We are in the process of developing differential calculus for vector-valued functions. Presently, we are focusing on the special case of scalar inputs, where the derivative is easy to define. Let us review the basic setup
Let be an interval, and let be a Euclidean space. We shall consider continuous functions
The image of under the mapping , i.e. the subset of defined by
is called a curve in . It is commonplace to refer to the function itself as a curve, though strictly speaking this is abuse of language because different functions may have the same image. For example, consider the functions
These functions are different, because they do not produce the same outputs for all inputs but their images are the same curve in and this curve is the unit circle
Definition 1: For any fixed , the Newton quotient of at is the function
Here are a few remarks on the Newton quotient of the function at the time .
First, the domain of consists of all nonzero numbers such that is contained in the domain of . For example, the Newton quotient of at the left endpoint of the domain has domain and the Newton quotient of at the left endpoint has domain
Second, is a vector in : this vector may be visualized as the directed line segment from the point to the point , scaled by the factor , as in the schematic figures below.
Third, if we view as the position of a particle moving in at time , the Newton quotient has a physical interpretation as the average velocity of the particle over the time period if , or over if The orientation of the vector is compatible with the flow of time. The idea of the derivative is that, if the curve is sufficiently smooth, then for small the vectors and should be nearly identical, and as we shrink they become an exact match: the instantaneous velocity of the particle at time
Definition 2: A function is said to be differentiable at if there exists a vector such that
and when this is the case is called the derivative of at , or the tangent vector to at . The curve is said to be smooth if it is differentiable for all and the derivative is continuous.
The main technical tool for checking differentiability and calculating derivatives is component functions. Recall that if is a basis of , then the corresponding component functions
are determined by the condition
We have the following useful theorem at our disposal.
Theorem 1: The curve is differentiable at time if and only if each of its component functions is differentiable at , and in this case the tangent vector is given by
Theorem 1 is useful for proving theorems about vector-valued functions, since it reduces the differentiation of a vector-valued function to the differentiation of its component functions, which map scalars to scalars, . This is exactly what makes curves easier to handle then general functions , whose component functions are instead mappings (as yet we do not even know how to define the derivative for functions whose domain is not one-dimensional, since this involves the ill-defined concept of dividing by a vector). But curves are relatively non-problematic because we can use Theorem 1 to reduce many statements to single variable calculus. For example, we have the following.
Propostion 1: A differentiable curve is continuous.
Proof: Since is differentiable, its component functions relative to any basis of are differentiable functions But we know from single variable calculus that a differentiable function is continuous, so each of the component functions is continuous. Now, since is continuous if and only if all of its component functions are continuous, we may conclude that is continuous.
Proposition 2: The function is constant if and only if for all
Proof: Suppose first that is constant, i.e. there exists a number such that for all In this case it is clear that the Newton quotient of at any is identically zero on its domain, so exists and equals the zero vector in
Next, suppose that is differentiable on with . Then, for each we have that
and since is a linearly independent set this forces
According to single-variable calculus, a function whose derivative is identically zero is a constant function. Therefore, there are constants such that for all , and consequently
Using exactly the same reasoning, one easily establishes the following properties of differentiable curves.
Proposition 3: Let be curves, and suppose both are differentiable at time Then the sum and the product are differentiable at and their tangent vectors at are given by and respectively.
Theorem 1 is also useful for doing concrete computations. For example, let us apply it to the functions defined by
Relative to the standard basis of the component functions of are
From single variable calculus, we know that
and therefore Theorem 1 allows us to conclude that the curve is differentiable for all times with tangent vector at time given by
these equations say that the tangent vector to the unit circle is a unit vector, and that it is orthogonal to the vector joining the origin to the point . This orthogonality is the basic ingredient in, for example, ancient weapons like the sling which David used to slay Goliath, or the mangonel developed in ancient China during the Warring States Period, which evolved into the medieval catapult that used counterweights rather than direct human strength.
Let us repeat the above computations for the curve The component functions of relative to the standard basis of are
so that by the chain rule from scalar calculus we have
for all This means that is a differentiable curve with tangent vector at time given by
Observe that we still have orthogonality,
but that is not a unit vector,
Physically, this is because describes the motion of a particle which is traveling twice as fast at the particle whose motion is described by : it traverses the unit circle counterclockwise twice as time flows from to , rather than just once. Indeed, in physics the tangent vector to a curve is known as the instantaneous velocity, and its norm is the instantaneous speed. The above calculations show that a sling swung twice as fast produces a projectile which travels twice as fast in exactly the same direction.