Определение и дифференцирование вектор-значных функций

Packaging motion into a single arrow

• A parametric curve gives $x(t)$ and $y(t)$ separately. Bundle them into one object: a vector-valued function 向量值函数.
• $\mathbf{r}(t)=\langle x(t),\,y(t)\rangle$ is the position vector — an arrow from the origin to the moving point.
• It describes planar motion compactly, tracking both coordinates at once.
• Everything from parametric curves reappears here, in vector form.

Differentiate component by component

• To differentiate a vector-valued function, just differentiate each component separately:
• $$\mathbf{r}'(t)=\langle x'(t),\,y'(t)\rangle$$
• No new rules — the derivative of a vector is the vector of the derivatives.
• $\mathbf{r}'(t)$ is the velocity vector, pointing in the direction of motion.

A vector-valued line: start at a, then slide by t lots of the direction b

Velocity and acceleration vectors

• $\mathbf{v}(t)=\mathbf{r}'(t)=\langle x'(t),\,y'(t)\rangle$ — the velocity vector.
• $\mathbf{a}(t)=\mathbf{r}''(t)=\langle x''(t),\,y''(t)\rangle$ — the acceleration vector.
• Each is found by differentiating the components again.
• The velocity vector is tangent to the path; its length is the speed.

Speed is the magnitude of velocity

• Speed is not a vector — it's the magnitude $|\mathbf{v}(t)|=\sqrt{\big(x'(t)\big)^2+\big(y'(t)\big)^2}$.
• That's the same speed integrand as parametric arc length.
• So velocity carries direction and size; speed is just the size.
• Differentiate to go position → velocity → acceleration, component by component.