Source: math2130-fall2016-traces.mw

>	restart; with(plots):

 

Let's explore the specific surface .  

To begin with, I'll plot this surface over the domain . 

>	plot3d(4-x^2-y^2,x=-2..2,y=-sqrt(4-x^2)..sqrt(4-x^2),scaling=constrained);

 

Below... 

P is a point (actually a tiny sphere) located at x=1, y=1, and z=4-1^2-1^2. 

Q is a point (actually a tiny sphere) located at x=1, y=-1, and z=4-1^2-(-1)^2. 

A is a plot of our surface (in blue). 

B is a plot of a the trace through the plane y=1 (in yellow). 

C is a plot of the plane y=1 (in gray). 

E is a plot of the the trace through the plane x=1 (in green). 

F is a plot of the plane x=1 (in gray). 

>	P := plot3d([0.15*cos(t)*sin(p)+1,0.15*sin(t)*sin(p)+1,0.15*cos(p)+4-1^2-1^2],t=0..2*Pi,p=0..Pi,color=red,numpoints=10): Q := plot3d([0.15*cos(t)*sin(p)+1,0.15*sin(t)*sin(p)-1,0.15*cos(p)+4-1^2-(-1)^2],t=0..2*Pi,p=0..Pi,color=red,numpoints=10):

 

>	A := plot3d(4-x^2-y^2,x=-2..2,y=-sqrt(4-x^2)..sqrt(4-x^2),scaling=constrained,color=blue):

 

>	B := spacecurve(<t,1,4-t^2-1^2>,t=-2..2,thickness=3,color=yellow):

 

>	C := plot3d(<x,1,z>,x=-2..2,z=-1..5,color=gray):

 

>	E := spacecurve(<1,t,4-t^2-1^2>,t=-2..2,thickness=3,color=green):

 

>	F := plot3d(<1,y,z>,y=-2..2,z=-1..5,color=gray):

 

>	display({P,A,B,C,E,F});

 

Looking at the trace through y=1 (the yellow curve below), we see that at (x,y)=(1,1) [i.e. the red ball] the curve is going down [follow the positive x-axis direction]. This means that   (the partial of with respect to x at (1,1) is negative). 

>	display({P,A,B});

 

Looking at the trace through x=1 (the green curve below), we see that at (x,y)=(1,1) [i.e. the red ball] the curve is going down [follow the positive y-axis direction]. This means that   (the partial of with respect to y at (1,1) is negative). 

>	display({P,A,E});

 

Looking at the trace through x=1 (the green curve below), we see that at (x,y)=(1,-1) [i.e. the red ball] the curve is going up [follow the positive y-axis direction]. This means that   (the partial of with respect to y at (1,1) is positive). 

>	display({Q,A,E});

 

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