Differential Geometry (Mixed Use of Vector Calculus & Linear
Algebra)
1A. Prove that if p=(x,y) is in the set where y<x and if
r=distance from p to the line y=x then the ball about p of radius r
does not intersect with the line y=x.
1B. Prove that the set where y<c is an open set.
State, without proofs all theorems of
neutral geometry that permit to construct a midpoint of a given
segment.
State, without proofs all theorems of
Euclidean geometry that permit to execute your construction in the
Poincare the Model of the Hyperbolic geometry.
** Please answer the question without using Calculus.
This is an algebra based Physics course. Thank you! **
a) Over 2 millenniums ago, intellectuals in the ancient world
concluded the earth's radius. This determined value is about
RE = 6400 km.
More than 70 years after Isaac Newton formulated his law of
universal gravity, Cavendish made a direct measurement of the
gravitational constant. This value is about G = 6.7 x
10-11 Nm2/kg2.
Show how to calculate the mass of...
QUESTION: In each part, find a formula for a vector field
consistent with the description. Provide at least one numeric
example showing the consistency of the formula and the description
(an example follows the descriptions).
1.All vectors are parallel to the x-axis and all vectors on a
vertical line have the same magnitude.
2.All vectors point toward the origin and have constant
length.
3.All vectors are of unit length and are orthogonal to the
position vector at that point.
Linear Algebra
we know that x ∈ R^n is a nonzero vector and C is a real
number.
find all values of C such that ( In − Cxx^T ) is nonsingular and
find its inverse
knowing that its inverse is of the same form
Differential Geometry
3. Evaluate the 1-form f = x2 dx - y2 dz
on the vector fields V = xU1 + yU2 + zU3, W = xy (U1 - U3) + yz (U1
- U2), and (1/x)V + (1/y)W.
Use differential calculus to find the maximum/minimum for an
engineering problem.
A sheet of metal is 340 mm x 225 mm and has four equal squares
cut out at the corners so that the sides and edges can be turned up
to form an open topped box shape.
Calculate: (a) The lengths of the sides of the cut-out squares
for the volume of the box to be as big as possible.
(b) The maximum volume of the box.
Chapter 3.6, Problem 20E in Introduction to Linear Algebra (5th
Edition)
Find the basis for the null space and the range of the given
matrix. Then use Gram-Schmidt to obtain the orthagonal bases.
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