# Definition

The vector function $r(t) = \left\langle x(t), y(t), z(t) \right\rangle$  has a derivative (is differentiable) at $t$ if $x$, $y$, and $z$ have derivatives at $t$. The derivative is the vector function: $$r\prime (t) = \frac{dr}{dt} = \lim_{\Delta t\to 0} \frac{r(t + \Delta t) - r(t)}{\Delta t} = \left\langle x\prime (t), y\prime (t), z\prime (t) \right\rangle$$

For a differentiable vector valued function $r(t) = \left\langle x(t), y(t), z(t) \right\rangle$ representing the position vector of a particle at time $t$, its velocity vector is given as: $$v(t) = \frac{dr}{dt} = r\prime (t) = \left\langle x\prime (t), y\prime (t), z\prime (t) \right\rangle$$

# Bird's Eye View

We will be discussing a very specific case of differentiation through the course of these notes: with respect to space curves (which, as we already know, are the set of points $(x(t), y(t), z(t))$, obtained as $t$ varies over a given parameter interval), to analyse how and when we can (and cannot!) differentiate on a given space curve.

# Context of the Definition

Formally, the domain of $f$ is the set of points in the domain of $f$ for which the limit exists, and the domain may be the same or smaller than the domain of $f$. If $f$ exists at a particular $t$, we say that $f$ is differentiable (has a derivative) at $t$. If $f$ exists at every point in the domain of $f$, we call $f$ differentiable. Mathematically, $$r\prime (t) = \frac{dr}{dt} = \lim_{\Delta t\to 0} \frac{r(t + \Delta t) - r(t)}{\Delta t} = \left\langle x\prime (t), y\prime (t), z\prime (t) \right\rangle$$

As we have studied in Equations of Planes and Lines, we know that the slope of a line in $R^{3}$ is given by a direction, rather than a scalar value. The direction vector given by $r\prime (t)$ represents the slope of the curve at a given $t$.

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Figure 1: Interpretation of $r\prime (t)$

There are multiple interpretations of the derivative:

### 1. Rate of Change

If $f(x)$ represents a quantity at any $x$ then the derivative $f\prime (a)$represents the instantaneous rate of change of $f(x)$ at $x = a$.

Illustration:
Suppose that the volume of water in a tank for $0\leq t \leq 6$ is given by $Q(t)=10+5t−t^{2}$
1. Is the volume of water increasing or decreasing at $t=0$ ?
2. Is the volume of water increasing or decreasing at $t=6$ ?
3. Does the volume of water ever stop changing? If yes, at what times(s) does the volume stop changing?

Hint: From Figure 1, we can infer that if the derivative of a function at a given point is positive, the function is increasing.

### 2. Slope of a Tangent Line

The slope of the tangent line to $f(x)$ at $x=a$ is $f\prime (a)$. The tangent line then is given by, $$y = f(a) + f\prime (a)(x - a)$$

Illustration:
What is the equation of the tangent line to $f(x)= \frac{5}{x}$ at $x=\frac{1}{2}$ ?

### 3. Velocity

This is a special case of the rate of change interpretation. If the position of an object is given by $r(t)$ after t units of time the velocity of the object at $t=a$ is given by $r\prime (a)$.

Illustration:
The position of an object at any time $t$ is given by $s(t) = \frac{t + 1}{t + 4}$
1. Determine the velocity of the object at any time $t$.
2. Does the object ever stop moving? If yes, at what time(s) does the object stop moving?

Hint: From Figure 1, we can infer that if the derivative of a function at a given point is zero, the function neither increases nor decreases.

### Can velocity be obtained for any space curve?

A mathematical version of the above question would be: " For a given curve $r(t)$, can the curve be differentiated for any and every value of $t$ in the parameter interval?". This property of the curve is called differentiability. A function can fail to be differentiable at a point if:

1. The function is not continuous at that point.
2. The graph has a sharp corner at the point.
3. The graph has a vertical line at the point.

The animation given below depicts all three cases.

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Figure 2: Non-differentiable curves

### Smooth Curves

A curve $x = f(t), y = g(t)$ is called a smooth curve if $f\prime (t), g\prime(t)$ are continuous and for no value of $t$ are $f\prime (t)$ and $g\prime (t)$ simultaneously equal to zero.

The number of continuous derivatives of the position function for a curve to be considered smooth depends on the problem at hand and may vary from two to infinity.

To understand this better, consider the following example: $$r(t) = \left\langle t^{2}e^{-t}, 2(t - 1)^{2}\right\rangle$$

Computing $r\prime (t)$, we get:

$r\prime (t) = \left\langle 2te^{-t} - t^{2}e^{-t}, 4(t - 1)\right\rangle$

$r_{x}\prime = 0 \implies t = 0, 2$

$r_{y}\prime = 0 \implies t = 1$

Hence, since for a given value of $t$, $r_{x}\prime$ and $r_{y}\prime$ are not simultaneously equal to zero, i.e. $r(t) \neq \overrightarrow{0}$, the given curve $r(t)$ is a smooth curve.

Visually, if a curve on zooming in infinitely, looks like a straight line, it is a smooth curve. Zooming in on $\left | x \right |$ at $x = 0$, will always show the sharp corner:

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Figure 3: Smooth curves | source: Eric Duminil (https://math.stackexchange.com/users/386794/eric-duminil), Intuitively, why are the curves of exponential, log, and parabolic functions all smooth, even though the gradient is being changed at every point?, URL (version: 2018-05-23): https://math.stackexchange.com/q/2791202

### Tangent Lines to Space Curves

If $r(t) = \left\langle f(t), g(t), h(t) \right\rangle$ is a position vector along a curve in $\mathbb{R}^{3}$, then $r\prime (t) = \left\langle f\prime (t), g\prime (t), h\prime (t)\right\rangle$ is a vector in the direction of the tangent line to the curve.

Would this mean the velocity vector of a point on a curve is always tangential to the curve? Something to think about. (or not!)

Note that it is required that $r(t)\neq\overrightarrow{0}$  to have a tangent vector, as $r(t) = \overrightarrow{0}$ implies that the vector has no magnitude and hence would not give the direction of the tangent.

The video below depicts a tangent line to the given space curve at the given point.

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Figure 4: Tangent line to a space curve

### Speed and Acceleration

As seen above, the first derivative of the position vector $r(t)$ is called velocity, often denoted by $v(t) = r\prime (t)$. Speed is defined as the length of velocity, i.e. $\left|\left| r\prime (t)\right|\right|$ and is a non-negative scalar.

While studying motion of objects, the total acceleration is often broken up as a tangential component, $a_{T}$ and a normal component, $a_{N}$. The tangential component is that part of the acceleration which is tangential to the curve and the normal component is the part of the acceleration normal to the curve.

$$a_{T} = v\prime (t) = \frac{r\prime(t)\cdot r\prime\prime(t)}{\left|\left|r\prime(t)\right|\right|}$$

$$a_{N} = k\hspace{0.2cm}\left|\left|r\prime (t)\right|\right|^{2} = \frac{\left|\left|r\prime(t)\times r\prime\prime(t)\right|\right|}{\left|\left|r\prime(t)\right|\right|}$$

where $k$ is the curvature of $r(t)$.

The total acceleration is given by: $$a = a_{T}\overrightarrow{T} + a_{N}\overrightarrow{N}$$ where $\overrightarrow{T}$ and $\overrightarrow{N}$ are the unit tangent and unit normal vectors for $r(t)$ respectively.

# References

[2] https://ximera.osu.edu/mklynn2/multivariable/content/02_2_velocity_and_speed/velocity_and_speed
[3] https://opentextbc.ca/calculusv1openstax/chapter/the-derivative-as-a-function/

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 Contributor: Mentor & Editor: Verified by: Approved On:

The following notes and their corrosponding animations were created by the above-mentioned contributor and are freely avilable under CC (by SA) licence. The source code for the said animations is avilable on GitHub and is licenced under the MIT licence.

The work under this website is licenced under a Creative Commons Attribution-Share Alike 4.0 International License CC BY-SA