Ordinary Differentiation Of Vectors

1. Core Concept Overview & Intuition

1.1 Lesson Learning Outcomes

By the end of this unit the learner should be able to:

• Evaluate ordinary derivatives of vector functions of a single real variable.
• Determine tangent and normal vectors (including the unit tangent, principal unit normal, and binormal vectors) to a space curve.
• Determine the velocity and acceleration vectors of a moving particle from its position vector.

1.2 The Central Idea

Ordinary differentiation of vectors extends the ordinary rules of differential calculus to vector-valued functions of one real variable. A vector function of a real variable $t$ has the form

$$
\vec{r}(t)=f_1(t)\,\hat{i}+f_2(t)\,\hat{j}+f_3(t)\,\hat{k}
$$

where $f_1(t)$, $f_2(t)$ and $f_3(t)$ are real functions of the real variable $t$, and $\hat{i}$, $\hat{j}$, $\hat{k}$ are the fixed unit vectors along the coordinate axes. As $t$ increases, the position vector $\vec{r}(t)$, drawn from the origin to a point $P$, traces out a curve in space.

The derivative is defined by the same limit process used in scalar calculus. If $\Delta t$ denotes an increment in $t$, the derivative of $\vec{r}(t)$ with respect to $t$ is the limit of the difference quotient as the increment tends to zero. Geometrically, this derivative is a new vector that is tangent to the traced curve at the point in question.

The key structural fact is that the ordinary rules of differentiation carry over to vector calculus, with the important qualification that the dot product and the vector (cross) product retain their nature: the product rule for a dot product produces a scalar, and the product rule for a cross product produces a vector, and in the cross-product case the order of the factors must be preserved.

1.3 Magnitude, Unit Vector and Resultant

Three elementary vector operations underpin every calculation in this unit.

• The magnitude of a vector $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ is the non-negative scalar $\lVert\vec{a}\rVert=\sqrt{a_1^2+a_2^2+a_3^2}$.
• The unit vector in the direction of a non-zero vector $\vec{a}$ is $\hat{a}=\vec{a}/\lVert\vec{a}\rVert$, a vector of magnitude one pointing in the same direction.
• The resultant of two or more vectors is their vector sum; its magnitude is computed by first forming the sum componentwise and then applying the magnitude formula.

These operations are used repeatedly: the magnitude of a derivative vector appears in the unit tangent and unit normal formulas, and the magnitude of the velocity and acceleration vectors is required in kinematics problems.

1.4 Tangent, Normal and Binormal Vectors

For a differentiable vector-valued function $\vec{r}(t)$, the derivative $\vec{r}'(t)$ is a tangent vector to the curve. Dividing by its magnitude gives the unit tangent vector $\hat{T}(t)$. The derivative of the tangent vector, $\vec{T}'(t)$, is normal to the tangent and hence to the curve; dividing by its magnitude gives the principal unit normal vector $\hat{N}(t)$, which for a non-straight curve is the unique unit vector pointing into the curve. The cross product $\hat{B}(t)=\hat{T}(t)\times\hat{N}(t)$ is the binormal vector, orthogonal to both the tangent and the normal.

1.5 Velocity and Acceleration

The motion of a particle is described by three vectors: position, velocity and acceleration. The position vector runs from the origin of the reference frame to the particle. Its Cartesian components are time dependent because the particle is in motion; the time dependence is often suppressed in notation. The velocity vector is the time derivative of the position vector and is always tangent to the trajectory at each point. The acceleration vector is the time derivative of the velocity vector, that is, the second time derivative of the position vector.

2. Mathematical Formalization & Core Definitions

> Definition (Vector function of a real variable). A vector function of a real variable $t$ is a mapping of the form $\vec{r}(t)=f_1(t)\hat{i}+f_2(t)\hat{j}+f_3(t)\hat{k}$, where $f_1$, $f_2$, $f_3$ are real-valued functions of the real variable $t$.

> Definition (Ordinary derivative of a vector function). Let $\vec{r}(t)$ be a position vector depending on the single scalar variable $t$, and let $\Delta t$ denote an increment in $t$. The derivative of $\vec{r}(t)$ with respect to $t$ is defined by the limit of the difference quotient as the increment tends to zero.

2.1 Differentiation Formulae

The following differentiation formulae apply to vector functions of a real variable $t$. In each line $\vec{a}$ and $\vec{b}$ denote differentiable vector functions of $t$, $c$ denotes a constant vector, $\phi(t)$ denotes a differentiable real (scalar) function, and $\vec{u}$ denotes a differentiable vector function.

1. Derivative of a constant vector:

$$
\frac{d}{dt}\left(\vec{c}\right)=\vec{0}
$$

2. Derivative of a scalar multiple:

$$
\frac{d}{dt}\left(\phi(t)\,\vec{a}\right)=\phi'(t)\,\vec{a}+\phi(t)\,\vec{a}'(t)
$$

3. Derivative of a sum or difference:

$$
\frac{d}{dt}\left(\vec{a}\pm\vec{b}\right)=\vec{a}'(t)\pm\vec{b}'(t)
$$

4. Derivative of a dot product (the result is a scalar):

$$
\frac{d}{dt}\left(\vec{a}\cdot\vec{b}\right)=\vec{a}'(t)\cdot\vec{b}+\vec{a}\cdot\vec{b}'(t)
$$

5. Derivative of a vector (cross) product (the result is a vector, and order is preserved):

$$
\frac{d}{dt}\left(\vec{a}\times\vec{b}\right)=\vec{a}'(t)\times\vec{b}+\vec{a}\times\vec{b}'(t)
$$

6. Chain rule for a vector function of a scalar function:

$$
\frac{d}{dt}\left[\vec{u}\left(\phi(t)\right)\right]=\phi'(t)\,\vec{u}'\left(\phi(t)\right)
$$

Higher-order derivatives are obtained by repeated differentiation:

$$
\frac{d^{2}}{dt^{2}}\left(\vec{a}\times\vec{b}\right)=\frac{d}{dt}\left[\frac{d}{dt}\left(\vec{a}\times\vec{b}\right)\right]
$$

2.2 Magnitude and Unit Vector

For $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ the magnitude is

$$
\lVert\vec{a}\rVert=\sqrt{a_1^{2}+a_2^{2}+a_3^{2}}
$$

and, provided $\vec{a}\neq\vec{0}$, the unit vector in the direction of $\vec{a}$ is

$$
\hat{a}=\frac{\vec{a}}{\lVert\vec{a}\rVert}
$$

The resultant of vectors $\vec{a}$ and $\vec{b}$ is $\vec{a}+\vec{b}$, formed componentwise, and its magnitude is obtained by applying the magnitude formula to the sum.

2.3 Tangent and Normal Vectors

> Definition (Tangent vector). Let $\vec{r}(t)$ be a differentiable vector-valued function. The tangent vector to the curve defined by $\vec{r}(t)$ is

$$
\vec{T}(t)=\vec{r}'(t)
$$

> Definition (Unit tangent vector). Let $\vec{T}(t)$ be a tangent vector. The unit tangent vector is

$$
\hat{T}(t)=\frac{\vec{T}(t)}{\lVert\vec{T}(t)\rVert}
$$

> Definition (Normal vector). Let $\vec{r}(t)$ be a differentiable vector-valued function and let $\vec{T}(t)$ be the tangent vector. The normal vector is the derivative of the tangent vector:

$$
\vec{N}(t)=\vec{T}'(t)
$$

> Definition (Principal unit normal vector). The principal unit normal vector is

$$
\hat{N}(t)=\frac{\vec{N}(t)}{\lVert\vec{N}(t)\rVert}
$$

> Definition (Binormal vector). The binormal vector is the cross product of the unit tangent and the principal unit normal:

$$
\hat{B}(t)=\hat{T}(t)\times\hat{N}(t)
$$

The binormal vector is orthogonal to both the tangent vector and the normal vector.

A normal vector is a perpendicular vector. Given a vector in space there are infinitely many perpendicular vectors; the construction above selects the special vector that is normal to the unit tangent vector and, for a non-straight curve, is the unique vector pointing into the curve.

2.4 Velocity and Acceleration

> Definition (Position, velocity and acceleration). The position vector $\vec{r}(t)$ runs from the origin of the reference frame to the position of the particle, with time-dependent Cartesian components. The velocity vector is the time derivative of the position vector:

$$
\vec{v}(t)=\frac{d\vec{r}}{dt}
$$

The velocity vector is always tangent to the trajectory of the particle at each point. The acceleration vector is the time derivative of the velocity vector:

$$
\vec{a}(t)=\frac{d\vec{v}}{dt}=\frac{d^{2}\vec{r}}{dt^{2}}
$$

The magnitudes of velocity and acceleration are obtained by applying the magnitude formula to the respective derivative vectors.