determine the length of a particle’s path in space by using the arc-length function
explain the meaning of the curvature of a curve in space and state its formula
describe the meaning of the normal and binormal vectors of a curve in space
additional notes
arc length and arc length parameterization
if \( C: \vec{r}(t) \) is smooth over an interval \( a \le t \le b \),
then \( \displaystyle{s(t)=\int_{a}^{t}\|\vec{v}(u)\|\,du } \)
is strictly increasing on the interval \( a \le t \le b \), which
means that \( s(t) \) is an invertible function, and so we can
"parameterize \( C: \vec{r}(t) \) by arc length"
an arc length parameterization, \( C: \vec{r}(s) \), of
a curve \( C \) has many advantages, just knowing that an arc length
parameterization exists, even without knowing what it is, will
facilitate several calculations
the definition of a smooth curve
\( C: \vec{r}(t) \) is called smooth at \( t=t_{0} \) if
\( \vec{r}(t) \) is continuous at \( t=t_{0} \) and
\( \vec{r}'(t_{0})\neq\vec{0} \)
if \( C: \vec{r}(t) \) is smooth at \( t=t_{0} \),
then we can define a tangent line to \( C: \vec{r}(t) \) at
\( \vec{r}(t_{0}) \)
if \( C: \vec{r}(t) \) is smooth over an entire interval
\( a \le t \le b \), then there are no cusps or corners, no
stopping or doubling-back on the path over that interval, and we
can define the arc length of \( C \) from \( \vec{r}(a) \) to
\( \vec{r}(b) \)
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