0,0))
        
        show(P+rects)
    else:
        if View == 'Filled 2d Contour Plot':
            C = contour_plot(f, (x,xmin,xmax), (y,ymin,ymax), plot_points=30, contours=15, cmap='autumn')
        elif View == '2d Plot with Contour Lines':
            C = contour_plot(f, (x,xmin,xmax), (y,ymin,ymax), plot_points=30, contours=15, fill=False)    
        V = plot_vector_field( [f.diff(x),f.diff(y)], (x,xmin,xmax), (y,ymin,ymax) )
    
        show(C+V)
///
}}}

{{{id=2|

///
}}}

{{{id=3|
# For the following procedure, you have to specify the domain of the function in the first four lines.
# I am sorry for this inconvenience.

# Convert irrational numbers to float, e.g., use "xmin=float(-pi)" instead of "xmin=-pi".
///
}}}

{{{id=4|
xmin=float(-pi)
xmax=float(pi)
ymin=float(-pi)
ymax=float(pi)

var('x,y')
@interact
def example(f=sin(x)*sin(y), Start_x=((xmin+xmax)/2,(xmin, xmax)), Start_y=((ymin+ymax)/2,(ymin, ymax)), View = selector(['Filled 2d Contour Plot', '2d Plot with Contour Lines', '3d Plot'], nrows=1, label="View")):
    
    step_size=min((xmax-xmin)/100,(ymax-ymin)/100)
    steps=1000
    
    ff = fast_float(f.diff(x), 'x', 'y')
    gg = fast_float(f.diff(y), 'x', 'y')
    hh = fast_float(f, 'x', 'y')
    
    if View == '3d Plot':
        P = plot3d(f, (x, xmin, xmax), (y, ymin, ymax))
    
        rects = Graphics()
    
        step = 20

        xborder = (xmax-xmin)/(2*step)
        yborder = (ymax-ymin)/(2*step)

        for i in range(step):
            for j in range(step):
 
                xx=xmin+i*(xmax-xmin)/(step-1)
                yy=ymin+j*(ymax-ymin)/(step-1)
            
                vx=ff(xx,yy)
                vy=gg(xx,yy)
            
                norm=sqrt(vx^2/xborder^2+vy^2/yborder^2)
 
                if norm != 0:
                    vx=vx/norm
                    vy=vy/norm
 
                rects += line3d([(xx,yy,hh(xx,yy)), (xx+vx,yy+vy,hh(xx+vx,yy+vy))], arrow_head=True, rgbcolor=(0,0,0))
        
        steps = int(steps)

        points = Graphics()
        xx = Start_x
        yy = Start_y
    
        for i in range(steps):
            
            nx = xx + step_size * ff(xx,yy)
            ny = yy + step_size * gg(xx,yy)
            
            if nx >= xmin:
                if nx <= xmax:
                    if ny >= ymin:
                        if ny <= ymax:
                            points += line([(xx,yy,hh(xx,yy)),(nx,ny,hh(nx,ny))],color='red',thickness=3)
            
            xx = nx
            yy = ny
                
        starting_point = point3d([(Start_x,Start_y,hh(Start_x,Start_y))],size=10,color='red')
  
        show(P+rects+points+starting_point)
    
    
    else:
        if View == 'Filled 2d Contour Plot':
            C = contour_plot(f, (x,xmin,xmax), (y,ymin,ymax), plot_points=30, contours=15, cmap='autumn')
        elif View == '2d Plot with Contour Lines':
            C = contour_plot(f, (x,xmin,xmax), (y,ymin,ymax), plot_points=30, contours=15, fill=False)    
        V = plot_vector_field( [f.diff(x),f.diff(y)], (x,xmin,xmax), (y,ymin,ymax) )
    
        steps = int(steps)

        points = Graphics()
        xx = Start_x
        yy = Start_y
    
        for i in range(steps):
            
            nx = xx + step_size * ff(xx,yy)
            ny = yy + step_size * gg(xx,yy)
            
            if nx >= xmin:
                if nx <= xmax:
                    if ny >= ymin:
                        if ny <= ymax:
                            points += line([(xx,yy),(nx,ny)])
            
            xx = nx
            yy = ny
              
            
        starting_point = point([(Start_x,Start_y)], pointsize=50)
        
    
        show(C+V+starting_point+points)
///
}}}

{{{id=5|

///
}}}