Question

Find a piecewise smooth parametrization of the path C. r(t) = ti + tj 0 ≤ t ≤ 1 _____????______ 1 ≤ t ≤ 2?

Answer 1

`r(t) = { (that(i)+that(j) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ " for " 0 <= t <= 1), ((2-t)hat(i)+(2-t)^2hat(j) " for " 1 <= t <= 2) :}`

Explanation:

Note that the curved portion of the curve `C` is a parabola with horizontal axis.

If it were parameterised from `(0, 0)` to `(1, 1)` for `t in [0, 1]` then we could write:

`r(t) = that(i)+t^2hat(j)`

As it is, we want to parameterise it in the opposite direction for `t` in the range `[1, 2]`.

Hence we want:

`r(t) = (2-t)hat(i)+(2-t)^2hat(j)`

So the piecewise parameterisation can be written:

`r(t) = { (that(i)+that(j) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ " for " 0 <= t <= 1), ((2-t)hat(i)+(2-t)^2hat(j) " for " 1 <= t <= 2) :}`