$\newcommand{\R}{\mathbb R }$ $\newcommand{\bfa}{\mathbf a}$ $\newcommand{\bfb}{\mathbf b}$ $\newcommand{\bff}{\mathbf f}$ $\newcommand{\bfu}{\mathbf u}$ $\newcommand{\bfx}{\mathbf x}$ $\newcommand{\bfy}{\mathbf y}$ $\newcommand{\ep}{\varepsilon}$
Standard pre-calculus skills such as sketching ellipses and hyperbolas.
Know, and be able to use, basic geometric properties of dot product, cross product, and Euclidean norm. See Section 0.P for some examples.
Identify (without proof) sets of the form $\{ \bfx\in \R^n : A\bfx = {\bf 0} \}$ or $\{ A\bfx : \bfx \in \R^n \}$ as subspaces of dimension $*$ in $\R^*$, when $A$ is a matrix for which you know, or can easily see, that the rows/columns are linearly independent. (You should be able to easily see this if the matrix has at most 2 rows/columns.) See Section 0.2.
Translate between different visual representations of functions of two variables, such as graphs and level sets, and also match formulas for functions with either graphs or level sets. See Section 0.P for some examples.
Determine (without proof) the interior, boundary, and closure of some set(s). Determine (without proof) whether some set(s) are
Questions about basic aspects of open/closed sets, boundary, interior etc.. See Section 1.1 for some examples.
Determine whether $\lim_{\bfx\to \bfa} f(\bfx)$ exists for specific (reasonable) functions $f$, and determine the limit if it exists. Explain your answer.
See Section 1.2 for some examples.
What do we mean by
In this class, explain
?Explain your answer
means that you are not being necessarily asked for a full mathematical proof, but you are being asked to communicate how you know your answer is correct. For example, for a limit that does not exist, a typical explanation might be something like This limit does not exist, because $\lim_{x\to 0^+} f(x^a, x^b) = c$, whereas $\lim_{x\to 0^+} f(x^a, 2x^b) = d$.
For a limit that does exist, an explanation might involve using the squeeze theorem.
Use basic properties of continuity and open/closed sets to identify functions as continuous or sets as open/closed. See Section 1.2 for a discusion.
Answer questions requiring (only) basic knowledge of the Bounded Sequence Theorem. See Section 1.3 for an example.
Determine whether a sequence $\ \ldots \ $ has a convergent subsequence; or, a subsequence that converges to a limit in a set $K = \ \ldots \ $.
Does the Extreme Value Theorem guarantee that the problem of minimizing/maximizing $f = \ldots$ with the constraints $\ldots$ must have a solution?
Show that the equation $f(\bfx)= 0$ has a solution on the set $S = \ldots$. This is a straightforward question when $S$ is known to be path-connected, $f$ is continuous, and one can rather easily find points where $f$ is positive/negative.
Determine all points where a function $f = \ldots$ is differentiable, and determine $\nabla f$ at all such points. We could also ask the same question about a vector-valued function $\bff:\R^n\to \R^m$. In this case we would write $D\bff$ rather than $\nabla f$. See Section 2.1.
Given a function $f = \ldots$, find the direction in which $f$ is increasing/decreasing most rapidly at the point $\bfx = \ldots$.
Determine the directional derivatives at the point $\bfa = \ldots$ of a function $f:\R^n\to \R$ in the direction $\bfu = \ldots$. This may require using the definition of directional derivative, if the function is not $C^1$. See Section 2.1 for examples.
(See Section 2.2). Let $\bff:\R\to \R^n$ be defined by $\ldots$, and consider the curve parametrized by $\bff$.
Use the differential to compute the approximate value of the function $f = \ldots$ at the point $\ldots$. See Section 2.2 for examples.
IMPORTANT!! Compute derivatives using the chain rule. See Section 2.3 for examples.
Use the chain rule to find relations between different partial derivatives of a function. See Section 2.3 for examples.
Find the tangent plane to a set of the form $\{ \bfx\in \R^n : f(\bfx) = c\}$ at the point $\bfa = \ldots$. See Section 2.3 for examples.
IMPORTANT!! Compute second partial derivatives of composite functons using the chain rule. See Section 2.5 for examples.
compute the first- or second-order Taylor polynomials of the function $f = \cdots$ at the point $\bfa = \cdots$. See Section 2.6
Find all critical points of the function $f= \cdots$ and determine whether each nondegenerate critical point is a local min, local max, or saddle point. We might also ask you to classify critical points (which means the same thing), and we might also ask about degenerate critical points. See Section 2.7
Use the Lagrange multiplier technique to solve constrained optimization problems. See Section 2.8 for examples.
Determine whether, for an equation of the form
${\bf F}(\bfx, \bfy) = \bf 0$, the Implicit Function Theorem
guarantees that it is (in principle) possible to solve for
$\bfy$ as a $C^1$ function of $\bfx$. If so, compute
derivatives of the implicitly-defined function.
See Section 3.1 for examples.
Also, answer geometric questions (for example about critical points and contour plots) related to the Implicit Function Theorem.
See Section 3.2 for examples.
Determine whether a $C^1$ function $\bf F:\R^n\to \R^n$ is locally invertible near a point $\bf a\in \R^n$. Be able to compute the derivative of the inverse function and to determine facts about local invertibilty from contour plots of the components of $\bf F$. See Section 3.3 for examples.
Be able to set up and evaluate integrals of reasonable functions over reasonable subsets of $\R^2$ or $\R^3$, if necessary by changing the order of integration in iterated integrals. See Section 4.3 for examples.
Recognize integrals that can be simplified by a transformation to polar, cylindrical, or spherical coordinates, then carry out the relevant transformation and evaluate the integral. See Section 4.4 for examples.
Recognize integrals that can be simplified by a custom-designed change of variables, then design and carry out the change of variables and evaluate the integral. See Section 4.4 for examples.
Determine whether improper integrals are absolutely convergent. Justify your answer by appealing to a theorem and checking that its hypotheses are satisfied. See Section 4.5 for examples.
Be able to exchange differentiation and integration, when justified, to carry out computations. See Section 4.5 for examples.
Be able to compute the following. (See Section 5.1 and Section 5.3).
Be able to use Green's Theorem. See Section 5.2
Be able to compute the curl or divergence of a vector field.
Be able to use the Divergence Theorem. See Section 5.5
You will eventually be resonsible for using Stokes' Theorem, but not until online notes have appeared on the topic.
Given a vector field ${\bf G}$,