John Travis
Mississippi College
Showing the gradient vector at a student-entered point along with a tangent vector in the gradient direction.
The tangent vector is plotted using the vector $ \left \langle f_x,f_y,\left || \bigtriangledown f \right ||^2 \right \rangle $.
var('x,y')
@interact
def _(f = cos(x+1)*sin(x^2+y^2),x0=slider(-6/5,6/5,1/20,0),y0=slider(-6/5,6/5,1/20,1),In3D = checkbox(default=True)):
H = plot3d(f,(x, -1.5, 1.7), (y, -1.5, 1.9),opacity=0.8)
fx = derivative(f,x)
fy = derivative(f,y)
H += point3d([(x0,y0,f(x=x0,y=y0))],size=20,zorder=2,color='red')
gradnorm = (fx(x=x0,y=y0)^2+fy(x=x0,y=y0)^2)
H += arrow((x0,y0,f(x=x0,y=y0)),(x0+fx(x=x0,y=y0),y0+fy(x=x0,y=y0),f(x=x0,y=y0)),color='green')
H += arrow((x0,y0,f(x=x0,y=y0)),(x0+fx(x=x0,y=y0),y0+fy(x=x0,y=y0),f(x=x0,y=y0)+gradnorm),color='lightgreen')
if In3D:
H.show(spin=True,stereo='redcyan')
else:
H.show(spin=True)
Click to the left again to hide and once more to show the dynamic interactive window |
# showing that the gradient vectors always point in the direction of steepest ascent
var('x,y')
f(x,y) = cos(x+1)*sin(x^2+y^2)
G = contour_plot(f, (x, -1.4, 1.7), (y, -1.5, 1.9),contours= 21, plot_points=150,colorbar=1,cmap='Greys')
H = plot3d(f,(x, -1.4, 1.7), (y, -1.5, 1.9),opacity=0.8)
fx = derivative(f,x)
fy = derivative(f,y)
xs = [0,1.12,1.375,0.57,-0.05,-0.5,0.1]
ys=[-0.69,0.78,0,1.68,1,0.9,0]
show_grads = True
for k in range(len(xs)):
x0 = xs[k]
y0 = ys[k]
G += point2d([(x0,y0)],size=60,zorder=2,color='red')
if show_grads:
G += arrow((x0,y0),(x0+fx(x0,y0),y0+fy(x0,y0)),color='green')
H += point3d([(x0,y0,f(x=x0,y=y0))],size=20,zorder=2,color='red')
if show_grads:
gradnorm = (fx(x=x0,y=y0)^2+fy(x=x0,y=y0)^2)
H += arrow((x0,y0,f(x=x0,y=y0)),(x0+fx(x=x0,y=y0),y0+fy(x=x0,y=y0),f(x=x0,y=y0)),color='green')
H += arrow((x0,y0,f(x=x0,y=y0)),(x0+fx(x=x0,y=y0),y0+fy(x=x0,y=y0),f(x=x0,y=y0)+gradnorm),color='lightgreen')
G.show()
H.show(figsize=(10,15),spin=True)
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