331 Real Parameter
A real parameter is a real number variable that is used to define a family of functions. It is used to select a specific member from a set.
Note: Also called parameter. Also described as a real number constant in the same role.
definition [d] (Real Parameter = Parameter) From do Carmo: a parametrized differentiable curve is a differentiable map
- \(\boldsymbol{\alpha}: I \rightarrow \mathbb{R}^{3}\)
of an open interval \(I = (a,b)\) of the real line \(\mathbb{R}\) into \(\mathbb{R}^{3}\). Differentiable means that each \(t \in I\) is mapped to a point
- \(\boldsymbol{\alpha}(t) = \bigl(x(t),\, y(t),\, z(t)\bigr)\)
in such a way that the functions \(x(t)\), \(y(t)\), \(z(t)\) are differentiable. The variable \(t\) is called the parameter of the curve.
where
- \(t\) is the real parameter.
- \(I = (a,b)\) is an open interval of \(\mathbb{R}\).
- \(\boldsymbol{\alpha}\) is the parametrized differentiable curve.
- \(x(t)\), \(y(t)\), \(z(t)\) are differentiable real-valued functions.
331.1 References
- do Carmo, M. P. Differential Geometry of Curves and Surfaces. — parametrized differentiable curve \(\boldsymbol{\alpha}: I \to \mathbb{R}^{3}\); “The variable \(t\) is called the parameter of the curve.”
- 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