333 Smooth Curve
A curve whose derivative is continuous and never the zero vector that is used to define a unique tangent direction along the path.
definition [d] (Smooth Curve = Regular Curve) A curve with a parametrization \(\mathbf{r}(t)\) on an interval \(I\) such that
- \(\dfrac{d\mathbf{r}}{dt}\) is continuous on \(I\) .
- \(\dfrac{d\mathbf{r}}{dt}(t) \neq \mathbf{0}\) for all \(t \in I\) except possibly at the endpoints.
where
- \(\mathbf{r}(t)\) is the position vector of the curve.
- \(\dfrac{d\mathbf{r}}{dt}(t)\) is the tangent vector.
- \(t\) is the parameter.
- \(I\) is the parameter interval.
- \(\mathbf{0}\) is the zero vector.
Note:
- \(\dfrac{d\mathbf{r}}{dt}(t)\) is also called the velocity vector.
- geometrically there are no sharp corners.
- geometrically there are no cusps.
- the tangent turns continuously.
333.1 References
- Stewart, J. Calculus. — smooth parametrization: \(\dfrac{d\mathbf{r}}{dt}\) continuous and \(\dfrac{d\mathbf{r}}{dt}(t)\neq\mathbf{0}\).
- do Carmo, M. P. Differential Geometry of Curves and Surfaces. — regular curve with \(\dfrac{d\alpha}{dt}(t)\neq 0\).
- Basic Derivation of Jacobian
- Chain Rule
- Cross Product
- Curl
- Curve
- Definite Integral
- Derivation of Curl
- Derivation of Divergence
- Derivation of Grad
- Derivation of Jacobian
- Derivation of Taylor Expansion
- Divergence
- Dot Product
- Envelope of Tangent Lines
- General Functions
- Gradient
- Improper Integral
- Indefinite Integral
- Jacobian
- Legendre Transform
- Legendre Transform Derivation
- Limit
- Line Integral
- Notes
- Parameterization
- Partial Derivative
- Real Parameter
- Scalar Projection
- Smooth Curve
- Vector Calculus
- Vector Differential Operator
- Vector Field
- Vector Projection