Math 215 - Multivariate Calculus - Sage

var('t','x','y')      #Define your variables
r(t) = (t,4-t^2)      #State your parametrization  
bounds = (t,-1,2)     #Give bounds for the parametrization
F(x,y) = (2*x+y,-x)   #State the vector field
xbounds = (x,-1,2)    #These bounds are useful if you want to make a good plot.
ybounds = (y,0,4)    #These bounds are useful if you want to make a good plot.

dr = r.diff(t)        #Compute the derivative
dW=F(*r(t)).dot_product(dr(t)) #Find a little bit of work.  The code r[0] gives the first component, and r[1] gives the second. 
W=integrate(dW,bounds)

pretty_print(html("""The work done by $F=%s$ along the curve r=$%s$ 
     over the bounds $%s$ is $%s$"""%tuple(map(latex, [F(x,y), r(t), bounds, W]))))
show(table([
[r"$\vec r(t)$", "$(x,y)$", r(t)],
[r"$d\vec r$", "$(dx,dy)$", dr(t)],
["$F(x,y)$", F(x,y), F(*r(t))],
["$M$", F(x,y)[0], F(*r(t))[0]],
["$N$", F(x,y)[1], F(*r(t))[1]],
[r"$dW=F\cdot d\vec r$", "$Mdx+Ndy$", dW],
[r"$W=\int_C F\cdot d\vec r$",W,W]
]
))

p=parametric_plot(r(t),bounds)
p+=plot_vector_field(F,xbounds,ybounds)
show(p)
Work Calculator
