Prerequisite concepts
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 $x$.
Technically, the derivative is a ratio of small changes in the limit that the change in the denominator goes to zero: $\frac{df}{dx}=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$
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."
Visualizing Curl
The traditional discussion of curl in vector calculus derives an algebraic expression in rectangular coordinates. As with divergence (see the Visualizing Divergence activity), we prefer to give a geometric derivation as circulation per unit area around an appropriately shaped closed curve. In this activity, students learn how to predict the value of the curl at any point by looking at the vector field near that point. The activity is an excellent one for fostering representational fluency, as students can be asked to engage with vector field maps on paper, plots in Mathematica, and symbolic expressions for vector fields. A nice extension is to have students consider vector fields that have different fundamental symmetries and to recognize that they can adjust the shape of the infinitesimal box to take advantage of such symmetry.
Representations used
