︠a3b6c1b0-d8ab-44eb-b56b-f77ee1a23e4ci︠
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Computing the line integral of a vector field in 3 dimensions.
John Travis
Mississippi College

︡e2b1a30d-f908-495a-85e4-b74d849e2cc1︡{"html": "Computing the line integral of a vector field in 3 dimensions.\nJohn Travis\nMississippi College"}︡
︠77afb780-a86b-4c2a-a140-191ccaf93c94asi︠
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## This worksheet interactively computes and displays the line integral of a 3D vector field 
## over a given smooth curve C
## 
## John Travis
## Mississippi College
## 06/16/11
##

var('x,y,z,t,s')

@interact
def _(M=input_box(default=x*y*z,label="$M(x,y,z)$"),
 N=input_box(default=-y*z,label="$N(x,y,z)$"),
 P=input_box(default=z*y,label="$P(x,y,z)$"),
 u=input_box(default=cos(t),label="$x=u(t)$"),
 v=input_box(default=2*sin(t),label="$y=v(t)$"),
 w=input_box(default=t*(t-2*pi)/pi,label="$z=w(t)$"),
 tt = range_slider(-2*pi, 2*pi, pi/6, default=(0,2*pi), label='t Range'),
 xx = range_slider(-5, 5, 1, default=(-1,1), label='x Range'),
 yy = range_slider(-5, 5, 1, default=(-2,2), label='y Range'),
 zz = range_slider(-5, 5, 1, default=(-3,1), label='z Range'),
 in_3d=checkbox(true)):

# setup the parts and then compute the line integral
 dr = [derivative(u(t),t),derivative(v(t),t),derivative(w(t),t)]
 A = (M(x=u(t),y=v(t),z=w(t))*dr[0]
 +N(x=u(t),y=v(t),z=w(t))*dr[1]
 +P(x=u(t),y=v(t),z=w(t))*dr[2])
 global line_integral
 line_integral = integral(A(t=t),t,tt[0],tt[1])
 
 html(r'<h2 align=center>$ \int_{C} \left \langle M,N,P \right \rangle dr $ = $ %s $ </h2>'%latex(line_integral))
 G = plot_vector_field3d((M,N,P),(x,xx[0],xx[1]),(y,yy[0],yy[1]),(z,zz[0],zz[1]),plot_points=6)
 G += parametric_plot3d([u,v,w],(t,tt[0],tt[1]),thickness='5',color='yellow')
 if in_3d:
 show(G,stereo='redcyan',spin=true)
 else:
 show(G,perspective_depth=true)
︡c46c473d-d58e-4065-9a17-e1fe3941e028︡{"auto":true}︡{"stdout":"(x, y, z, t, s)\n"}︡{"interact":{"style":"None","flicker":false,"layout":[[["M",12,null]],[["N",12,null]],[["P",12,null]],[["u",12,null]],[["v",12,null]],[["w",12,null]],[["tt",12,null]],[["xx",12,null]],[["yy",12,null]],[["zz",12,null]],[["in_3d",12,null]],[["",12,null]]],"id":"abe1cfed-600f-44e1-86d6-e830e42c3f7c","controls":[{"control_type":"input-box","default":"x*y*z","label":"$M(x,y,z)$","nrows":1,"width":null,"readonly":false,"submit_button":null,"var":"M","type":null},{"control_type":"input-box","default":"-y*z","label":"$N(x,y,z)$","nrows":1,"width":null,"readonly":false,"submit_button":null,"var":"N","type":null},{"control_type":"input-box","default":"y*z","label":"$P(x,y,z)$","nrows":1,"width":null,"readonly":false,"submit_button":null,"var":"P","type":null},{"control_type":"input-box","default":"cos(t)","label":"$x=u(t)$","nrows":1,"width":null,"readonly":false,"submit_button":null,"var":"u","type":null},{"control_type":"input-box","default":"2*sin(t)","label":"$y=v(t)$","nrows":1,"width":null,"readonly":false,"submit_button":null,"var":"v","type":null},{"control_type":"input-box","default":"-(2*pi - t)*t/pi","label":"$z=w(t)$","nrows":1,"width":null,"readonly":false,"submit_button":null,"var":"w","type":null},{"control_type":"range-slider","default":[12,24],"var":"tt","width":null,"vals":["-2*pi","-11/6*pi","-5/3*pi","-3/2*pi","-4/3*pi","-7/6*pi","-pi","-5/6*pi","-2/3*pi","-1/2*pi","-1/3*pi","-1/6*pi","0","1/6*pi","1/3*pi","1/2*pi","2/3*pi","5/6*pi","pi","7/6*pi","4/3*pi","3/2*pi","5/3*pi","11/6*pi","2*pi"],"animate":true,"label":"t Range","display_value":true},{"control_type":"range-slider","default":[4,6],"var":"xx","width":null,"vals":["-5","-4","-3","-2","-1","0","1","2","3","4","5"],"animate":true,"label":"x Range","display_value":true},{"control_type":"range-slider","default":[3,7],"var":"yy","width":null,"vals":["-5","-4","-3","-2","-1","0","1","2","3","4","5"],"animate":true,"label":"y Range","display_value":true},{"control_type":"range-slider","default":[2,6],"var":"zz","width":null,"vals":["-5","-4","-3","-2","-1","0","1","2","3","4","5"],"animate":true,"label":"z Range","display_value":true},{"default":true,"var":"in_3d","readonly":false,"control_type":"checkbox","label":"in_3d"}]}}︡
︠c6e579e5-4764-4bf3-8886-b4d9e22ff02b︠
# Evaluate this cell to get a numerical approximation to the line integral evaluated above...
line_integral.n()
︡0e1565c3-5001-405a-9e9a-ce93beecd879︡{"stdout": "13.5010967590442"}︡

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