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 in general for
• Pythagoras for any inner product space
• Al Khashi for any triangle
• Thiqueness of row reduction: is unique in
• The dot product formula
• The cross product formula
• Image of transpose is kernel
• Cauchy-Binet formula
• Arc length for differentiable
• Curvature formulas
• Euler formula and special case
• Distortion formula for

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
• 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
• Column and row vectors
• Parametrized curves
• Parametrized surfaces
• Functions
• Level surfaces
• Linear manifolds
• Quadratic manifolds
• Kernel of a linear map
• Image of a linear map

14.1.5 Differentiation

• Velocity
• Acceleration
• Jerk
• Free fall: given
• TNB frame, , ,
• derivative of a map
• Jacobian matrix of a map
• Distortion factor
• Distortion factor for simplifies to
• Example: , is speed
• Curvature , In also

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

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

• , ,
• dot product
• angle
• cross product ,
• area parallelogram
• triple scalar product
• volume of parallelepiped:
• parallel vectors , orthogonal vectors:
• scalar projection
• vector projection
• completion of square: gives
• unit vector direction: vector of length

14.1.11 Lines, Planes, Functions

• parametric equation for plane containing
• plane
• parametric equation for line containing
• graph in the domain of
• plane has normal
• line contains
• plane through , , : find normal vector

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 , or
• level surface : Example: graph
• linear equation like defines plane
• quadric: ellipsoid, paraboloid, hyperboloid, cylinder, cone

14.1.13 Distance formulas

• distance
• distance point-plane:
• distance point-line:
• distance line-line:
• distance parallel lines , distance point where is in .
• distance parallel planes: where is in first plane.

14.1.14 Functions

• graph:
• contour curve: is a curve in the plane
• contour map: draw curves for various
• contour surface: in space

14.1.15 Curves

• plane and space curves
• circle: ,
• ellipse: ,
• velocity , acceleration , speed
• unit tangent vector
• integration: get from and by integration
• integration: get from acceleration as well as and
• is tangent to the curve at the point
• polar curve to polar graph
• , arc length of parametrized curve
• normal vector, is perpendicular to
• bi-normal vector, is perpendicular to and
• curvature
• and arc length are independent of parametrization

14.1.16 Coordinates

• Cartesian coordinates
• polar coordinates ,
• cylindrical coordinates ,
• spherical coordinates
• radius: and spherical radius
• radius: important relation
• Jacobian matrix
• Distortion factor

14.1.17 Surfaces

• polar curve, especially , polar graphs
• cylindrical surface, surface of revolution
• spherical surface: example sphere
• level curves of
• plane: , ,
• surface of revolution: ,
• graph: ,
• rotated graph ,
• ellipsoid:
• unit sphere: ,

14.2 First Hourly (Practice A)

Problem 14A.1 (10 points):

The Fibonacci numbers are defined recursively as follows: start with , then define , so that , , , etc. Prove that for every positive integer .

Problem 14A.2 (10 points):

Let

1. (4 points) Compute and .
2. (4 points) Now row reduce both and and form .
3. (2 points) Is the statement true for all , ?

Problem 14A.3 (10 points):

1. (2 points) Parametrize the line through and in .
2. (2 points) Parametrize the ellipse in .
3. (2 points) Parametrize the graph in .
4. (2 points) Parametrize the circle , in .
5. (2 points) Parametrize the line in .

Problem 14A.4 (10 points):

Find the arc length of the curve for .

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 has a finite arc length?

Problem 14A.6 (10 points):

1. (2 points) Find .
2. (2 points) What is ?
3. (2 points) Convert from cylindrical to Cartesian.
4. (2 points) What are the spherical coordinates of ?
5. (2 points) What surface is in spherical coordinates given as ?

Problem 14A.7 (10 points):

1. (5 points) You are given and and and . Find .
2. (5 points) What is the curvature of at ?

Problem 14A.8 (10 points):

1. (5 points) Find a parametrization of the cylinder .
2. (5 points) Find for the paraboloid .

Problem 14A.9 (10 points):

Let

1. (2 points) The image of is a plane. By using the cross product, write it as .
2. (2 points) What is the first fundamental form ?
3. (2 points) From a) you have . Find .
4. (2 points) Find the distortion factor of .
5. (2 points) What theorem was involved to see ?

Problem 14A.10 (10 points):

1. (5 points) What is the Jacobian matrix of the map
2. (5 points) Find the distortion factor .

14.3 First Hourly (Practice B)

Problem 14B.1 (10 points):

Prove that for every positive integer .

Problem 14B.2 (10 points):

1. (5 points) Row reduce the matrix
2. (5 points) Compute the matrix product

Problem 14B.3 (10 points):

1. (2 points) Parametrize the curve in .
2. (2 points) Parametrize the curve in .
3. (2 points) Parametrize the curve , in .
4. (2 points) Parametrize the line in .
5. (2 points) Parametrize the ellipse in .

Problem 14B.4 (10 points):

Find the arc length of the curve for .

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 and ?
3. (2 points) What formula gives the volume of a parallelepiped spanned by three vectors , , ?
4. (2 points) Who invented the quaternions?
5. (2 points) Assume . Does this mean ?

Problem 14B.6 (10 points):

1. (2 points) Write the complex number in the form .
2. (2 points) Which point has the cylindrical coordinates ?
3. (2 points) What are the spherical coordinates of the point ?
4. (2 points) What surface is ? Give the name and write it in Cartesian coordinates.
5. (2 points) What surface is given in cylindrical coordinates by the equation ?

Problem 14B.7 (10 points):

1. (5 points) You are given Find .
2. (5 points) What is the curvature of at ?

Problem 14B.8 (10 points):

1. (2 points) Find a parametrization of the cone .
2. (2 points) Find a parametrization of .
3. (2 points) Find a parametrization of the surface .
4. (2 points) Find a parametrization of the plane .
5. (2 points) Find a parametrization of the cylinder .

Problem 14B.9 (10 points):

1. (5 points) Find the dot product between the two matrices
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 of the coordinate change
2. (5 points) What is the distortion factor of the map 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 the formula holds.

Problem 14.2 (10 points):

1. (5 points) Row reduce the matrix using basic row reduction steps.
2. (5 points) For compute either or depending on which of the two makes sense.

Problem 14.3 (10 points):

1. (2 points) Parametrize the curve in .
2. (2 points) Parametrize the curve in .
3. (2 points) Parametrize the curve , in .
4. (2 points) Parametrize the line , in .
5. (2 points) Parametrize the circle , in .

Problem 14.4 (10 points):

1. (8 points) Compute arc length of for .
2. (2 points) Without doing any calculation, what is the arc length of the new parametrization with ?

Problem 14.5 (10 points):

1. (2 points) Formulate the Khashi formula.
2. (2 points) We have seen a theorem of Heine-. Fill in the second name!
3. (2 points) The linear space is also called the of .
4. (2 points) Give the Euler’s formula and deduce the "most beautiful formula in math".
5. (2 points) Is row reduced?