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<p>Gradient Fields - Example 1: Paraboloid (Bowl) &nbsp; <span style="color: #990000;"> $z=x^2-4x+y^2+2y$</span><br />YouTube PlayList on <a href="http://www.youtube.com/playlist?list=PLF4CB7AB822F21236" target="_blank">Gradient Vector Fields</a></p>

︡cdfc70cd-9279-4933-9585-0dfe4e00580e︡{"html": "<p>Gradient Fields - Example 1: Paraboloid (Bowl) &nbsp; <span style=\"color: #990000;\"> $z=x^2-4x+y^2+2y$</span><br />YouTube PlayList on <a href=\"http://www.youtube.com/playlist?list=PLF4CB7AB822F21236\" target=\"_blank\">Gradient Vector Fields</a></p>"}︡
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var('x y')
g(x,y)=(x^2-4*x+y^2+2*y)
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V=plot_vector_field(g.gradient(),(x,-5,5),(y,-5,5),color='red')
show(V, figsize=5)
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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)
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C= contour_plot(g,(x,-5,5),(y,-5,5), cmap='hsv',fill=False, contours=10) 
show(C , figsize=5)
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show(C+V, figsize=5)
︡45ac27ca-ac18-4ef7-9eee-bf240bdc9a45︡{"html": "<font color='black'><img src='cell://sage0.png'></font>"}︡