Concept map

Difference / Change

Understanding both difference (how far apart two values are at one time) and change (how far apart the value of a single parameter is at two different times) is necessary for understanding derivatives.

The derivative relates how much $f$ changes as $x$ changes

Conceptually, a derivative can be thought of as how much one value changes as another value changes. FIXME: Representations

*The Magnitude of the Derivative

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$\frac{\Delta f}{\Delta x}$

The derivative is a ratio of small changes

One can view $\frac{df}{dx}$ as approximately given by a fraction where $df$ is a small change in $f$ and $dx$ is a small change in $x$.

The derivative of a constant is zero

The derivative of a constant is always zero. This can be understood by thinking of 'The derivative is a ratio of small changes,' as the amount by which a constant changes is zero.

$\vec \nabla f$

$d \vec r$

The gradient can tell you a small change in a function in any direction (differentials edition)

$df=\vec \nabla f \cdot d \vec r$

$\Delta x$

$dx$

Experiment

There are experimental limits to how $small$ of a change can be measured

While it is relatively easy to imagine very, very small changes in physical values, there are often experimental limits on how small of a change can be measured. When designing and conducting experiments, there is a tension between these experimental limitations and normative representations of functions as smooth lines or

The derivative is a limit

Technically, the derivative is a ratio of small changes in the limit that the change in the denominator goes to zero: $\frac{df}{dx}=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

$\left(\frac{\partial f}{\partial x}\right)_y$

$\frac{d f}{d x}$

$\frac{\partial f}{\partial x}$

You can flip a derivative

In other words, $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$. Note that this concept is challenging to express in Newton's notation, but arises naturally if implicit differentiation is covered.

*The Magnitude of a Partial Derivative

FIXME: Add for Contours: "The magnitude of a partial derivative is proportional to the density of contour lines along the path of the partial derivative."[]FIXME: Add for Surfaces: "The magnitude of a partial derivative is the slope of the surface along the path of the partial derivative."[]FIXME: Add current concept: "All partial derivatives can be found as a slope of a tangent plane" and change to "The Magnitude of a partial derivative a slope of the tangent plane in the direction of the partial derivative FIXME: Direction Language?"

$\frac{d f}{d x}$

"With respect to what" matters

Taking derivatives of the same function with respect to different variables will produce different results. In physics, these different results might have different dimensions and thus the derivative can have wildly different physical interpretations based off of "'with respect to' what".

$df$

$đf$

Differentials are small changes or differences

One typical notation used for derivatives is Leibniz notation ($e.g.,$ $df/dx$). Physicists often think of the $df$ and $dx$ as distinct quantities (``differentials'') that can be measured, calculated, and manipulated independent of one another. In particular, a differential can be thought of as the (small) change in a quantity when a small change is made to a physical system. A differential can also be thought of as the difference between the values of a quantity between two different (nearby) physical states. This line of thinking is also valuable for thinking about integration in physical situations.

There is a partial derivative in every direction at any point

For a smooth, two dimensional function, a partial derivative can be found at any point and in any direction.

$\left(\frac{\partial f}{\partial x}\right)_y$

$\frac{\partial f}{\partial x}$

The coefficients in a differentials equation are partial derivatives

FIXME

$d \vec r$

$d \vec r$ is a small displacement vector

$d \vec r$ represents a small displacement vector (i.e., it points along the direction of a step with a magnitude equal to the length of that step).

$\vec \nabla f$

The gradient is a vector

For an $n-$dimensional function $f$, the gradient of $f$ at a point is an $n-$dimensional vector

$d \vec r$

The direction of $d\vec r$ is tangent to a path between two nearby points

FIXME: Tangent to what path? How is this useful?

$d \vec r$

The magnitude of $d\vec r$ is the distance between two nearby points

FIXME: What's the point of this concept?

$\vec \nabla f$

*The Magnitude of the Gradient

The magnitude of the gradient is the value of the slope in the direction of greatest increase

$\vec \nabla f$

*The Direction of the Gradient

The Master Formula states that a small change in a function $df$ is the dot product of the gradient of the function with a small step $d \vec r$ through the domain of the function: \[df = \vec \nabla f \cdot d \vec r.\] In order to maximize the change in $f$ one must maximize this dot product, which happens when the small step $d \vec r$ is parallel to the gradient. Or, turning this statement around, the gradient points in the direction in which the function is increasing the most.

$\left(\frac{\partial f}{\partial x}\right)_y$

$\vec \nabla f$

The components of the gradient are partial derivatives

FIXME

$df$

Differentials are small chunks

$df$ can be viewed as a small chunk of $f$. When combined with 'The derivative is a ratio of small changes', it's easy to relate differentials to derivatives.

$df$

Total differentials are linear

No matter how complicated a multivariable function is when written as an equation, the total differential of that function will be linear in the differential terms. In other words: $dF(x,y,z)=A dx + B dy + C dz.$

Individual differentials can be manipulated algebraically

FIXME: What? One can do algebra with differentials and reinterpret them as partial derivatives. Can treat like a variable

$df$

Equations can be 'zapped with d' to relate differentials

FIXME: Does the difference between zapping with d and taking the total differential need to be articulated?

The derivative can be approximated by the slope of a secant line

The value of the derivative at a point on a curve can be approximated by the slope of a line secant with the curve near that point.

$df$

Differentials allow the finding of partial derivatives when a variable cannot be solved for

Because total differentials are linear, one can find the total differential and then solve algebraically to obtain partial derivatives that might not be obtainable though differentiation.

The magnitude of the derivative at a point is the slope of a tangent line at that point

All partial derivatives can be found as a slope of a tangent plane

Just as the derivative of a one-dimensional function at a point is equal to the slope of a tangent line at that point, and partial derivative of a multivariable function can be found as a slope of a tangent plane.

The derivative is a function

One of Zandieh's process-object layers for derivatives is that derivatives are functions. This means that the value of a derivative depends on where in the domain of the function one is looking. The derivative of a function is itself a function, with the same domain as the original function. Both the (derivative) function and the value of the derivative at a point are commonly referred to as "the derivative."

$\left(\frac{\partial f}{\partial x}\right)_y$

$\frac{\partial f}{\partial x}$

Partial derivatives are functions

The partial derivative of a function is itself a function (although taking a partial derivative at a point will just give a number). This means that the value of the partial derivative at different points in the domain will, in general, be different.

$\vec \nabla f$

The gradient is a function

The gradient of a function (unless taken at a single point) is itself a function

$\vec \nabla \cdot \vec v$

The divergence is a function

The divergence of a function (unless taken at a single point) is itself a function

$\vec \nabla \times \vec v$

the curl is a (vector) function

FIXME

Variables can be held constant

When taking a derivative of a multivariable function, it is possible to consider a simpler situation in which some of the variables are considered to be constant.

$\left(\frac{\partial f}{\partial x}\right)_y$

Derivatives can be found while holding one or more variables constant

Derivatives of multivariable functions can be found by holding one or more variables constant (subject to physical limitations).

$\left(\frac{\partial f}{\partial x}\right)_y$

The value of a partial derivative depends on the value(s) of what is held constant

FIXME: Is this really a statement about paths? I don't think we have path-related concepts.

$\frac{d f}{d x}$

Single variable chain rule

If $f=f(y)$ and $y=y(x), then \frac{df}{dy}=\frac{df}{dx}\frac{dx}{dy}.$

$\left(\frac{\partial f}{\partial x}\right)_y$

Partial derivatives that do not have the same variable(s) held constant (at the same values?) are not the same derivative

FIXME

$\left(\frac{\partial f}{\partial x}\right)_y$

$\frac{\partial f}{\partial x}$

Chain Rule Diagrams

A partial derivative can be expressed in terms of other partial derivatives

FIXME

$\frac{d f}{d x}$

The derivative is a local quantity

The value of the derivative does not depend on what is happening far away from the point at which the derivative is being evaluated: it can be determined by what is happening in the neighborhood of the point. Geometrically, the derivative (slope) of a curve at a point is not a statement about the general shape of the curve but rather about the behavior of the curve immediately around that point.

$\vec \nabla f$

The gradient is a local quantity

The magnitude and direction of the gradient depends on the behavior of the function infinitesimally near the point of interest. One implication of this is that the gradient does not point in the direction of maxima a finite distance away (except sometimes accidentally), but rather in the direction of steepest slope at the point of interest.

$\vec \nabla \cdot \vec v$

The divergence is a local quantity

FIXME

$\vec \nabla \times \vec v$

The curl is a local quantity

FIXME

Partial derivatives are local quantities

$\vec \nabla \cdot \vec v$

Formulas for the Divergence

In each coordinate system, there exists a formula for the divergance. The standard examples are: \[\begin{align}\vec\nabla\cdot\vec v &= \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z} \text{ in rectangular coodinates} \\ \vec\nabla\cdot\vec v &= \frac{1}{s}\frac{\partial \left(s v_s\right)}{\partial s} + \frac{1}{s}\frac{\partial v_\phi}{\partial \phi}+\frac{\partial v_z}{\partial z} \text{ in cylindrical coodinates} \\ \vec\nabla\cdot\vec v &= \frac{1}{r^2}\frac{\partial \left(r^2 v_r\right)}{\partial r} + \frac{1}{r \sin(\theta)}\frac{\partial \left(\sin(\theta)\ v_\theta\right)}{\partial \theta}+\frac{1}{r \sin(\theta)}\frac{\partial v_\phi}{\partial \phi} \text{ in spherical coodinates}\end{align}\]

Components of the Divergence

For each partial derivative term in the divergence (in orthogonal coordinate systems), the direction of the component involved and the direction of the change agree (e.g. the $\textbf{x}$'s match in $\frac{\partial v_\textbf{x}}{\partial \textbf{x}}$.)

$\vec \nabla \cdot \vec v$

The divergence is a scalar field

The divergence at a point is a scalar. Taking the divergence of a function yields a scalar at every value in the domain of that function: a scalar field

The Divergence is Geometric

The divergence is geometric (i.e. it is independent of coordinate system)

The Divergence is a Particular Geometric Combination of Partial Derivatives

$\vec \nabla \times \vec v$

Formulas for the Curl

In any coordinate system, there exists a formula for the curl. Examples include:\[\vec\nabla\times\vec v = \left(\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z}\right)\hat x + \left(\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x}\right)\hat y + \left(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}\right)\hat z \text{ in rectangular coodinates}\]\[\vec\nabla\times\vec v = \left(\frac{1}{s}\frac{\partial v_z}{\partial \phi} - \frac{\partial v_\phi}{\partial z}\right) \hat s + \left(\frac{\partial v_s}{\partial z} - \frac{\partial v_z}{\partial s}\right) \hat \phi + \frac{1}{s}\left(\frac{\partial \left(s\ v_\phi\right)}{\partial s} - \frac{\partial v_s}{\partial \phi}\right) \hat z \text{ in cylindrical coodinates}\]\[\vec\nabla\times\vec v = \frac{1}{r \sin(\theta)}\left(\frac{\partial \left(\sin(\theta)\ v_\phi\right)}{\partial \theta} - \frac{\partial v_\theta}{\partial \phi}\right) \hat r + \frac{1}{r}\left(\frac{1}{\sin(\theta)}\frac{\partial v_r}{\partial \phi} - \frac{\partial \left(r\ v_\phi\right)}{\partial r}\right)\hat \theta +\frac{1}{r} \left(\frac{\partial \left(r\ v_\theta\right)}{\partial r} - \frac{\partial v_r}{\partial \theta}\right) \hat \phi \text{ in spherical coodinates}\]

Components of the curl

For each partial derivative term in the curl (in orthogonal coordinate systems), the direction of the component involved and the direction of the change differ (e.g. $\textbf{$\alpha$} \neq \textbf{$\beta$}$ in $\frac{\partial v_\textbf{$\alpha$}}{\partial \textbf{$\beta$}}$.)

The Curl is a particular geometric combination of partial derivatives

The curl is a particular geometric combination of partial derivatives

The Curl is Geometric

The curl is geometric (i.e. it is independent of coordinate system)

The curl is a vector field

ENTER LONG DESCRIPTION

The Curl is a vector field

The derivative can be interpreted physically

While the constituents of a derivative (the $f$ and $x$ in $\frac{df}{dx}$) can have physical interpretations, the derivative itself can have a different physical interpretation. For example, $\frac{dx}{dt}$ is a velocity. FIXME: find a better example

$\vec \nabla \times \vec v$

The curl of the magnetic vector potential is the magnetic field

FIXME

$\vec \nabla \cdot \vec v$

The divergence of the electric field is equal to $\rho / \epsilon_0$

FIXME

$\vec \nabla f$

The negative gradient of the electric potential is the electric field

While the magnitude of the electric field is equal to the gradient of the electric potential, the electric field points in the opposite direction of the gradient of the electric potential, and thus $\vec E = - \vec \nabla V$.

$\vec \nabla \cdot \vec v$

$\vec \nabla \times \vec v$

The divergence of the curl is equal to $\mu_0$ times the current (in magneto-statics)

FIXME

$\vec \nabla \times \vec v$

The curl of the electric field is zero in electrostatics

FIXME: Long description + Does this really need the pre-req of the derivative of a constant is zero. What is the thing that is constant if the curl is zero?

The thermodynamic identity defines temperature and pressure as partial derivatives

[T = \left(\frac{\partial U}{\partial S}\right)_V] [p = -\left(\frac{\partial U}{\partial V}\right)_S]

Maxwell's Equations

The derivative is a linear operator

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Power law

The derivative with respect to $x$ of $x^A$ is $A \cdot x^{A-1}$.

The derivative at a cusp is undefined

At a cusp (or kink) in a curve, the derivative (and it's value) are not defined.

$\frac{d f}{d x}$

Product rule

When taking the derivative of a product with $n$ factors, the product rule gives that this derivative is equal a sum of $n$ terms, the $i$th of which is the original function but with the $i$th factor replaced by the derivative of the $i$th factor: $\frac{d}{dt}(xy)=\frac{dx}{dt} y+x \frac{dy}{dt}$

There is a tangent line in every direction at every point

For a smooth, two dimensional function, a tangent line exists at any point and in any direction.

Maxwell's Relations

$\int\vec\nabla\cdot\vec F dV= \oint\vec F\cdot d\vec A$

The total divergence in a volume is the same as the total flux out of its surface.[]2) The divergence theorem is used to move between the integral and differential forms of Gauss's law.[]3) Dimensions are the same

$\oint \vec F\cdot d\vec r = \int \vec\nabla\times\vec F\cdot d\vec A$

FIXME

$d \vec r$

Differential form of $\vec r$ in spherical and cylindrical coordinates

FIXME

$\vec \nabla f$

The gradient lives in the domain and has the same number of spatial dimensions as the original function

The gradient of an $n$ dimensional function is $n$ dimensional. This is not to be confused with the gradient having the same physical dimensions (e.g., Newtons) as the original function.

$\vec \nabla \cdot \vec v$

The divergence is related to the total flux through a closed surface

If the total flux through a closed surface is not zero, then the divergence of the function at that point is non-zero.

$\vec \nabla \cdot \vec v$

The divergence of the magnetic field is zero

FIXME

$\vec \nabla \times \vec v$

FIXME

Thermodynamic potentials are related through Legendre transformations

Experiment

There might be experimental limits on which quantities you can measure

While it may be possible to conceptualize a particular derivative, such as $\left(\frac{\partial U}{\partial S}\right)_{V},$ Where $U$ is internal energy, $S$ is entropy, and $V$ is volume, that does not mean that the particular derivative can be directly measured. In the above example, the entropy $S$ is not measurable.

Legendre Transformations

$đQ = TdS$ (edit later)

$đW = - pdV$ (edit later)

Developing physical interpretations of partial derivatives (dimensions)

Second partial Derivatives