Lecture 8

Behind us

• Curves via parametric equations
• Vector-valued functions

Coming up

• Today: derivatives and integrals of vector functions
• Next: explosive investigation of curvature and arclength
• Read Sections 10.2, 13.2, and 10.3 of the book. Failure to do so may result in lack of knowledge and general misery.

Logistics

• Midterm 1 is Feb 5 in section!

Questions!

Question

A fun-loving electron is traveling in a spiral path around the surface of a torus.

• The radius of the torus (i.e., the radius of the circle at the center of the tube) is $2$ and the diameter of the circular cross-section is $1$.
• The electron starts at position $(1,0,0)$, travels at a constant angular velocity around the vertical axis of $1$ radian per second, and its path winds up and around the torus $4$ times before it returns to its starting position.

What is the position and velocity of the electron at time $t$?

Fun-loving electron in action

It moves!

Using the tools from last time, here is a parametric description of the motion:

$$(x,y,z)=(\cos(t)(2-\cos(4t)),\sin(t)(2-\cos(4t)),\sin(4t))$$

In vector form:

$$\mathbf f(t)=\langle \cos(t)(2-\cos(4t)),\sin(t)(2-\cos(4t)),\sin(4t)\rangle$$

What is the velocity? What should it be?

Velocity is the derivative of position, right?

We should have that the velocity of the electron is $$\mathbf v(t)=\mathbf f'(t).$$ But what is this?

Classical definition of the derivative still works for vector-valued functions:

$$\mathbf f'(t)=\lim_{h\to 0}\frac{\mathbf f(t+h)-\mathbf f(t)}{h}$$

As usual, the derivative of the position vector is the velocity vector .

Calculating the derivative in practice

Given a vector function

$$\mathbf f(t)=\langle x(t),y(t),z(t)\rangle$$

the derivative is just

$$\mathbf f'(t)=\langle x'(t),y'(t),z'(t)\rangle,$$

the component-wise derivative .

The usual caveat applies: the derivative must exist for this to make sense! I.e., this formula works when all three derivatives exist, and when they don't neither does the derivative of $\mathbf f(t)$!

Practice

• calculate the velocity of the electron as a function of $t$
• How does the speed vary over time?
• (What is the speed , anyway? The magnitude of the velocity vector? That sounds right....)

Help the Piglet

The piglet of calculus tried the question on the previous slide and got the following answer:

The derivative of $\mathbf f(t)=\langle \cos(t)(2-\cos(4t)),\sin(t)(2-\cos(4t)),\sin(4t)\rangle$ is:

$$\begin{align*}\frac{d\mathbf f}{dt}=\langle -\sin(t)(2-\cos(4t))+&4\cos(t)\sin(4t),\\ &\quad 4\cos(t)\sin(4t),\sin(4t)\rangle\end{align*}$$

Every time the piglet enters the answer in webassign, it is marked wrong. What mistakes did the piglet make?

The piglet

• Forgot to differentiate the third component!
• Wrong: $\sin(4t)$
• Right: $4\cos(4t)$
• Used the product rule incorrectly in the second component (by just taking the product of the derivatives)!
• Wrong: $4\cos(t)\sin(4t)$
• Right: $\cos(t)(2-\cos(4t))+4\sin(t)\sin(4t)$
• Anything else?
• Have you ever made mistakes like these?

A new problem

A joey (baby kangaroo) is riding in her mother's pouch. She has a smartphone with an accelerometer that continuously reports the acceleration vector. Her friend wrote an app that calculates the velocity at any moment in time $t$. She is too small to see out of the pouch, but, like all infant kangaroos, she is interested in calculating her position as a function of time. Sadly, her somewhat stupid friend never figured out how to get the position before he started a new app that mimics the hilarious sounds that giraffes make when embarrassed.

• The joey records the velocity at time $t$ as $$\mathbf v(t)=\langle 1,t,\sin(t)\rangle$$
• The joey starts at the point $(0,0,1)$
• Where is the joey at time $t$?

Integrate!

Just like one variable calculus: reconstruct position from velocity by integrating .

• Riemann sums: $\sum\mathbf v(t_i)\Delta(t)$
• Practical: $$\int\langle x(t),y(t),z(t)\rangle dt=\left\langle\int x(t)dt,\int y(t)dt,\int z(t)dt\right\rangle.$$
• Example: $$\int\langle 1,t,t^2\rangle dt=\langle a,b,c\rangle+\left\langle t,\frac{1}{2}t^2,\frac{1}{3}t^3\right\rangle$$
• The constant of integration is now a vector!

Help the joey

Using vector integration, compute the path that the joey takes, starting at $t=0$. Initial position: $(0,0,1)$, velocity $\langle 1,t,\sin(t)\rangle$. A looping animation (for $t=0$ to $t=15$ in slo-mo):

Next time: curvature and arclength!