- “Flatland” movie
- This afternoon (Feb. 10), 2:30, Newton 201
- Section 13.1
- Curves in space and their tangents
- ⟨x,y,z⟩ = ⟨ f(t), g(t), h(t) ⟩
- Path of particle
- Represented as vector r(t) = f(t)i + g(t)j + h(t)k
- Limits and continuity
- r(t) = f(t)i + g(t)j + h(t)k
- Epsilon-delta definition
- r(t) = f(t)i + g(t)j + h(t)k
- Derivatives and motion
- Long equation for dr/dt = df/dt i + dg/dt j + dh/dt k
- Velocity = derivative of position
- Speed = magnitude of velocity
- Acceleration = derivative of velocity
- Unit vector gives direction of motion at t
- Differentiation rules
- Vector functions of constant length
- Vector-valued functions in muPad
- Define as vectors for best results
- Can take derivatives
- Most interestingly for now, can plot with
plot::Curve3d
- How about plotting helix from text: ⟨ cos t, sin t, t ⟩, 0 ≤ t ≤ 4π
- Examples
- Function that looks like upside-down parabola in z, above the line from (-1,-1,0) to (1,1,0)
- Maybe y = x, z = 1 - x2?
- Relationships between x, y, and z are right, but want parametric vector-valued function form r(t) = ⟨ f(t), g(t), h(t) ⟩
- So use ⟨ t, t, 1-t2 ⟩
- Check shape with muPad
- Also notice how setting t to a specific value causes muPad to evaluate vector at that t
- Function that looks like upside-down parabola in z, above the line from (-1,-1,0) to (1,1,0)
- More examples of derivatives, limits, etc.
- No new reading