First Hourly

14.1 Keywords for First Hourly

This is a bit of a checklist. Make your own list. But here is a checklist which tries to be comprehensive. Check off the topics you know and check back with things you do not recall. You will need to have the following on your finger tips.

14.1.1 Theorems

• Cauchy-Schwarz \(|v \cdot w| \leq|v||w|\) in general for \(M(n, m)\)
• Pythagoras \(c^{2}=a^{2}+b^{2}\) for any inner product space
• Al Khashi \(c^{2}=a^{2}+b^{2}-2 a b \cos (\alpha)\) for any triangle
• Thiqueness of row reduction: \(\operatorname{rref}(A)\) is unique in \(M(n, m)\)
• The dot product formula \(v \cdot w = |v||w|\cos(\alpha)\)
• The cross product formula \(|v \times w|=|v||w| \sin (\alpha)\)
• Image of transpose \(\operatorname{im}(A^{T})\) is kernel \(\operatorname{ker}(A)\)
• Cauchy-Binet formula \(|v \times w|^{2}=|v|^{2}|w|^{2}-(v \cdot w)^{2}\)
• Arc length \(\int_{a}^{b}|r^{\prime}(t)|\,d t\) for differentiable \(r\)
• Curvature formulas \(|T^{\prime}| /|r^{\prime}|=|r^{\prime} \times r^{\prime \prime}| /|r^{\prime}|^{3}\)
• Euler formula \(e^{it} = \cos(t) + i\sin(t)\) and special case
• Distortion formula \(\sqrt{\det(dr^Tdr)} = |r_u \times r_v|\) for \(r: \mathbb{R}^2 \to \mathbb{R}^3\)

14.1.2 Proofs

• The use of precise definitions and notation
• Be able to argue by contradiction
• Think visually, make good pictures
• Use algebra to tackle geometric problems
• Master the method of induction
• Know the benefits and risks of intuition
• Be aware of computer assisted verification
• Believe in your creativity

14.1.3 Algorithms

• Find the angle between vectors or matrices
• Find the area of parallelogram
• Find the volume of parallelepiped
• Row reduce a matrix in \(M(n, m)\)
• Get position from velocity or acceleration
• Find the vector perpendicular to a plane
• Find the length of a curve or matrix
• Find the curvature at some point
• Compute with complex numbers
• Switch between coordinate systems
• Compute the distortion factor
• Get distances between objects

14.1.4 Objects

• Matrices \(A\)
• Column and row vectors
• Parametrized curves \(r(t)=[x(t), y(t), z(t)]^{T}\)
• Parametrized surfaces \(r(u, v)=[x(u, v), y(u, v), z(u, v)]^{T}\)
• Functions \(f(x, y, z)\)
• Level surfaces \(f(x, y, z)=d\)
• Linear manifolds \(\{x \mid A x=d\}\)
• Quadratic manifolds \(\left\{x \mid x^{T} B x+A x=d\right\}\)
• Kernel of a linear map \(\{x \mid A x=0\}\)
• Image of a linear map \(\left\{A x \mid x \in \mathbb{R}^{n}\right\}\)

14.1.5 Differentiation

• Velocity \(r^{\prime}\)
• Acceleration \(r^{\prime \prime}\)
• Jerk \(r^{\prime \prime \prime}\)
• Free fall: \(r^{\prime \prime}=v\) given
• TNB frame, \(T=r^{\prime} /|r^{\prime}|\), \(N=T^{\prime} /|T^{\prime}|\), \(B=T \times N\)
• derivative \(d r \in M(n, m)\) of a map \(\mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\)
• Jacobian matrix \(d r\) of a map \(\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\)
• Distortion factor \(\sqrt{\operatorname{det}(d r^{T} d r)}\)
• Distortion factor for \(n=m\) simplifies to \(|\operatorname{det}(d r)|\)
• Example: \(r^{\prime}(t)=d r\), \(\sqrt{\operatorname{det}(d r^{T} d r)}=|r^{\prime}|\) is speed
• Curvature \(|T^{\prime}| /|r^{\prime}|\), In \(\mathbb{R}^{3}\) also \(|r^{\prime} \times r^{\prime \prime}| /|r^{\prime}|^{3}\)

14.1.6 Integration

• Integrate to get arc length.
• Integrate to get position from velocity etc.
• Integration technique: substitution
• Integration technique: integration by parts
• Integration technique: partial fractions
• Integration technique: simplification

14.1.7 Coordinate systems

• Cartesian coordinates
• Polar coordinates
• Cylindrical coordinates
• Spherical coordinates
• General coordinate change
• Distortion factor \(|\operatorname{det}(d r)|=\sqrt{\operatorname{det}(d r^{T} d r)}\)

14.1.8 Parametrized Surfaces

• Spheres
• Surfaces of revolution
• Graphs
• Planes
• Torus
• Helicoid

14.1.9 People

• Mandelbrot
• Hamilton
• Descartes
• Cauchy
• Binet
• Schwarz
• Euler
• Heine
• Cantor
• Bolzano
• Archimedes
• Newton
• Einstein
• Napoleon

14.1.10 Geometry of Space

• \(v = [v_1, v_2, v_3]^T\), \(w = [w_1, w_2, w_3]^T\), \(v + w = [v_1 + w_1, v_2 + w_2, v_3 + w_3]^T\)
• dot product \(v \cdot w = v_1w_1 + v_2w_2 + v_3w_3 = |v||w|\cos(\alpha)\)
• angle \(\cos(\alpha) = (v \cdot w)/|v||w|\)
• cross product \(v \cdot (v \times w) = 0\), \(w \cdot (v \times w) = 0\)
• area parallelogram \(|v \times w| = |v||w|\sin(\alpha)\)
• triple scalar product \(u \cdot (v \times w)\)
• volume of parallelepiped: \(|u \cdot (v \times w)|\)
• parallel vectors \(v \times w = 0\), orthogonal vectors: \(v \cdot w = 0\)
• scalar projection \(\operatorname{comp}_w(v) = v \cdot w / |w|\)
• vector projection \(\operatorname{proj}_w(v) = (v \cdot w)w / |w|^2\)
• completion of square: \(x^2 - 4x + y^2 = 1\) gives \((x - 2)^2 + y^2 = 5\)
• unit vector \(=\) direction: vector of length \(1\)

14.1.11 Lines, Planes, Functions

• parametric equation for plane \(r(t, s)=p+t v+s w\) containing \(p\)
• plane \(A^{T}[x, y, z]=a x+b y+c z=d\)
• parametric equation for line \(r(t)=p+t v\) containing \(p\)
• graph \(G=\{(x, y, f(x, y)) \mid(x, y)\) in the domain of \(f\}\)
• plane \(a x+b y+c z=d\) has normal \(n=[a, b, c]^{T}\)
• line \(\frac{x-x_{0}}{a}=\frac{y-y_{0}}{b}=\frac{z-z_{0}}{c}\) contains \(v=[a, b, c]^{T}\)
• plane through \(A\), \(B\), \(C\): find normal vector \((a, b, c)=A B \times C B\)

14.1.12 Level surfaces

• intercepts: intersections of a surface with coordinate axis
• traces: intersections of a surface with coordinate planes
• generalized traces: intersections with \(\{x=c\}\), \(\{y=c\}\) or \(\{z=c\}\)
• level surface \(g(x, y, z)=c\): Example: graph \(g(x, y, z)=z-f(x, y)\)
• linear equation like \(2 x+3 y+5 z=7\) defines plane
• quadric: ellipsoid, paraboloid, hyperboloid, cylinder, cone

14.1.13 Distance formulas

• distance \(d(P, Q)=|P Q|=\sqrt{(P_{1}-Q_{1})^{2}+(P_{2}-Q_{2})^{2}+(P_{3}-Q_{3})^{2}}\)
• distance point-plane: \(d(P, \Sigma)=|(P Q) \cdot n| /|n|\)
• distance point-line: \(d(P, L)=|(P Q) \times u| /|u|\)
• distance line-line: \(d(L, M)=|(P Q) \cdot(u \times v)| /|u \times v|\)
• distance parallel lines \(L\), \(M\) distance point \(d(P, M)\) where \(P\) is in \(L\).
• distance parallel planes: \(d(P, \Sigma)\) where \(P\) is in first plane.

14.1.14 Functions

• graph: \(z=f(x, y)\)
• contour curve: \(f(x, y)=c\) is a curve in the plane
• contour map: draw curves \(f(x, y)=c\) for various \(c\)
• contour surface: \(f(x, y, z)=c\) in space

14.1.15 Curves

• plane and space curves \(r(t)\)
• circle: \(x^{2}+y^{2}=r^{2}\), \(r(t)=[r \cos t, r \sin t]^{T}\)
• ellipse: \((x-x_{0})^{2} / a^{2}+(y-y_{0})^{2} / b^{2}=1\), \(r(t)=[x_{0}+a \cos t, y_{0}+b \sin t]^{T}\)
• velocity \(r^{\prime}(t)\), acceleration \(r^{\prime \prime}(t)\), \(|r^{\prime}(t)|\) speed
• unit tangent vector \(T(t)=r^{\prime}(t) /|r^{\prime}(t)|\)
• integration: get \(r(t)\) from \(r^{\prime}(t)\) and \(r(0)\) by integration
• integration: get \(r(t)\) from acceleration \(r^{\prime \prime}(t)\) as well as \(r^{\prime}(0)\) and \(r(0)\)
• \(r^{\prime}(t)\) is tangent to the curve at the point \(r(t)\)
• \(r(t)=[f(t) \cos (t), f(t) \sin (t)]^{T}\) polar curve to polar graph \(r=f(\theta)\)
• \(\int_{a}^{b}|r^{\prime}(t)| \,d t\), arc length of parametrized curve
• \(N(t)=T^{\prime}(t) /|T^{\prime}(t)|\) normal vector, is perpendicular to \(T(t)\)
• \(B(t)=T(t) \times N(t)\) bi-normal vector, is perpendicular to \(T\) and \(N\)
• \(\kappa(t)=|T^{\prime}(t)| /|r^{\prime}(t)|\) curvature \(=|r^{\prime}(t) \times r^{\prime \prime}(t)| /|r^{\prime}(t)|^{3}\)
• \(\kappa(t)\) and arc length are independent of parametrization

14.1.16 Coordinates

• Cartesian coordinates \((x, y, z)\)
• polar coordinates \((x, y)=(r \cos (\theta), r \sin (\theta))\), \(r \geq 0\)
• cylindrical coordinates \((x, y, z)=(r \cos (\theta), r \sin (\theta), z)\), \(r \geq 0\)
• spherical coordinates \((x, y, z)=\big(\rho \cos (\theta) \sin (\phi), \rho \sin (\theta) \sin (\phi), \rho \cos (\phi)\big)\)
• radius: \(r=x^{2}+y^{2}\) and spherical radius \(\rho=x^{2}+y^{2}+z^{2}\)
• radius: important relation \(r=\rho \sin (\phi)\)
• Jacobian matrix
• Distortion factor

14.1.17 Surfaces

• \(g(r, \theta)=0\) polar curve, especially \(r=f(\theta)\), polar graphs
• \(r=f(z, \theta)\) cylindrical surface, \(r=r(z)\) surface of revolution
• \(g(\rho, \theta, \phi)=0\) spherical surface: example \(\rho=1\) sphere
• \(f(x, y)=c\) level curves of \(f(x, y)\)
• plane: \(a x+b y+c z=d\), \(r(s, t)=r_{0}+s v+t w\), \([a, b, c]^{T}=v \times w\)
• surface of revolution: \(x^{2}+y^{2}=r(z)^{2}\), \(r(\theta, z)=[r(z) \cos (\theta), r(z) \sin (\theta), z]^{T}\)
• graph: \(g(x, y, z)=z-f(x, y)=0\), \(r(x, y)=[x, y, f(x, y)]^{T}\)
• rotated graph \(g(x, y, z)=y-f(x, z)=0\), \(r(x, z)=[x, f(x, z), z]^{T}\)
• ellipsoid: \(r(\theta, \phi)=[a \cos \theta \sin \phi, b \sin \theta \sin \phi, c \cos \phi]^{T}\)
• unit sphere: \(x^2 + y^2 + z^2 = 1\), \(r(u, v) = [\cos{u}\sin{v}, \sin{u}\sin{v}, \cos{v}]^T\)

14.2 First Hourly (Practice A)

Problem 14A.1 (10 points):

The Fibonacci numbers are defined recursively as follows: start with \(F_{0}=0\), \(F_{1}=1\) then define \(F_{n+1}=F_{n}+F_{n-1}\), so that \(F_{2}=1\), \(F_{3}=2\), \(F_{4}= 3\), \(F_{5}=5\) etc. Prove that \[F_{0}+F_{1}+\cdots+F_{n}=F_{n+2}-1\] for every positive integer \(n\).

Problem 14A.2 (10 points):

Let \[A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right], \quad B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 1 \end{array}\right].\]

1. (4 points) Compute \(A B\) and \(\operatorname{rref}(A B)\).
2. (4 points) Now row reduce both \(A\) and \(B\) and form \(\operatorname{rref}(A) \operatorname{rref}(B)\).
3. (2 points) Is the statement \(\operatorname{rref}(A B)=\operatorname{rref}(A) \operatorname{rref}(B)\) true for all \(A\), \(B\)?

Problem 14A.3 (10 points):

1. (2 points) Parametrize the line through \((1,1,1)\) and \((4,3,1)\) in \(\mathbb{R}^{3}\).
2. (2 points) Parametrize the ellipse \(x^{2} / 16+y^{2} / 25=1\) in \(\mathbb{R}^{2}\).
3. (2 points) Parametrize the graph \(y=x^{5}+x\) in \(\mathbb{R}^{2}\).
4. (2 points) Parametrize the circle \(x^{2}+(y-2)^{2}=1\), \(z=4\) in \(\mathbb{R}^{3}\).
5. (2 points) Parametrize the line \(x=y=z\) in \(\mathbb{R}^{3}\).

Problem 14A.4 (10 points):

Find the arc length of the curve \[r(t)=\left[t \cos (t^{2}), t \sin (t^{2}), t^{2}\right]\] for \(0 \leq t \leq 2\).

Problem 14A.5 (10 points):

1. (2 points) What is the Heine-Cantor theorem?
2. (2 points) Formulate the triangle inequality.
3. (2 points) What is the Al Kashi identity?
4. (2 points) Give the name of a nowhere differentiable function.
5. (2 points) Is it true that a continuous curve \(r(t)\) has a finite arc length?

Problem 14A.6 (10 points):

1. (2 points) Find \((3+i)(4+2i)\).
2. (2 points) What is \(e^{i 3 \pi / 4}\)?
3. (2 points) Convert from cylindrical \((r, \theta, z)=(2, \pi / 2,1)\) to Cartesian.
4. (2 points) What are the spherical coordinates of \((1, \sqrt{3}, 2)\)?
5. (2 points) What surface is in spherical coordinates given as \(\rho \sin (\phi)=1\)?

Problem 14A.7 (10 points):

1. (5 points) You are given \(r^{\prime \prime \prime}(t) = (3, 4, 5)\) and \(r(0)=(7,8,9)\) and \(r^{\prime}(0)=(1,0,0)\) and \(r^{\prime \prime}(0)=(0,1,0)\). Find \(r(1)\).
2. (5 points) What is the curvature of \(r(t)=[t, t+t^{2}, t+t^{2}+t^{3}]\) at \(t=0\)?

Problem 14A.8 (10 points):

1. (5 points) Find a parametrization \(r(u, v)\) of the cylinder \(x^{2}+z^{2}=9\).
2. (5 points) Find \(r(u, v)\) for the paraboloid \(y^{2}+3 z^{2}=x\).

Problem 14A.9 (10 points):

Let \[A=\left[\begin{array}{ll}1 & 1 \\ 2 & 1 \\ 1 & 1\end{array}\right].\]

1. (2 points) The image of \(A\) is a plane. By using the cross product, write it as \(a x+b y+c z=d\).
2. (2 points) What is the first fundamental form \(g=A^{T} A\)?
3. (2 points) From a) you have \([a, b, c]^{T}=v \times w\). Find \(\sqrt{a^{2}+b^{2}+c^{2}}\).
4. (2 points) Find the distortion factor \(\|A\|=\sqrt{\operatorname{det}(A^{T} A)}\) of \(A\).
5. (2 points) What theorem was involved to see \(\|A\|=|v \times w|\)?

Problem 14A.10 (10 points):

1. (5 points) What is the Jacobian matrix \(d f\) of the map \[f(x, y, z)=\left[x^{2}+y^{2}+z^{2}, x+y,-x^{2}\right]^{T}?\]
2. (5 points) Find the distortion factor \(\operatorname{det}(d f)\).

14.3 First Hourly (Practice B)

Problem 14B.1 (10 points):

Prove that \(1+2+4+8+\cdots+2^{n}=2^{n+1}-1\) for every positive integer \(n\).

Problem 14B.2 (10 points):

1. (5 points) Row reduce the matrix \[A=\left[\begin{array}{llll}1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 5\end{array}\right].\]
2. (5 points) Compute the matrix product \[\left[\begin{array}{lll}3 & 4 & 5\end{array}\right] A\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1\end{array}\right].\]

Problem 14B.3 (10 points):

1. (2 points) Parametrize the curve \(x=\sin (y)\) in \(\mathbb{R}^{2}\).
2. (2 points) Parametrize the curve \(r=\sin ^{2}(5 \theta)\) in \(\mathbb{R}^{2}\).
3. (2 points) Parametrize the curve \(y=x^{5}+x\), \(z=4\) in \(\mathbb{R}^{3}\).
4. (2 points) Parametrize the line \(2 x+y=4\) in \(\mathbb{R}^{2}\).
5. (2 points) Parametrize the ellipse \((x-1)^{2}+y^{2}/4=1\) in \(\mathbb{R}^{2}\).

Problem 14B.4 (10 points):

Find the arc length of the curve \[r(t)=\left[\begin{array}{c} e^{t} \\ e^{-t} \\ \sqrt{2} t \end{array}\right]\] for \(0 \leq t \leq 1\).

Problem 14B.5 (10 points):

1. (2 points) Formulate the Cauchy-Schwarz inequality.
2. (2 points) What formula gives the area of the parallelogram spanned by two vectors \(v\) and \(w\)?
3. (2 points) What formula gives the volume of a parallelepiped spanned by three vectors \(u\), \(v\), \(w\)?
4. (2 points) Who invented the quaternions?
5. (2 points) Assume \(\operatorname{rref}(A)=\operatorname{rref}(B)\). Does this mean \(A=B\)?

Problem 14B.6 (10 points):

1. (2 points) Write the complex number \(z=e^{-i \pi / 2}\) in the form \(z=a+i b\).
2. (2 points) Which point \((x, y, z)\) has the cylindrical coordinates \((r, \theta, z)=(1, \pi / 2,0)\)?
3. (2 points) What are the spherical coordinates \((\rho, \phi, \theta)\) of the point \((x, y, z)=(\sqrt{2}, \sqrt{2},-2)\)?
4. (2 points) What surface is \(\rho \sin ^{2}(\phi)=\cos (\phi)\)? Give the name and write it in Cartesian coordinates.
5. (2 points) What surface is given in cylindrical coordinates by the equation \(r \sin (\theta)=2\)?

Problem 14B.7 (10 points):

1. (5 points) You are given \[r^{\prime \prime}(t)=\left[\begin{array}{l}0 \\ 3 \\ t\end{array}\right], \quad r(0)=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right], \quad r^{\prime}(0)=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right].\] Find \(r(1)\).
2. (5 points) What is the curvature of \[r(t)=\left[\begin{array}{c}\cos (t) \\ \sin (t) \\ t\end{array}\right]\] at \(t=0\)?

Problem 14B.8 (10 points):

1. (2 points) Find a parametrization of the cone \(x^{2}+y^{2}=z^{2}\).
2. (2 points) Find a parametrization of \(x^{2} / 4+y^{2} / 9+z^{2} / 16=1\).
3. (2 points) Find a parametrization of the surface \(x^{2}-y^{2}=z\).
4. (2 points) Find a parametrization of the plane \(z=2\).
5. (2 points) Find a parametrization of the cylinder \(x^{2}+z^{2}=1\).

Problem 14B.9 (10 points):

1. (5 points) Find the dot product \(A \cdot B=\operatorname{tr}(A^{T} B)\) between the two matrices \[\begin{aligned} & A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{array}\right], \\ & B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{array}\right]. \end{aligned}\]
2. (5 points) Find the cosine of the angle between these two matrices.

Problem 14B.10 (10 points):

1. (5 points) What is the Jacobian matrix \(d f\) of the coordinate change \[f\left(\left[\begin{array}{l} x \\ y \end{array}\right]\right)=\left[\begin{array}{c} 2 x-y+\sin (x) \\ x \end{array}\right].\]
2. (5 points) What is the distortion factor \(\operatorname{det}(d f)\) of the map \(f\) which by the way is called the Chirikov map.

14.4 First Hourly

Problem 14.1 (10 points):

Prove by induction that that for every \(n \geq 1\) the formula \(2 \sum_{k=0}^{n-1} 3^{k}=\) \(3^{n}-1\) holds.

Problem 14.2 (10 points):

1. (5 points) Row reduce the matrix \[A=\left[\begin{array}{lllll}0 & 1 & 1 & 1 & 1 \\ 4 & 4 & 4 & 4 & 4 \\ 2 & 2 & 2 & 2 & 2\end{array}\right]\] using basic row reduction steps.
2. (5 points) For \(B=\left[\begin{array}{lll}1 & 1 & 1\end{array}\right]\) compute either \(A B\) or \(B A\) depending on which of the two makes sense.

Problem 14.3 (10 points):

1. (2 points) Parametrize the curve \(4 x^{2}+y^{2}=1\) in \(\mathbb{R}^{2}\).
2. (2 points) Parametrize the curve \(y-e^{x}=0\) in \(\mathbb{R}^{2}\).
3. (2 points) Parametrize the curve \(x=y^{3}\), \(z=4\) in \(\mathbb{R}^{3}\).
4. (2 points) Parametrize the line \(x+y=4\), \(z=2\) in \(\mathbb{R}^{3}\).
5. (2 points) Parametrize the circle \(x^{2}+y^{2}+z^{2}=4\), \(z=1\) in \(\mathbb{R}^{3}\).

Problem 14.4 (10 points):

1. (8 points) Compute arc length of \(r(t)=\left[\frac{t^{3}}{3}, \sqrt{2} \frac{t^{4}}{4}, \frac{t^{5}}{5}\right]\) for \(0 \leq t \leq 1\).
2. (2 points) Without doing any calculation, what is the arc length of the new parametrization \(r(t^{3})\) with \(0 \leq t \leq 1\)?

Problem 14.5 (10 points):

1. (2 points) Formulate the \(\mathrm{Al}\) Khashi formula.
2. (2 points) We have seen a theorem of Heine-\(\ldots\ldots\). Fill in the second name!
3. (2 points) The linear space \(\{x \mid A x=0\}\) is also called the \(\ldots\ldots\) of \(A\).
4. (2 points) Give the Euler’s formula \(e^{i t}=\ldots\ldots\) and deduce the "most beautiful formula in math".
5. (2 points) Is \[\operatorname{rref}(A)=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]\] row reduced?

Problem 14.6 (10 points):

1. (2 points) Express \(z=e^{i \pi / 2}+3 e^{i \pi}\) in the form \(z=a+i b\).
2. (2 points) Write \((r, \theta, z)=(2,-\pi / 2,0)\) in Cartesian coordinates.
3. (2 points) Write \((x, y, z)=(2,2,0)\) in spherical coordinates \((\rho, \phi, \theta)\).
4. (2 points) Write the surface \(\rho \cos (\phi)=2\) in Cartesian coordinates.
5. (2 points) Write the surface \(r \cos (\theta)=2\) in Cartesian coordinates.

Problem 14.7 (10 points):

1. (5 points) You are given \[r^{\prime \prime}(t)=\left[\begin{array}{c}0 \\ 1 \\ \cos (t)\end{array}\right], \quad r(0)=\left[\begin{array}{l}2 \\ 3 \\ 4\end{array}\right], \quad r^{\prime}(0)=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right].\] Find \(r(1)\).
2. (2 points) Is there a time \(t\) such that the curve \(r(t)\) ever reaches the ground \(z=0\)?
3. (3 points) What is the curvature of \[r(t)=\left[\begin{array}{c}t^{2} \\ \cos (t) \\ \sin (t)\end{array}\right]\] at \(t=0\)?

Problem 14.8 (10 points):

We parametrize some surfaces. Choose the parameters on your own.

1. (2 points) Find a parametrization of the hyperboloid \(x^{2}+y^{2}-z^{2}=1\).
2. (2 points) Find a parametrization of the cylinder \((x-1)^{2} / 4+y^{2} / 9=1\).
3. (2 points) Find a parametrization of the surface \(z=\cos (x y)\).
4. (2 points) Find a parametrization of the plane \(x+y-3 z=1\).
5. (2 points) Find a parametrization of the cylinder \(x^{2} / 9+(y-2)^{2}=1\).

Problem 14.9 (10 points):

1. (4 points) Compute the dot product (inner product) \(A \cdot B=\operatorname{tr}(A^{T} B)\) of the two matrices \[A=\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right], \quad B=\left[\begin{array}{lll} 0 & 1 & 2 \\ 2 & 3 & 3 \end{array}\right]\]
2. (4 points) Now determine the cosine of the angle between \(A\) and \(B\).
3. (2 points) Finally find the distance \(|A-B|\) between \(A\) and \(B\).

Problem 14.10 (10 points):

1. (4 points) What is the Jacobian matrix \(d r\) of the coordinate change \[r\left(\left[\begin{array}{l} x \\ y \end{array}\right]\right)=\left[\begin{array}{c} 4 x+y \\ y^{2} \end{array}\right]?\]
2. (2 points) Now find the first fundamental form \(g=d r^{T} d r\).
3. (2 points) Compute the distortion factor \(|\operatorname{det}(d r)|\).
4. (2 points) Check in this case that \(|\operatorname{det}(d r)|=\sqrt{\operatorname{det}(g)}\).