key term - Orientation

Definition

Orientation refers to the direction in which a curve is traced as a parameter increases. It determines the order of traversal along the curve.

5 Must Know Facts For Your Next Test

1. Orientation is determined by the direction in which the parameter $t$ increases.
2. For parametric equations, changing the orientation can reverse the direction of traversal along the curve.
3. In polar coordinates, orientation affects how angles are traced from the positive x-axis.
4. Orientation plays a crucial role in calculating line integrals and determining curve properties like tangent vectors.
5. The orientation of a parametric curve can be visualized by plotting points for increasing values of $t$.

Review Questions

Related terms

Parametric Equations:

Equations that express coordinates as functions of a parameter, typically written as $(x(t), y(t))$.

Polar Coordinates:

A coordinate system where each point on a plane is determined by an angle and a distance from a fixed point.

Tangent Vector: $\vec{r}'(t)$ represents the derivative of position vector with respect to parameter $t$, indicating direction and rate of change.