Final Exam

39.1 Keywords for the Final (see also Units 14+28)
39.2 Final Exam (Practice A)
39.3 Final Exam (Practice B)
39.4 Final Exam

39.1 Keywords for the Final (see also Units 14+28)

39.1.1 Discrete Calculus

$G=(V, E)$ graph with vertex set $V$ and edge set $E$
$0$-form: function on $V$. Discrete scalar function
$1$-form: function on oriented $E$. Discrete vector field
$2$-form: function on oriented triangles $T$
$d(f)=\operatorname{grad}(f)$ is a function on edges $a \to b$ defined by $f(b)-f(a)$
$H=d F=\operatorname{curl}(F)$ is a function on triangles obtained by summing $F$ along the triangle
For a $1$-form $F$, $d^{*} F$ is a function on vertices. Add up the attached edge values
For a $2$-form $H$, $d^{*} H$ is a function on edges. Add up the attached triangle values

39.1.2 New People

Mentioned: Cartan, Maxwell, Stokes, Green, Gauss, Newton, Einstein, Kirchhoff, Menger, Koch, Escher, Peirce

39.1.3 Partial Derivatives

$L(x, y)=f(x_{0}, y_{0})+f_{x}(x_{0}, y_{0})(x-x_{0})+f_{y}(x_{0}, y_{0})(y-y_{0})$ linear approximation
$Q(x, y)=L(x_{0}, y_{0})+f_{x x}(x-x_{0})^{2} / 2+f_{y y}(y-y_{0})^{2} / 2+f_{x y}(x-x_{0})(y-y_{0})$
Use $L(x, y)$ to estimate $f(x, y)$ near $f(x_{0}, y_{0})$. The result is \[f(x_{0}, y_{0})+a(x-x_{0})+b(y-y_{0})\]
tangent plane: $a x+b y+c z=d$ with $a=f_{x}$, $b=f_{y}$, $c=f_{z}$, $d=a x_{0}+b y_{0}+c z_{0}$
Estimate $f(x, y)$ by $L(x, y)$ or $Q(x, y)$ near $(x_{0}, y_{0})$
$f_{x y}=f_{y x}$ Clairaut’s theorem for functions which are in $C^{2}$
$r_{u}(u, v)$, $r_{v}(u, v)$ tangent to surface parameterized by $r(u, v)$

39.1.4 Parametrization

$r: G \subset \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$, $d r$ Jacobian
$g=d r^{T} d r$ first fundamental form, $|d r|=\sqrt{g}$ distortion factor
$\operatorname{curl}(F)(r(u, v)) \cdot(r_{u} \times r_{v})=F_{u} \cdot r_{v}-F_{v} \cdot r_{u}$ important formula

39.1.5 Partial Differential Equations

$f_{x y}=f_{y x}$, Clairaut
$f_{t}=f_{x x}$, heat equation
$f_{t t}-f_{x x}=0$, wave equation
$f_{x}-f_{t}=0$, transport equation
$f_{x x}+f_{y y}=0$, Laplace equation
$f_{t}+f f_{x}=f_{x x}$, Burgers equation
$d F^{*}=j$, $d F=0$, Maxwell equations
$\operatorname{div} (F)=4 \pi \sigma$, Gravity equation

39.1.6 Gradient

$\nabla f(x, y)=[f_{x}, f_{y}]^{T}$, $\nabla f(x, y, z)=[f_{x}, f_{y}, f_{z}]^{T}$, gradient
$D_{v} f=\nabla f \cdot v$ directional derivative
$\frac{d}{d t} f(r(t))=\nabla f(r(t)) \cdot r^{\prime}(t)$ chain rule
$\nabla f(x_{0}, y_{0})$ is orthogonal to the level curve $f(x, y)=c$ containing $(x_{0}, y_{0})$
$\nabla f(x_{0}, y_{0}, z_{0})$ is orthogonal to the level surface $f(x, y, z)=c$ containing $(x_{0}, y_{0}, z_{0})$
$\frac{d}{d t} f(x+t v)=D_{v} f$ by chain rule
$(x-x_{0}) f_{x}(x_{0}, y_{0})+(y-y_{0}) f_{y}(x_{0}, y_{0})=0$ tangent line
$(x-x_{0}) f_{x}(x_{0}, y_{0}, z_{0})+(y-y_{0}) f_{y}(x_{0}, y_{0}, z_{0})+(z-z_{0}) f_{z}(x_{0}, y_{0}, z_{0})=0$ tangent plane
$D_{v} f(x_{0}, y_{0})$ is maximal in the $v=\nabla f(x_{0}, y_{0}) /|\nabla f(x_{0}, y_{0})|$ direction
$f(x, y)$ increases in the $\nabla f /|\nabla f|$ direction at points which are not critical points
if $D_{v} f(x)=0$ for all $v$, then $\nabla f(x)=0$
$f(x, y, z)=c$ defines $y=g(x, y)$, and $g_{x}(x, y)=-f_{x}(x, y, z) / f_{z}(x, y, z)$ implicit diff

39.1.7 Extrema

$\nabla f(x, y)=[0,0]^{T}$, critical point
$D=\operatorname{det}(d^{2} f)=f_{x x} f_{y y}-f_{x y}^{2}$ discriminant
Morse: critical point and $D \neq 0$, in $2$D looks like $x^{2}+y^{2}$, $x^{2}-y^{2}$, $-x^{2}-y^{2}$
$f(x_{0}, y_{0}) \geq f(x, y)$ in a neighborhood of $(x_{0}, y_{0})$ local maximum
$f(x_{0}, y_{0}) \leq f(x, y)$ in a neighborhood of $(x_{0}, y_{0})$ local minimum
$\nabla f(x, y)=\lambda \nabla g(x, y)$, $g(x, y)=c$, $\lambda$ Lagrange equations
$\nabla f(x, y, z)=\lambda \nabla g(x, y, z)$, $g(x, y, z)=c$, $\lambda$ Lagrange equations
second derivative test: $\nabla f=(0,0)$, $D>0$, $f_{x x}<0$ local max , $\nabla f=(0,0)$, $D>0$, $f_{x x}>0$ local min, $\nabla f=(0,0)$, $D<0$ saddle point
$f(x_{0}, y_{0}) \geq f(x, y)$ everywhere, global maximum
$f(x_{0}, y_{0}) \leq f(x, y)$ everywhere, global minimum

39.1.8 Double Integrals

$\iint_{R} f(x, y) \,d y \,d x$ double integral
$\int_{a}^{b} \int_{c}^{d} f(x, y) \,d y \,d x$ integral over rectangle
$\int_{a}^{b} \int_{c(x)}^{d(x)} f(x, y) \,d y \,d x$ bottom-to-top region
$\int_{c}^{d} \int_{a(y)}^{b(y)} f(x, y) \,d x \,d y$ left-to-right region
$\iint_{R} f(r, \theta) \fbox{$r$} \,d r \,d \theta$ polar coordinates
$\iint_{R}|r_{u} \times r_{v}| \,d u \,d v$ surface area
$\int_{a}^{b} \int_{c}^{d} f(x, y) \,d y \,d x=\int_{c}^{d} \int_{a}^{b} f(x, y) \,d x \,d y$ Fubini
$\iint_{R} \fbox{$1$} \,dx \,dy$ area of region $R$
$\iint_{R} f(x, y) \,d x \,d y$ signed volume of solid bound by graph of $f$ and $x y$-plane

39.1.9 Triple Integrals

$\iiint_R f(x, y, z) \,dz \,dy \,dx$ triple integral
$\int_a^b \int_c^d \int_u^v f(x, y, z) \,dz \,dy \,dx$ integral over rectangular box
$\int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x, y)}^{h_2(x, y)} f(x, y) \,dz \,dy \,dx$ type I region
$\iiint_R f(r, \theta, z) \fbox{$r$} \,dz \,dr \,d\theta$ integral in cylindrical coordinates
$\iiint_R f(\rho, \theta, \phi) \fbox{$\rho^2 \sin(\phi)$} \,d\rho \,d\phi \,d\theta$ integral in spherical coordinates
$\int_a^b \int_c^d \int_u^v f(x, y, z) \,dz \,dy \,dx = \int_u^v \int_c^d \int_a^b f(x, y, z) \,dx \,dy \,dz$ Fubini
$V = \iiint_E \fbox{$1$} \,dz \,dy \,dx$ volume of solid $E$
$M = \iiint_E \sigma(x, y, z) \,dz \,dy \,dx$ mass of solid $E$ with density $\sigma$

39.1.10 Line Integrals

$F(x, y)=[P(x, y), Q(x, y)]^{T}$ vector field in the plane
$F(x, y, z)=[P(x, y, z), Q(x, y, z), R(x, y, z)]^{T}$ vector field in space
$\int_{C} F \cdot d r=\int_{a}^{b} F(r(t)) \cdot r^{\prime}(t) \,d t$ line integral
$F(x, y)=\nabla f(x, y)$ gradient field $=$ potential field $=$ conservative field

39.1.11 Fundamental theorem of line integrals

FTLI: $F(x, y)=\nabla f(x, y)$, $\int_{a}^{b} F(r(t)) \cdot r^{\prime}(t) \,d t=f(r(b))-f(r(a))$
Closed loop property $\int_{C} F \,d r=0$, for all closed curves $C$
Always equivalent: closed loop property, path independence and gradient field
Mixed derivative test $\operatorname{curl}(F) \neq 0$ assures $F$ is not a gradient field
In simply connected regions: $\operatorname{curl}(F)=0$ implies that field $F$ is conservative
Conservative field: can not be used for perpetual motion.

39.1.12 Green’s Theorem

$F(x, y)=[P, Q]^{T}$, curl in two dimensions: $\operatorname{curl}(F)=Q_{x}-P_{y}$
Green’s theorem: $C$ boundary of $R$, then $\int_{C} F \cdot d r=\iint_{R} \operatorname{curl}(F) \,d x \,d y$
Area computation: Take $F$ with $\operatorname{curl}(F)=Q_{x}-P_{y}=1$ like $F=[-y, 0]^{T}$ or $F=[0, x]^{T}$
Green’s theorem is useful to compute difficult line integrals or difficult $2$D integrals

39.1.13 Flux integrals

$F(x, y, z)$ vector field, $S=r(R)$ parametrized surface
$r_u \times r_{v} \,d u \,d v=d S$ is a $2$-form on surface
$\iint_{S} F \cdot d S=\iint_{S} F(r(u, v)) \cdot(r_{u} \times r_{v}) \,d u \,d v$ flux integral

39.1.14 Stokes Theorem

$F(x, y, z)=[P, Q, R]^{T}$, $\operatorname{curl}([P, Q, R]^{T})=[R_{y}-Q_{z}, P_{z}-R_{x}, Q_{x}-P_{y}]^{T}=\nabla \times F$
Stokes’s theorem: $C$ boundary of surface $S$, then $\int_{C} F \cdot d r=\iint_{S} \operatorname{curl}(F) \cdot d S$
Stokes theorem allows to compute difficult flux integrals or difficult line integrals

39.1.15 Grad Curl Div

$\nabla=[\partial_{x}, \partial_{y}, \partial_{z}]^{T}$, $F=\nabla f$, $\operatorname{curl}(F)=\nabla \times F$, $\operatorname{div}(F)=\nabla \cdot F$
$\operatorname{div}(\operatorname{curl}(F))=0$ and $\operatorname{curl}(\operatorname{grad}(f))=0$
$\operatorname{div}(\operatorname{grad}(f))=\Delta f$ Laplacian
Incompressible $=$ divergence free field: $\operatorname{div}(F)=0$ everywhere. Implies $F=\operatorname{curl}(H)$
Irrotational $=\operatorname{curl}(F)=0$ everywhere. Implies $F=\operatorname{grad}(f)$

39.1.16 Divergence Theorem

$\operatorname{div}([P, Q, R]^{T})=P_{x}+Q_{y}+R_{z}=\nabla \cdot F$
Divergence theorem: solid $E$, boundary $S$ then $\iint_{S} F \cdot d S=\iiint_{E} \operatorname{div}(F) \,d V$
The divergence theorem allows to compute difficult flux integrals or difficult 3D integrals

39.1.17 Some topology

Simply connected region $D$: can deform any closed curve within $D$ to a point
Interior of a region $D$: points in $D$ for which small neighborhood is still in $D$
Boundary of curve $C$: the end points of the curve
Boundary of $S$ points on surface not in the interior of the parameter domain
Boundary of solid $G$: points in $G$ which are not in the interior of $D$
Closed surface: a surface without boundary like a sphere
Closed curve: a curve with no boundary like a knot

39.1.18 Some surface parameterizations

Sphere of radius $\rho$: $r(u, v) = [\rho\cos(u)\sin(v), \rho\sin(u)\sin(v), \rho\cos(v)]^T$
Graph of function $f(x, y)$: $r(u, v) = [u, v, f(u, v)]^T$
Example: Paraboloid $r(u, v) = [u, v, u^2 + v^2]^T$
Plane containing $P$ and vectors $u$, $v$: $r(s, t) = P + su + tv$
Surface of revolution: distance $g(z)$ of $z - \mathrm{axis}$: $r(u, v) = [g(v)\cos(u), g(v)\sin(u), v]^T$
Example: Cylinder $r(u, v) = [\cos(u), \sin(u), v]^T$
Example: Cone $r(u, v) = [v\cos(u), v\sin(u), v]^T$
Example: Paraboloid $r(u, v) = [\sqrt{v}\cos(u), \sqrt{v}\sin(u), v]^T$

39.1.19 Integration for integral theorems

Double and triple integral: $\iint_{G} f(x, y) \,d A$, $\iiint_{G} f(x, y, z) \,d V$
Line integral: $\int_{a}^{b} F(r(t)) \cdot r^{\prime}(t) \,d t$
Flux integral: $\iint_{S} F(r(u, v)) \cdot(r_{u} \times r_{v}) \,d u \,d v$

39.1.20 Differential forms

A tensor of type $(p, q)$ is a multi-linear map $(E^{*})^{p} \times E^{q} \rightarrow \mathbb{R}$.
A $\boldsymbol{k}$-form is a field, which attaches at every point a multi-linear anti-symmetric map of $k$ variables.
$F=5 x^{3} \,d y \,d z+7 \sin (y) x \,d x \,d z+3 \cos (x y) \,d x \,d y$ is an example of a $2$-form. In calculus this is identified with a vector field $F=[5 x^{3}, 7 \sin (y) x, 3 \cos (x y)]$.
The exterior derivative of a term like $F=P \,d x \,d y$ is \[\begin{aligned} d F&=(P_{x} \,d x+P_{y} \,d y+P_{z} \,d z) \,d x \,d y\\ &=P_{z} \,d z \,d x \,d y\\ &=P_{z} \,d x \,d y \,d z. \end{aligned}\]
The General Stokes theorem tells $\int_{G} d F=\int_{d G} F$, where $d G$ is the boundary of $G$.

39.2 Final Exam (Practice A)

Problem 39A.1 (10 points):

On the graph $G$ in Figure (39.1) we are given a $1$-form $F$ on a graph $G=(V, E)$.

(3 points) Write the values of the curl $d F$. As a $2$-form it is a function on the set $T$ of triangles.
(3 points) Compute the "discrete divergence" $d^{*} F$, which is a $0$-form, a function on the vertices.
(4 points) Find the value of the Laplacian $d^{*} d F+d d^{*} F$ and enter the values near the edges in Figure (39.2).

**Figure 1.** A graph with a $1$-Form $F$. Enter here the result for a) and b).

**Figure 2.** Enter here the result for c).

Problem 39A.2 (10 points, each question is one point):

Who formulated the law of gravity in the form the partial differential equation $\operatorname{div}(F)=4 \pi \sigma$?
The expression $5 x \,d x \,d z \,d x+77 \,d y \,d z \,d y+3 \,d x \,d y+6 \,d y \,d x$ simplifies to$\ldots\ldots$
What value is $\iint_{S}[x, y, z] \cdot d S$ if $S$ is the unit sphere oriented outwards?
What is the distance between the point $(0,0,3)$ and the $x y$-plane?
Is it true that if $|r^{\prime}(t)|=1$ everywhere, then $r^{\prime \prime}(t)$ is perpendicular to the velocity $r^{\prime}(t)$?
What is the distortion factor $|d r|$ for the change of coordinates $r(u, v)=[-2 v, 3 u]$?
If $r(u, v)$ parametrizes a surface in $\mathbb{R}^{3}$, is it true that $r_{u} \times(r_{u} \times r_{v})$ tangent to the surface?
Yes or no: if $(0,0,0)$ is a maximum of $f(x, y, z)$ then $f_{x x}(0,0,0)<0$.
Write down the quadratic approximation of $1+x+y+\sin (x^{2}-y^{2})$?
If $S: f(x, y, z)=x^{2}+y^{2}+z^{2}=1$ is oriented outwards, then the flux of $\nabla f$ through $S$ is either negative, zero or positive. Which of the three cases is it?

Problem 39A.3 (10 points, each problem is one point):

Which of the triangles in Figure (39.3) is integrated over in $\int_{0}^{1} \int_{y}^{1} f(x, y) \,d x \,d y$?
We have seen a counter example for Clairaut’s theorem. This function $f(x, y)$ was in $C^{k}$ but not in $C^{k+1}$. The integer $k$ indicated how many times we could differentiate $f$ continuously. What was the $k$?
To what group of partial differential equations belongs $\operatorname{div}(E)=$ $4 \pi j+E_{t}$?
Write down the Cauchy-Schwarz inequality.
Let $G$ be the first stage of the Menger sponge (with $20$ cubes from $27$ cubes present). Is it simply connected?
Take a exterior derivative of the differential form $F=\sin (x z) \,d x \,d y$.
Parametrize the surface $x=z^{2}-y^{3}$.
Parametrize the curve obtained by intersecting of the ellipsoid $x^{2} / 4+y^{2}+z^{2} / 9=1$ with the plane $y=0$.
What surface is given in spherical coordinates as $\sin (\phi) \cos (\theta)=\cos (\phi)$?
Write down the general formula for the area of a triangle with vertices $(0,0,0)$, $(a, b, c)$, $(u, v, w)$.

Problem 39A.4 (10 points):

(6 points) Find the equation of the plane which contains the line \[r(t)=[1+t, 2+t, 3-t]\] and which is perpendicular to the plane $\Sigma: x+2 y-z=4$.
(4 points) What is the angle between the normal vectors of $\Sigma$ and the plane you just found?

Problem 39A.5 (10 points):

(8 points) Find the critical points of the function $f(x, y)=\cos (x)+y^{5}-5 y$ and classify them using the second derivative test. You can assume that $0 \leq x<2 \pi$.
(2 points) Does the function $f$ have a global maximum or a global minimum?

Problem 39A.6 (10 points):

(5 points) Use the Lagrange method to find the maximum of $f(x, y)=$ $y^{2}-x$ under the constraint $g(x, y)=x+x^{3}-y^{2}=2$.
(5 points) The Lagrange equations fail to find the maximum of $f(x, y)=$ $y^{2}-x$ under the constraint $g(x, y)=x^{3}-y^{2}=0$. Still, the Lagrange theorem still allows you to find the maximum. How?

Problem 39A.7 (10 points):

(6 points) Find the tangent plane at the point $P=(4,2,1,1)$ of the surface \[x^{2}-2 y^{2}+z^{3}+w^{2}=2.\]
(4 points) Parametrize the line $r(t)$ which passes through $P$ which is perpendicular to the hyper surface at that point. Then find $\big(r(1)+r(-1)\big) / 2$.

Problem 39A.8 (10 points):

Estimate $f(0.012,0.023)$ for $f(x, y)=\log (1+x+3 x y)$ using linear approximation.
Estimate $f(0.012,0.023)$ for $f(x, y)=\log (1+x+3 x y)$ using quadratic approximation.

Problem 39A.9 (10 points):

Lets look at the curve which satisfies the acceleration \[r^{\prime \prime}(t)=[-2 \cos (t),-2 \sin (t),-2 \cos (t),-2 \sin (t)],\] has the initial position $[2,0,2,0]$ and initial velocity $[0,2,0,2]$. Find $r(t)$.
What is the curvature $|T^{\prime}(t)| /|r^{\prime}(t)|$ of $r(t)$ at $t=0$?

Problem 39A.10 (10 points):

Integrate the function $f(x, y)=x+x^{2}-y^{2}$ over the region $1, \(x y>0$.
Find the surface area of \[r(t, s)=[\cos (t) \sin (s), \sin (t) \sin (s), \cos (s)]\] where $0 \leq t \leq 2 \pi$ and $0 \leq s \leq t / 2$.

Problem 39A.11 (10 points):

Let $E$ be the solid \[x^{2}+y^{2} \geq z^{2}, \quad x^{2}+y^{2}+z^{2} \leq 9, \quad y \geq|x|.\]

(7 points) Integrate \[\iiint_{E} (x^{2}+y^{2}+z^{2}) \,d x \,d y \,d z.\]
(3 points) Let $F$ be a vector field \[F=[x^{3}, y^{3}, z^{3}]\] Find the flux of $F$ through the boundary surface of $E$, oriented outwards.

Problem 39A.12 (10 points):

What is the line integral of the force field \[F(x, y, z, w)=[1,5 y^{4}+z, 6 z^{5}+y, 7 w^{6}]^{T}+[y-w, 0,0,0]^{T}\] along the path $r(t)=[t^{3}, \sin (6 t), \cos (8 t), \sin (6 t)]$ from $t=0$ to $t=2 \pi$?
Hint: We have written the field by purpose as the sum of two vector fields.

Problem 39A.13 (10 points):

Find the area of the region $|x|^{2 / 5}+|y|^{2 / 5} \leq 1$. Use an integral theorem.

Problem 39A.14 (10 points):

What is the flux of the vector field \[F(x, y, z, w)=[x+\cos (y), y+z^{2}, 2 z, 3 w]\] through the boundary of the solid \[E: 1 \leq x \leq 3, \quad 3 \leq y \leq 5, \quad 0 \leq z \leq 1, \quad 4 \leq w \leq 8\] oriented outwards?

Problem 39A.15 (10 points):

Find the flux of the curl of the vector field \[F(x, y, z)=[-z, z+\sin (x y z), x-3]^{T}\] through the twisted surface seen in Figure (39.5) is oriented inwards and parametrized by \[r(t, s)=\big[(3+2 \cos (t)) \cos (s),(3+2 \cos (t)) \sin (s), s+2 \sin (t)\big]\] where $0 \leq s \leq 7 \pi / 2$ and $0 \leq t \leq 2 \pi$.

**Figure 5.** The boundary of the surface is made of two circles $r(t, 0)$ and $r(t, 7 \pi / 2)$. The picture gives the direction of the velocity vectors of these curves (which in each case might or might not be compatible with the orientation of the surface).

39.3 Final Exam (Practice B)

Problem 39B.1 (10 points):

The graph $G=(V, E)$ in Figure (39.6) represents a discrete surface in which all triangles are oriented counterclockwise. The values of a $1$-form $=$ vector field $F$ are given.

(2 points) Find the line integral of $F$ along the boundary curve oriented counter clockwise.
(2 points) Compute the curl $H=d F$ and write its values into the triangles.
(2 points) What is the sum of all curl values? Why does it agree with the result in a)?
(2 points) Find also $g=d^{*} F$ and enter it near the vertices.
(1 point) True or False: $\sum_{x \in V} g(x)=0$.
(1 point) True of False: we called $L=d d^{*}$ the Laplacian of $G$.

**Figure 6.** A discrete $2$-dimensional region on which a $1$-form $F$ models a vector field. You compute the curl $d F$ and divergence $d^{*} F$ of $F$.

Problem 39B.2 (10 points, each question is one point):

Name the $3$-dimensional analogue of the Mandelbrot set.
If $A$ is a $5 \times 4$ matrix, then $A^{T}$ is a $m \times n$ matrix. What is $m$ and $n$?
Write down the general formula for the arc length of a curve \[r(t)=[x(t), y(t), z(t)]^{T}\] with $a \leq t \leq b$.
Write down one possible formula for the curvature of a curve \[r(t)=[x(t), y(t), z(t)]^{T}.\]
We have seen a parametrization of the $3$-sphere invoking three angles $\phi$, $\theta_{1}$, $\theta_{2}$. Either write down the parametrization or recall the name of the mathematician after whom it this parametrization is named.
The general change of variable formula for $\Phi: R \rightarrow G$ is \[\iiint_{R} f(u, v, w) \ldots\ldots\ldots \,d u \,d v \,d w=\iiint_{G} f(x, y, z) \,d x \,d y \,d z.\] Fill in the blank part of the formula.
What is the numerical value of $\log (-i)$?
We have used the Fubini theorem to prove that $C^{2}$ functions $f(x, y)$ satisfy a partial differential equation. Please write down this important partial differential equation as well as its name. (It was used much later in the course.)
What is the integration factor $|d r|$ for the parametrization \[r(u, v)=[a \cos (u) \sin (v), b \sin (u) \sin (v), c \cos (v)]^{T}?\]
In the first lecture, we have defined $\sqrt{\operatorname{tr}(A^{T} A)}$ as the length of a matrix. What is the length of the $3 \times 3$ matrix which contains $1$ everywhere?

Problem 39B.3 (10 points, each problem is one point):

Assume that for a Morse function $f(x, y)$ the discriminant $D$ at a critical point $(x_{0}, y_{0})$ is positive and that $f_{y y}(x_{0}, y_{0})<0$. What can you say about $f_{x x}(x_{0}, y_{0})$?
We have proven the identity $|d r|=|r_{u} \times r_{v}|$, where $r$ was a map from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$. For which $m$ and $n$ was this identity defined?
Which of the following is the correct integration factor when using spherical coordinates in $4$ dimensions?

$|d \Phi|=r$
$|d \Phi|=3+\cos (\phi)$
$|d \Phi|=\rho^{2} \sin (\phi)$
$|d \Phi|=\rho^{3} \sin (2 \phi) / 2$

Which of the following vector fields are gradient fields? (It could be none, one, two, three or all.)

$F=[x, 0]^{T}$
$F=[0, x]^{T}$
$F=[x, y]^{T}$
$F=[y, x]^{T}$

Which of the following four surfaces is a one-sheeted hyperboloid? (It could be none, one, two, three or all.)

$x^{2}+y^{2}=z^{2}-1$
$x^{2}-y^{2}=1-z^{2}$
$x^{2}+y^{2}=1-z^{2}$
$x^{2}-y^{2}=z^{2}+1$

Parametrize the surface $x^{2}+y^{2}-z^{2}=1$ as \[r(\theta, z)=[\ldots\ldots\ldots, \ldots\ldots\ldots, \ldots\ldots\ldots]^T.\]
Who was the creative person who discovered dark matter and proposed the mechanism of gravitational lensing?
What is the cosine of the angle between the matrices $A, B \in M(2,2)$, where $A$ is the identity matrix and $B$ is the matrix which has 1 everywhere? You should get a concrete number.
We have seen the identity $|v|^{2}+|w|^{2}=|v-w|^{2}$, where $v$, $w$ are vectors in $\mathbb{R}^{n}$. What conditions do $v$ and $w$ have to satisfy so that the identity holds?
Compute the exterior derivative $d F$ of the differential form \[F=e^{x} \sin (y) \,d x \,d y+\cos (x y z) \,d y \,d z.\]

Problem 39B.4 (10 points):

(4 points) Find the plane $\Sigma$ which contains the three points \[A=(3,2,1), \quad B=(3,3,2), \quad C=(4,3,1).\]
(3 points) What is the area of the triangle $A B C$?
(3 points) Find the distance of the origin $O=(0,0,0)$ to the plane $\Sigma$.

Problem 39B.5 (10 points):

(8 points) Find all the critical points of the function \[f(x, y)=x^{5}-5 x+y^{3}-3 y\] and classify these points using the second derivative test.
(2 points) Is any of these points a global maximum or global minimum of $f$?

Problem 39B.6 (10 points):

(8 points) Use the Lagrange method to find all the maxima and all the minima of \[f(x, y)=x^{2}+y^{2}\] under the constraint \[g(x, y)=x^{4}+y^{4}=16.\]
(2 points) In our formulation of Lagrange theorem, we also mentioned the case, where $\nabla g(x, y)=[0,0]^{T}$. Why does this case not lead to a critical point here?

Problem 39B.7 (10 points):

(5 points) The hyper surface \[S=\{f(x, y, z, w)=x^{2}+y^{2}+z^{2}-w=5\}\] defines a three-dimensional manifold in $\mathbb{R}^{4}$. It is poetically called a hyper-paraboloid. Find the tangent plane to $S$ at the point $(1,2,1,1)$.
(5 points) What is the linear approximation $L(x, y, z, w)$ of $f(x, y, z, w)$ at this point $(1,2,1,1)$?

Problem 39B.8 (10 points):

Estimate the value $f(0.1,-0.02)$ for \[f(x, y)=3+x^{2}+y+\cos (x+y)+\sin (x y)\] using quadratic approximation.

Problem 39B.9 (10 points):

(8 points) We vacation in the $\boldsymbol{5}$-star hotel called MOTEL $\boldsymbol{22}$ in $5$-dimensional space and play there ping-pong. The ball is accelerated by gravity \[r^{\prime \prime}(t)=[x(t), y(t), z(w), v(t), w(t)]=[0,0,0,0,-10]^{T}.\] We hit the ball at $r(0)=[4,3,2,1,2]^{T}$ and give it an initial velocity $r^{\prime}(0)=[5,6,0,0,3]^{T}$. Find the trajectory $r(t)$.
(2 points) At which positive time $t>0$ does the ping-pong ball hit the hyper ping-pong table $w=0$? (The points in this space are labeled $[x, y, z, v, w]$.)

Problem 39B.10 (10 points):

(5 points) Integrate the function $f(x, y)=(x^{2}+y^{2})^{22}$ over the region \[G=\{10\}.\]
(5 points) Find the area of the region enclosed by the curve \[r(t)=[\cos (t), \sin (t)+\cos (2 t)]^{T},\] with $0 \leq t \leq 2 \pi$.

Problem 39B.11 (10 points):

(7 points) Integrate \[f(x, y, z)=x^{2}+y^{2}+z^{2}\] over the solid \[G=\{x^{2}+y^{2}+z^{2} \leq 4,\ z^{2}<1\}.\]
(3 points) What is the volume of the same solid $G$?

Problem 39B.12 (10 points):

(8 points) Compute the line integral of the vector field \[F=[y z w+x^{6}, x z w+y^{9}, x y w-z^{3}, x y z+w^{4}]^{T}\] along the path \[r(t)=[t+\sin (t), \cos (2 t), \sin (4 t), \cos (7 t)]^{T}\] from $t=0$ to $t=2 \pi$.
(2 points) What is $\int_{0}^{2 \pi} r^{\prime}(t) \,d t$?

Problem 39B.13 (10 points):

(8 points) Find the line integral of the vector field \[F(x, y)=[3 x-y, 7 y+\sin \left(y^{4}\right)]^{T}\] along the polygon $A B C D E$ with \[A=(0,0), \quad B=(2,0), \quad C=(2,4), \quad D=(2,6), \quad E=(0,4).\] The path is closed. It starts at $A$, then reaches $B$, $C$, $D$, $E$ until returning to $A$ again.
(2 points) What is line integral if the curve is traced in the opposite direction?

Problem 39B.14 (10 points):

(8 points) What is the flux of the vector field \[F(x, y, z)=[y+x^{3}, z+y^{3}, x+z^{3}]^{T}\] through the sphere $S=\{x^{2}+y^{2}+z^{2}=9\}$ oriented outwards?
(2 points) What is the flux of the same vector field $F$ through the same sphere $S$ but where $S$ is oriented inwards?

Problem 39B.15 (10 points):

(7 points) What is the flux of the curl of the vector field \[F(x, y, z)=[-y, x+z(x^{2}+y^{5}), z]^{T}\] through the surface \[S=\big\{x^{2}+y^{2}+z^{2}+z\big(x^{4}+y^{4}+2 \sin (x-y^{2} z)\big)=1,\ z>0\big\}\] oriented upwards?
(3 points) The surface in a) was not closed, it did not include the bottom part \[D=\{z=0,\ x^{2}+y^{2} \leq 1\}\] Assume now that we close the bottom and orient the bottom disc $D$ downwards. What is the flux of the curl of the same vector field $F$ through this closed surface obtained by taking the union of $S$ and $D$?

39.4 Final Exam

Welcome to the final exam. Please don’t get started yet. We start all together at 9:00 AM after getting reminded about some formalities. You can fill out the attendance slip already. Also, you can already enter your name into the larger box above.

You only need this booklet and something to write. Please stow away any other material and any electronic devices. Remember the honor code.
Please write neatly and give details. Except for problems 2 and 3 we want to see details, even if the answer should be obvious to you.
Try to answer the question on the same page. There is additional space on the back of each page. If you must, use additional scratch paper at the end. But put your final result near the question and box the final result.
If you finish a problem somewhere else, please indicate on the problem page where we can find it.
You have 180 minutes for this final exam.

**Figure 7.** A two dimensional discrete sphere $S$.

Problem 39.1 (10 points):

In Figure (39.8) you see a discrete two dimensional region $G$ in which all triangles are oriented counter clockwise. The $1$-form $F$ as a function on oriented edges is given in the picture. Answer the following questions and give reasons:

(2 points) The curl $d F$ of $F$ is a function on oriented triangles. What can you say about the sum over all the curl values $d F$ in the graph $G$ of Figure (39.8)?
(2 points) Is $F$ a gradient field $F=d f$ for some function $f$ on vertices?
(2 points) What is the sum of the natural divergence values $d^{*} F$ on vertices?
(2 points) What was the name of the matrix $K=d^{*} d$ that acts on $0$-forms. It has been defined more than $150$ years ago.
(2 points) In Figure (39.7), you saw a two-dimensional discrete sphere $S$. which plays the role of a closed surface $x^{2}+y^{2}+z^{2}=1$ in $\mathbb{R}^{3}$. Given a $1$-form $F$, a function on oriented edges of $S$, what is the sum over all curls on $S$? The answer is a number but you have to justify the answer.

**Figure 8.** The region $G$ from problem 39.1.

Problem 39.2 (10 points, each question is one point):

Albert Einstein used the notation $v_{k} w^{k}$ for two vectors $v$, $w$. It is today called "Einstein notation". What did Einstein mean, when he wrote $v_{k} w^{k}$?
If $S=r(R)$ is a two-dimensional surface parametrized by \[r(u, v)=[x(u, v), y(u, v), z(u, v)]^{T},\] what is the relation between $|r_{u} \times r_{v}|$ and $\sqrt{\operatorname{det}(d r^{T} d r)}$?
What is the Newton method used for? We have seen this numerical tool in a proof seminar.
What is the curvature of a circle with radius $20$?
Define the $1 \times 5$ matrix $A=[1,1,1,1,1]$. One of the two matrices $A$, $B=A^{T}$ is row reduced. Which one?
What is the distortion factor of the coordinate change $\Phi(x, y)=(3 x+y, x+y)$?
What is the numerical value of $i^{22}$, if $i=\sqrt{-1}$ is the imaginary unit?
What is the name of the differential equation $i \hbar \frac{d}{d t} \psi=K \psi$, where $K$ is a matrix? It appears in a theory which also is called "matrix mechanics".
Why is the distance between two lines $r_{1}(t)=Q+t v$ and $r_{2}(t)=P+t w$ given by the formula \[|(v \times w) \cdot P Q| /|v \times w|?\]
You are given a Morse function $f$ on a $2$-torus and you count that $f$ has $11$ maxima and $11$ minima. How many saddle points are there?

**Figure 9.** Herr Einstein wishes you good luck!

Problem 39.3 (10 points, each question is one point):

In this problem, we work in hyperspace $\mathbb{R}^{4}$, where points have coordinates $(x, y, z, w)$.

Write down the exterior derivative $d F$ of the $2$-form \[F=x^{2} y^{2} z^{2} w^{2} \,d y \,d z.\]
Write down the exterior derivative of the $3$-form \[F=x^{2} y^{2} z^{2} w^{2} \,d x \,d z \,d w\]
Let $G$ be the two-dimensional torus $x^{2}+y^{2}=1$, $z^{2}+w^{2}=1$ embedded in $\mathbb{R}^{4}$. What does the general Stokes theorem tell about $\iint_{G} F \,d S$, where $F$ is the $2$-form from a)?
What is $d^{2} F=d d F$, where $F$ is the $2$-form given in a)?
What is $d^{2} F=d d F$, where $F$ is the $3$-form given in b)?
A $(1,1)$ tensor on $\mathbb{R}^{4}$ can be interpreted as a $4 \times 4$ $\ldots \ldots \ldots$.
A $(0,1)$ tensor on $\mathbb{R}^{4}$ can also be interpreted as a$\ldots\ldots\ldots$.
Is $\operatorname{grad}(\operatorname{grad}(f))$ defined if $f$ is a function?
Does $\operatorname{div}(\operatorname{div}(F))$ make sense for any field $F$?
You see $3$ contour maps of functions $f$, $g$ and $h$ of two variables. One of them is not Morse. Which one? The first the second or the third?

Problem 39.4 (10 points):

(3 points) Parametrize the line $L$ which contains the points \[A=(3,2,1), \quad B=(3,3,2).\]
(3 points) Given the additional point $P=(3,3,3)$, find the distance between $P$ and $L$.
(4 points) Write down the equation $a x+b y+c z=d$ of the plane containing $L$ and $P$.

Problem 39.5 (10 points):

(6 points) Find all the critical points of the function \[f(x, y)=x^{7}-7 x+x y-y\] and classify them using the second derivative test.
(2 points) The island theorem told us that the number of maxima plus the number of minima minus the number of saddle points of $f$ is $1$ on an island. In the current case this fails. Why does this not contradict the island theorem?
(2 points) Does the function $f$ have a global maximum or a global minimum?

Problem 39.6 (10 points):

(7 points) Use the Lagrange method to find the minimum of the function \[f(x, y, z, w)=x^{2}+2 y^{2}+3 z^{2}+w^{2}\] under the constraint \[g(x, y, z, w)=x+y+z+w=17.\]
(3 points) You saw in a) that in this case, the Lagrange equations are a system of linear equations for a couple of unknown. This can be written in matrix form as $A X=b$, where the vector $X$ encodes the unknown quantities and $b$ is a constant vector. What is the size of the matrix $A$?

Problem 39.7 (10 points):

(5 points) Find the tangent plane at the point $P=(3,1,3,-1)$ of the hyper cone \[S=\big\{f(x, y, z, w)=x^{2}+y^{2}-z^{2}-w^{2}=0\big\}\] in $\mathbb{R}^{4}$.
(5 points) Write down the linearization $L(x, y, z, w)$ of $f(x, y, z, w)$ at $(3,1,3,-1)$.

Problem 39.8 (10 points):

Estimate the value $f(0.1,-0.02)$ for $f(x, y)=e^{x+y}$ using quadratic approximation $Q(x, y)$ at $(x_{0}, y_{0})=(0,0)$.

Problem 39.9 (10 points):

(6 points) Find the curve $r(t)$ which satisfies \[r(0)=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right] \quad \text{and} \quad r^{\prime}(0)=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right] \quad \text{and} \quad r^{\prime \prime}(t)=\left[\begin{array}{c}1-\sin (t) \\ -4 \sin (2 t) \\ -9 \sin (3 t)\end{array}\right].\]
(4 points) What is the curvature of the curve at the point $r(0)$?

Problem 39.10 (10 points):

Find the area of the region enclosed by the curve \[r(t)=\left[\begin{array}{c} 3 \cos (t) \\ 2 \sin (t)+\cos (7 t) \end{array}\right],\] where $0 \leq t \leq 2 \pi$.

**Figure 10.** The region in problem 39.10.

Problem 39.11 (10 points):

Integrate \[f(x, y, z)=\frac{e^{x^{2}+y^{2}+z^{2}}}{\sqrt{x^{2}+y^{2}+z^{2}}}\] over the half avocado \[E=\big\{4 \leq x^{2}+y^{2}+z^{2} \leq 16,\ z \leq 0\big\}.\] In other words, compute $\iiint_{E} f \,d V$.

**Figure 11.** The avocado in problem 39.11.

Problem 39.12 (10 points):

Compute the line integral \[\int_{C} F \cdot d r=\int_{0}^{1} F(r(t)) \cdot r^{\prime}(t) \,d t\] of the vector field \[F=\left[\begin{array}{l} P \\ Q \\ R \end{array}\right]=\left[\begin{array}{c} 3 x^{2}+y z \\ 3 y^{2}+x z \\ 3 z^{2}+x y \end{array}\right]\] along the path $C$ parametrized by \[r(t)=\left[\begin{array}{c} \cos (7 \pi t) e^{t(1-t)} \\ \sin (11 \pi t) \\ e^{t(1-t)} \end{array}\right]\] from $t=0$ to $t=1$.

Problem 39.13 (10 points):

Find the line integral $\int_{C} F \cdot d r$ of the vector field \[F(x, y)=\left[\begin{array}{c} y+x^{4} \\ y^{3}+y^{4} \end{array}\right]\] along the boundary $C$ of the hexagon region shown in the picture. The curve $C$ is a closed polygon going counter clockwise from $(2,0)$ over $(1,2)$, $(-1,2)$, $(-2,0)$, $(-1,-2)$, $(1,-2)$ back to $(2,0)$.

**Figure 12.** The hexagon in Problem 39.13.

Problem 39.14 (10 points):

Find the flux $\iint_{S} \operatorname{curl}(F) \cdot d S$ of the curl of the vector field \[F=\left[\begin{array}{c} x^{7} \\ -x \\ \sin \left(z^{2}\right)+z^{3} x \end{array}\right]\] through the surface $S$ parametrized by \[r(s, t)=\left[\begin{array}{c} \big(6+2 \cos ^{2}(s / 2) \cos (t)\big) \cos (2 s) \\ 2 \cos ^{2}(s / 2) \sin (t)+2 s \\ \big(6+2 \cos ^{2}(s / 2) \cos (t)\big) \sin (2 s) \end{array}\right]\] with $0 \leq s \leq 7 \pi / 2$ and $0 \leq t<2 \pi$.
Hint: The surface has two boundary curves obtained by looking at $s=0$ or $s=7 \pi / 2$. We don’t tell you the orientation of the larger curve \[r_{1}(t)=r(0, t)=[6+2 \cos (t), 2 \sin (t), 0]^{T}\] is but you should know that the smaller curve \[r_{2}(t)=r(7 \pi / 2, t)=[-6-\cos (t), \sin (t)+7 \pi, 0]^{T}\] is correctly oriented.

**Figure 13.** The surface $S$ with two boundary circles in Problem 39.14.

Problem 39.15 (10 points):

Find the flux of \[\iint_{S} F \cdot d S\] the vector field \[F=\left[\begin{array}{c} \sin (z)+y^{3}+x \\ \sin (x)+z^{3}+y \\ \sin (y)+x^{3}+z \end{array}\right]\] through the boundary surface $S$ of the solid $E$ given in the picture. The solid is obtained by sculpuring a cube \[-1 \leq x \leq 1, \quad -1 \leq y \leq 1, \quad -1 \leq z \leq 1\] of side length $2$, by cutting away at each corner the points in distance less than $1$ from that corner. In other words, we look at the points in the cube which have distance larger than $1$ from any of the $8$ corners. The surface $S$ bounding the solid $E$ is oriented outwards.

**Figure 14.** The solid given in Problem 39.15.