SparrowProblem::usage="Let the temperature in a region of space be gven by T[x, y, z] = 3 + + 2 degrees.a A sparrow is flying along the curve r[s] = , 2s, at a constant speed of 3 m/s. What is the velocity of the sparrow when s = 1?b At what rate does the sparrow feel the temperature is changing at the point A, 2, 1 for which s = 1 .c At the point A, 2, 1 in what direction will the temperature be decreasing at maximum rate?d An eagle crosses the path of the sparrow at A, 2, 1, is moving at right angles to the path of the sparrow,and is also moving in a direction in which the temperature remains constant. In what directions could the eagle be flying as it passes through the point A?References: Internet: personal.math.ubc.ca, 2.7.2 Directional Derivatives And The Gradient, Exercise Stage 2, Prob.-24; Personal.";
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a) Find the velocity vector:
In[]:=
r[s_]:=,2s,;
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s
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s
In[]:=
r'[s]
Out[]=
{,2,2s}
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s
In[]:=
r'[1]
Out[]=
{1,2,2}
b) The temperature change:
In[]:=
f[x_,y_,z_]:=3++2;g[x_,y_,z_]=Grad[f[x,y,z],{x,y,z}]
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Out[]=
{6x,y,4z}
In[]:=
n=g,2,1
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Out[]=
{2,2,4}
In[]:=
{2,2,4}.{1,2,2}
14°/s
c)
AtthepointA,2,1inwhatdirectionwillthetemperaturebedecreasingatmaximumrate?
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-{2,2,4}
d) The eagle’s trajectory must be perpendicular to the gradient (so that the temperature remains constant) and attacks at a right angle against the direction of movement of the crow’s speed:
In[]:=
{1,2,2}{2,2,4}
Out[]=
{4,0,-2}
±{2,0,-1}
You can approach from the right or from the left.
In[]:=
ShowParametricPlot3D,2s,,{s,-5,5},ColorFunction->"CherryTones",Graphics3DBrown,Point,2,1,Arrowheads[.03],ArrowTube,2,1,,2,1+5{2,0,-1},Graphics3DBlack,Ball,2,1,0.5,Arrowheads[.03],ArrowTube,2,1,,2,1+5{1,2,2},ViewPoint{1.3,-2.4,2.},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize500
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Out[]=
