parametrization

Examples of parametrization in the following topics:

• Parametric Surfaces and Surface Integrals

  • A parametric surface is a surface in the Euclidean space $R^3$ which is defined by a parametric equation.
  • A parametric surface is a surface in the Euclidean space $R^3$ which is defined by a parametric equation with two parameters: $\vec r: \Bbb{R}^2 \rightarrow \Bbb{R}^3$.
  • Parametric representation is the most general way to specify a surface.
  • The curvature and arc length of curves on the surface can both be computed from a given parametrization.
  • The same surface admits many different parametrizations.

• Parametric Equations

  • is a parametric equation for the unit circle, where $t$ is the parameter.
  • Thus, one can describe the velocity of a particle following such a parametrized path as follows:
  • One example of a sketch defined by parametric equations.
  • Note that it is graphed on parametric axes.
  • Express two variables in terms of a third variable using parametric equations

• Calculus with Parametric Curves

  • Parametric equations are equations which depend on a single parameter.
  • Thus, one can describe the velocity of a particle following such a parametrized path as:
  • Writing these equations in parametric form gives a common parameter for both equations to depend on.
  • Writing in parametric form makes this easier to do.
  • A trajectory is a useful place to use parametric equations because it relates the horizontal and vertical distance to the time.

• Surfaces in Space

• Arc Length and Speed

  • Arc length and speed in parametric equations can be calculated using integration and the Pythagorean theorem.
  • Since there are two functions for position, and they both depend on a single parameter—time—we call these equations parametric equations.

• Linear Equations

  • The parametric form of a linear equation involves two simultaneous equations in terms of a variable parameter $t$, with the following values:

• Line Integrals

  • where $r: [a, b] \to C$ is an arbitrary bijective parametrization of the curve $C$ such that $r(a)$ and $r(b)$ give the endpoints of $C$ and $a$.
  • where $\cdot$ is the dot product and $r: [a, b] \to C$ is a bijective parametrization of the curve $C$ such that $r(a)$ and $r(b)$ give the endpoints of $C$.

• Arc Length and Surface Area

  • If a curve is defined parametrically by $x = X(t)$ and y = Y(t), then its arc length between $t = a$ and $t = b$ is:
  • The curve can be represented parametrically as $x=\sin(t), y=\cos(t)$ for $0 \leq t \leq \frac{\pi}{2}$.

• Vectors in the Plane

  • This plane may be described parametrically as the set of all points of the form$\mathbf R = \mathbf {R}_0 + s \mathbf{V} + t \mathbf{W}$, where $s$ and $t$ range over all real numbers, $\mathbf{V}$ and $\mathbf{W}$ are given linearly independent vectors defining the plane, and $\mathbf{R_0}$ is the vector representing the position of an arbitrary (but fixed) point on the plane.

• Equations of Lines and Planes