Vector Functions and Curves

A vector function of a single variable is much like a parametrized curve,
f(t)=f1(t)i+f2(t)j+f3(t)k
where i, j, k, are the vectors (1,0,0), (0,1,0), and (0,0,1), respectively. The functions fi are the component functions and much of calculus of vector functions of one variable is done by components, for example,
f'(t)=f1'(t)i+f2'(t)j+f3'(t)k.

A curve is simply defined as a vector function of a single variable r(t)=x(t)i+y(t)j+z(t)k

In[15]:=

  x[t_]:=2 Cos[t]
  y[t_]:=2 Sin[t]
  z[t_]:=1/3 t

In[16]:=

  ParametricPlot3D[{x[t],y[t],z[t]},{t,0,6 Pi}];

Up to Vector Differential Calculus