3905.sagews

Gradient Fields - Example 1: Paraboloid (Bowl)   $z=x^2-4x+y^2+2y$
YouTube PlayList on Gradient Vector Fields

var('x y')
g(x,y)=(x^2-4*x+y^2+2*y)

V=plot_vector_field(g.gradient(),(x,-5,5),(y,-5,5),color='red')
show(V, figsize=5)

W = plot3d(g,(x,-5,5),(y,-5,5), aspect_ratio=[1,1,.2], adaptive=True,color=['red','yellow']) 
S = sphere((2,-1,-5),size=1, color='blue', aspect_ratio=[1,1,1])
show(W + S, figsize=5)

C= contour_plot(g,(x,-5,5),(y,-5,5), cmap='hsv',fill=False, contours=10) 
show(C , figsize=5)

show(C+V, figsize=5)