1. Curves. We are interested in describing the motion of my car. Well, everyone is interested in the motion of my car. How can we describe it mathematically?
First we approximate the car as a point. The point-like car moves in time, so the value of its components are functions of time. More precisely, the position of my car is
where the functions , , and are sometimes called component functions. Another way to think about this is writing
Classical mechanics studies such curves under various circumstances. We will discuss some notions of kinematics, and study what it means to differentiate curves.
Example 1. Consider a point traveling in circular motion in the -plane. What does this look like?
Well, it’s a paramteric curve, using trigonometric functions we write
This descrivbes an anti-clockwise circular motion with radius 1, lying in the -plane.
Example 2. Suppose a particle travels along a parabolic curve, what does the curve look like? We can write it explicitly as
This is precisely aa parabola.
2. Calculus with Vector-Valued Functions. We should recall the construction of the tangent line to a curve at a point had us write
When we consider the situation when we work with instead of a function . We have be the base point for the tangent to the curve, then we have
The problem: what exactly is ?
3. We can let (or more generally the codomain can be for any positive integer ). We have
describe the rate of change of the position vector with respect to time. What does this look like? Well, writiing out
where primes denote differentiation with respect to time.
Remark 1. We can keep iterating this procedure to obtain higher order derivatives of a curve.
4. Kinematics. We have describe the position of a particle. The velocity of the particle is a vector-valued function
However, we also can consider the speed or the magnitude of the velocity
Observe the speed is a scalar quantity: it’s just some function of time. The velocity is a vector-valued function of time.
We have one last kinematical quantity to consider: the acceleration. This is just the rate of change of velocity with respect to time:
Observe we can reconstruct the position from the velocity by considering
which when we consider we have the integral evaluated “component-wise”:
We can similarly reconstruct velocity from acceleration.
Example 3. Consider the curve describing circular motion
What is its velocity vector, acceleration vector, and speed?
Solution: We find its velocity
From this we can compute its speed as
The acceleration is precisely the derivative of the velocity vector
That concludes our example.
Exercise 1. Find the velocity vector, speed, and acceleration of the parabolic curve
Exercise 2. Let and be differentiable vector-valued functions of time. Prove or find a counter-example that
Exercise 3. Calculate