SageMath syntax for Section 3.2

SageMath is a free open-source mathematics software system. You can use a SageMath cell online by going here.

• Define and differentiate vector-valued function $\vec{r}$:

x
var("t")
r=vector([t,t^2,t^3])
diff(r,t)

x
var("t")
r=vector([sin(t),cos(t^2),exp(-t^3)])
rp=diff(r,t)
rp(t=1.0)

• Compute a principal unit tangent vector.

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var("t")
r=vector([sin(2*t),-cos(2*t),t])
rp=diff(r,t)
T=rp/rp.norm()
T(t=pi/2)

• Compute and plot a 2d-curve and a principal unit tangent vector.

x
var("t")
r=vector([sin(2*t),sin(t)*cos(2*t)])
rp=diff(r,t)
T=rp/rp.norm()
p1=parametric_plot(r,(t,0,pi))
p2=plot(T(t=pi/3), start=r(t=pi/3), color="green")
show(p1+p2)

• Integrate a vector-valued function.

x
var("t")
r=vector([sin(t),cos(t),t])
integrate(r,t)

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var("t")
r=vector([t,t^2,t^3])
integrate(r,t,1,2)