$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \renewcommand{\SS}{\mathbb{S}} \newcommand{\DD}{\mathbb{D}} \newcommand{\pp}{^{\prime\prime}} \newcommand{\p}{^\prime} \newcommand{\proj}{\operatorname{proj}} \newcommand{\area}{\operatorname{area}} \newcommand{\len}{\operatorname{length}} \newcommand{\acc}{\operatorname{acc}} \newcommand{\ang}{\sphericalangle} \newcommand{\map}{\mathrm{map}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

Assignment 3

Problems

Linear Transformations

This excercise goes with similar examples in the text, on visualizing linear transformations action on the plane by what they do to the points of the unit square. For instance, we saw that the transformation $\smat{2&0\\0&1}$ scales the $x$ axis by a factor of $2$ and leaves the $y$ axis invariant, so it performs the following stretch to our little smiley face

Exercise 1 Choose your own image on the plane (hand-drawn is great!), and draw a reference image of it undistorted, inside the unit square. Then draw its image under each of the following linear transformations:

\[\pmat{2&0\\ 0&2}\hspace{1cm} \pmat{1&1\\ 0&1}\hspace{1cm} \pmat{2&1\\ 1&1}\hspace{1cm} \pmat{0&-1\\1 &0}\]

Determinants & Area

Recall the following definition: the determinant of a linear transformation $M=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$ is

\[\det M = \left|\begin{matrix}a &b\\c&d\end{matrix}\right| = ad-bc\]

The determinant measures the change in area under a linear map.

In class, said this measured the area change of the unit square under the linear transformation $M$, but now we will confirm it. We can actually find this area in a pretty satisfying way using just what we’ve proven about Euclidean geometry so far. We know the areas of squares, rectangles, and right triangles, so let’s try to write the area we are after as a difference of things we know:

A formula for the determinant can be found knowing only the area of squares, rectangles, and right triangles. (I learned this awesome diagram from Prof Daniel O’Connor!)

Exercise 2 Show the area of the parallelogram spanned by $\langle a,c\rangle$ and $\langle b,d\rangle$ is $ad-bc$, using the Euclidean geometry we have done, and the diagram above.

Calculating Derivatives

Practice calculating the derivative of multivariable functions as matrices, and applying them to vectors. No proofs, here, just some computations!

Exercise 3 Find the derivatives of the following functions, at the specified points.

The function $f(x,y)=(xy,x+y)$ at the point $p=(1,2)$.
The function $\phi(x,y)=\left(xy^2-3x,\frac{x}{y^2+1}\right)$ at the point $q=(3,0)$.

Now use these to compute the following quantities:

$Df_{(1,2)}\langle 3,4\rangle$
$D\phi_{(3,0)}\langle a,b\rangle$

Differentiating Compositions

This is another problem which focuses on the new computational skills, using linear algebra. No proofs here either!

Exercise 4 If $F,G,H$ are the following multivariate functions \[F(x,y)=(x-y,xy)\] \[G(x,y)=(-y,x)\] \[H(x,y)=(x^3,y^3)\]

Differentiate the following compositions:

$F\circ G$ at $(1,1)$
$G\circ G$ at $(0,2)$
$F\circ G\circ H$ at $(-1,3)$.

When the Derivative is Constant

In class, we proved that if a function is linear, then its derivative is constant. But is this the only time a function’s derivative is constant? Certainly no - the derivative of $(x,y)\mapsto(x+1,y)$ is constant (equal to the identity matrix!), even though this function is not linear.

We call a function affine if it is the composition of a linear function and addition of a constant. For instance, $2x+3$ or $5x+2y-7$ are affine functions. We call a multivariable function affine if each of its component functions is affine.

Exercise 5 (When the derivative is constant) Prove that a function $\phi\colon\RR^2\to\RR^2$ has a constant derivative if and only if the function is affine: that is, a linear map plus constants.

Hint: if the derivative is a constant matrix, can you integrate each entry (with respect to the right variable) to figure out what the original functions were?

OPTIONAL Problems

You do not need to turn these problems in. But they are all good things to know or review, so I recommend you attempt them. The first is about arithmetic with infintiesimals: there was more interest than I expected in class, so I wanted to showcase a little more of the power of this approach here.

Infinitesimal Algebra

In class we said that we do not allow arithmetic between of infinitesimals that live in different tangent spaces. And this is a good rule to follow! But in speical situations one can actually define this arithmetic consistently, and the line $\RR$ and plane $\RR^2$ are two of these situations. First, consider the line. If we choose two real numbers $a,b$ and infititesimals $u\in T_a\mathbb{R}$, $v\in T_b\RR$, we can multiply them using regular rules of multiplication, with one new law: the product of two infinitesimals is zero

\[(a+u)(b+v)=ab+av+bu+uv=ab+(av+bu)\]

We then interpret this as saying that the point the product is based at is $ab\in\RR$ (which makes sense, we multiplied $a$ and $b$) and the infinitesimal result in $T_{ab}\RR$ is $av+bu$.

The amazing thing about this multiplication is that when you use it, you can automatically compute the derivative of a function, just by applying it to a point and tangent vector at the same time! The “real part” of the answer is the function value, and the “infinitesimal part” is the derivative.

Exercise 6 (Optional)

Confirm this for yourself, by evaluating the function $f(x)=x^2$ at the point $x+\epsilon$, where $\epsilon$ is an infinitesimal in $T_x\RR$.
This also works for multivariable functions! Consider the following map from $\RR^2\to \RR^2$: can you find its derivative at $(x,y)$ as the infinitesimal part of applying it to the point $(x+u,y+v)\in T_{(x,y)}\RR^2$? Then, take the derivative the “normal way” (by differentiating each component function and forming a matrix) and apply to the infinitesimal vector $\langle u,v\rangle\in T_{(x,y)}\RR^2$ to see you get the same result.

\[L(x,y)=\left(x^2-y^2,2xy\right)\]

Calculus Practice

Another good thing to be doing this week is some calculus review! So, you should attempt the following calculus questions, and practice whichever skills you find yourself rusty with. (U-sub, integration by parts, and power series).

Exercise 7 (Optional) Compute the following integrals, as a refresher of your calculus skills:

\[\int \sin(2q-3)dq\hspace{1cm}\int \frac{x}{x+1}dx\] \[\int y^2 e^{y^3}dy\hspace{1cm}\int t^2e^tdt\]

Exercise 8 (Optional) Find Power series for the following functions

\[x\cos(2x^2)\]

\[\int e^{x^2}dx\]

\[\frac{4}{3+5x^2}\]

Solutions

1: Linear Transformations (Quinn)

\[ \begin{pmatrix} 2 & 0\\ 0 & 2 \end{pmatrix} \]

\[ \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix} \]

\[ \begin{pmatrix} 2 & 1\\ 1 & 1 \end{pmatrix} \]

\[ \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} \]