This mini-lecture demonstrates the relationship between \(df\) on the tangent plane to its “components“ in coordinate directions, leading to the multivariable chain rule.

Find the total differential of the following functions:

1. \(y=3u^2 + 4\cos 3v\)
2. \(y=3uv\)
3. \(y=3u^2\cos wv\)
4. \(y=u\cos(3v^2-2)\)

Find the total differential of the following functions:

1. \(y=3x^2 + 4\cos 2x\)
2. \(y=3x^2\cos kx\) (where \(k\) is a constant)
3. \(y=\frac{\cos 7x}{x^2}\)
4. \(y=\cos(3x^2-2)\)

The depth of a puddle in millimeters is given by \[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\] Your path through the puddle is given by \[x=3t \qquad y=4t\] and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.

1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
3. FOOD FOR THOUGHT (optional)
   There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.

Take-home messages from today's interactive lecture:

• You can think about the differential \(dx\) as a small amount of \(x\). While technically, it is an infinitesimally small amount, in practice it is almost always possible to think of it as just really, really small. By which, we mean small enough that any error you make by giving it a finite rather than infinitesmal size is too small to make a significant difference in your calculation. Much more about this in later activities.
• A differentials equation tells you how a small change in one variable is related to a small change in one or more other variables. You can interpret these equations geometrically with a figure.
• When doing algebra with differentials equations, you can think of each differential as a new variable.
• When you zap an equation with \(d\), the resulting differentials equation is linear in the differentials, i.e. each of the differentials appears to the first power. This is an open invitation to do linear algebra to rearrange the equation.
• The dimensions of \(dx\) are the same as the dimensions of \(x\). After all, \(dx\) is just a small amount of \(x\).
• Every equation involving differentials should have matching factors of smallness in every term. For example: \begin{align} dx&=2y^2\, dy&\hbox{one factor of $d$ in every term}\\ dx\, dy&=r\, dr\, d\theta &\hbox{two factors of $d$ in every term} \end{align} as opposed to: \[\cancel{dx=2y\, dy\, dz}\]

Instructor's Guide

Introduction

The vector differential \(d\vec{r}\) is a fundamental unifying feature of our approach to vector calculus and therefore to electrostatics and magnetostatics. It is used both to define derivatives like the gradient and directional derivatives and integrals in space, from integrals along curves to surfaces and volumes.

This interactive lecture is the first step in defining and evaluating the vector differential. It should introduce the scalar differential and how to calculate it by “zapping with \(d\)”, i.e. it is mainly procedural. Emphasize that zapping with \(d\) linearizes the equations so that it is easy to do substitution with differentials. Also emphasize the role of systems of equations.

An important non-procedural, geometric message is how the differentials (as opposed to differential) equation tells you how small changes in one variable are related to small changes in other variables.

The important takehome messages are listed in the summary handout for students.

You might want to have the student's build their understanding through a thoughtful series of SWBQ's (one at a time with class discussion after each one!)

Global prompt: “Zap the following expression with \(d\). Then draw a figure that shows you what the differentials equation is telling you about the relationship of small changes.”

• After they have done the following problem, remind them that most of them already know how to do this from \(u\)-substitution so that their knowledge about that is in their working memory. \[u=y^3\]
• Also a \(u\)-substitution problem, but they need to do chain rule. Mention that this is a great time for them to be reviewing basic calculus if they need to. Also, they must know, in any give expression, which letters are constants and which are variables. \[u=A e^{-(\frac{x}{a})^2}\]
• This example shows that you don't have to have an equation with an isolated variable on one side AND you can have more than two variables. Look at this example both for \(r=\) constant and \(r\) as a variable. Draw the figure for the case \(r=\) constant. How much harder is the figure if \(r\) is variable? \[x^2+y^2=r^2\]
• A system of equations. Ask the students to solve for \(dx\) and \(dy\) OR for \(dr\) and \(d\phi\). Which is easy and which is hard? Remind students that in \(u\)-substitution they probably know that they need to change ALL instances of the old variable to \(u\)'s, the same applies here. If they are changing to \(r\) and \(\phi\) from \(x\) and \(y\), they should, if possible, change ALL of the instances. Sometimes this is algebraically impossible to do, but they should at least keep trackin their minds that, if they still have \(x\)'s and \(y\)'s, they should now think of these as functions of \(r\) and \(\phi\) and not independent, i.e. \(x=x(r,\phi)\) and \(y=y(r,\phi)\) even if they can't write down explicitly what those functions are. These nuances are particularly important in thermodynamics. (Note: We like to use \(\phi\) for polar coordinates so that these coordinates agree with the physicists' use of spherical coordinates.) \begin{align} r^2&=x^2+y^2\\ \tan\phi&=\frac{y}{x} \end{align}
• (Optional:) Another system of equations. Ask the students to solve for \(dx\) and \(dy\) OR for \(dr\) and \(d\phi\). Which is easy and which is hard? \begin{align} x&=r\cos\phi\\ y&=r\sin\phi \end{align}

This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the \(r\)-direction and derivatives in \(x\)- and \(y\)-directions using the chain rule.

Students use chain rule diagrams to construct a multivariable chain rule in terms of differentials.

This small group activity is designed to provide practice with the multivariable chain rule. Students determine a particular rate of change using given information involving other rates of change. The discussion emphasizes the equivalence of a variety of approaches, including the use of differentials. Good “review” problem; can also be used as a homework problem.

• The gradient is perpendicular to the level curves.
• The gradient is a local quantity, i.e. it only depends on the values of the function at infinitesimally nearby points.
• Although students learn to chant that "the gradient points uphill," the gradient does not point to the top of the hill.
• The gradient path is not the shortest path between two points.

Students are gently introduced to the relationship between contour maps and surfaces.

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.

Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases (\(x\), \(y\), and \(z\)). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.

This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.