Vector Calculus

Calculus of Motion

Calculus originated as a mathematical way of studying change. Motion is the most natural sort of change, so it makes sense to start our journey there.

In 1D, we can find physical quantities like velocity and acceleration through differentiation of \( x(t), \) the position along the \(x\)-axis at time \(t.\) Life becomes more interesting when things move in more than one dimension.

In \(\mathbb{R}^n,\) we need \(n\) functions (representing the coordinates of the moving object) bundled together into an ordered list \( \vec{x}(t) = \big\langle x(t) , y(t), \dots \big\rangle \) called a vector. Velocity and acceleration are tied to the derivatives of these functions, as we'll see.

Beginning from this perspective shows us why the melding of vectors with calculus is so useful. No prior physics experience is needed, but an interest in simple mechanics would make the unit much more enjoyable!

Calculus of Motion

Intro to Vector Calculus

                 

Calculus of Motion

We first apply vectors in multivariable calculus to the problem of describing a line \(L\) in \( \mathbb{R}^n.\) Assuming \( L \) goes through \( p = (a,b,c) \) and is parallel to \( \vec{v} = \langle v_{x},v_{y},v_{z} \rangle, \) we found \( \vec{x}(t) = \vec{p} + t \vec{v} \) for the position vectors of points on the line.

If we think of \( t \) as time, \( \vec{x}(t) \) (the red arrow below with base at the origin and tip on the line) describes the position of an object moving along \(L.\)

In the visualization above, you have control over the direction of \( \vec{v} \) via angles \( \theta \) and \( \phi\), \(\vec{v}\)'s length \( l = \| \vec{v} \|,\) and the point \( p.\)

Experiment with the sliders and determine what quantity is most closely associated with the object's speed along the line.

Calculus of Motion

Intro to Vector Calculus

                 

Calculus of Motion

For linear motion in \(\mathbb{R}^3,\) we can explicitly find \( x(t), y(t),\) and \(z(t):\) \[ \vec{x}(t) = \vec{p} + t \vec{v} = \langle \underbrace{x_{0} + t v_{x}}_{x(t)}, \underbrace{y_{0} + t v_{y}}_{y(t)}, \underbrace{z_{0} + t v_{z}}_{z(t)} \rangle.\] The rate of change of the object's motion in the \(x\)-direction is \(x'(t);\) similar statements hold for \(y'(t)\) and \(z'(t).\) Bundling these derivatives together into the vector \[ \vec{v}(t) = \vec{x}'(t) = \big\langle x'(t), y'(t), z'(t) \big\rangle \] gives us complete information about the velocity of the object through space.

For example, we'll learn in the Vector-valued Functions chapter that the speed is just the magnitude of velocity \( \| \vec{v}(t) \|.\)

If an object moves along the line \[ \big\langle x(t) ,\ y(t) , \ z(t) \big\rangle = \langle 0, 2, -3 \rangle + t \langle 1, -2, 2 \rangle,\] compute its speed and enter your answer as an integer on the right.

Calculus of Motion

Intro to Vector Calculus

                 

Calculus of Motion

Similar logic applies when an object travels along a line with non-constant speed. In the visualization below, you once more have control over the direction of the line \((\)now called \( \hat{u} )\) and the position of a point \( p = (a,b,c)\) on \( L.\)

Now, however, instead of \( \vec{p} + t \vec{v} \) the position vector (red above) is given by \[ \vec{x}(t) = \vec{p} + \textcolor{red}{3 \sin( 2 \pi t) } \hat{u}.\] Compute the object's speed, which is the magnitude of \[ \vec{x}'(t) = \vec{v}(t) = \big\langle x'(t), y'(t), z'(t) \big\rangle.\]

Calculus of Motion

Intro to Vector Calculus

                 

Calculus of Motion

In addition to the position vector \( \vec{x}(t), \) it's often useful to display both the velocity \( \vec{v}(t) = \vec{x}'(t) \) and acceleration \( \vec{a}(t) = \vec{v}'(t) = \vec{x}''(t).\)

These vectors, however, are usually drawn attached to the object itself, not the origin. The velocity vector always points in the direction of motion, but the same isn't true for acceleration.

Use the visualization and/or the derivative definitions to determine which of the two vectors \( \vec{u}_{1}(t) \) (purple) and \( \vec{u}_{2}(t) \) (green) is the acceleration for \[ \vec{x}(t) = \big\langle -1+\cos(t) ,1, \cos(t) \big\rangle .\]

Calculus of Motion

Intro to Vector Calculus

                 

Calculus of Motion

Vectors also help us quantify nonlinear motion in the plane. Position vectors are formed by bundling \( x(t) \) and \( y(t) \) together into the ordered list \( \big\langle x(t), y(t) \big\rangle;\) velocity and acceleration vectors come from taking the first and second derivatives.

There's a great deal about vectors and motion we still have to learn but, to close out this unit, let's start unraveling nonlinear motion in 2D.

Planet X orbits along a circle under the influence of gravity. If the circle has radius \( r \) and is centered at the origin where the star sits, what's one possible position vector of the planet?


Hint: \( \sin^2( t) + \cos^2( t) = 1 \) for all real \( t. \)

Calculus of Motion

Intro to Vector Calculus

                 

Calculus of Motion

As Planet X orbits around its home star, it follows the position vector \[ \big\langle x(t) , y(t) \big\rangle = \big\langle r \cos( t) , r \sin(t) \big\rangle. \] The visualization below shows three arrows representing the position of Planet X, its velocity, and its acceleration. Note that the black and blue vectors overlap; the blue one points towards the origin from the circle, while the black one points from the origin to the circle.

Which one corresponds to Planet X's acceleration \( \big\langle x''(t) , y''(t) \big\rangle ?\)

Calculus of Motion

Intro to Vector Calculus

                 

Calculus of Motion

Now that we have some experience with 2D nonlinear motion, let's make the jump to 3D. Here's a visualization showing motion along a curve in 3D called a helix.

Here, \( b \) measures the helix's vertical spacing. Taking \( b = 0\) gives us circular motion in the plane much like Planet X's. When \( b > 0,\) the particle moves with constant speed parallel to the \( z\)-axis while also simultaneously circulating around it.


Which vector represents the acceleration?

Calculus of Motion

Intro to Vector Calculus

                 

Calculus of Motion

In the Vector Calculus in a Nutshell unit, we were introduced to vector fields \( \vec{V}(\vec{x}) \) where the number of components of the input matched that of the output. This unit, however, showed how very versatile vectors truly are: here, we worked with vectors \( \vec{x}(t), \vec{v}(t), \) and \( \vec{a}(t) \) depending only on a single variable \( t.\)

We'll explore the calculus of both kinds of vector functions (and others) in greater depth in Vector-valued Functions. There we'll apply our knowledge to save a magic harp from the clutches of a vile giant!

The next two units go beyond motion and give a glimpse of how calculus can be synthesized together with the vector concept to produce new types of integrals and derivatives.

Calculus of Motion

Intro to Vector Calculus

                 
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