Question

1) Convert the point (x,y,z)=(−2,5,3) to spherical coordinates.
Give answers as positive values, either as expressions, or decimals
to one decimal place.

(ρ,θ,ϕ)=

2) Convert the point (r,θ,z)=(2,2π,4) to Cartesian coordinates.
Give answers either as expressions, or decimals to at least one
decimal place.

(x,y,z)=

3) Convert the point (ρ,θ,ϕ)= (5,5π/3,3π/4) to Cartesian
coordinates. Give answers either as expressions, or decimals to at
least one decimal place.

(x,y,z) =

Answer #1

Give a formula for the area element in the plane in rectangular
coordinates x and y. (Answer:
dx dy, or more properly |dx ∧
dy|; either is acceptable, as are dy dx
and |dy ∧ dx|.)
Give a formula for the area element in the plane in polar
coordinates r and θ.
Give a formula for the volume element in space in rectangular
coordinates x, y, and z. (Answer:
dx dy dz, or more properly |dx
∧ dy ∧ dz|;...

Find the rectangular coordinates of the point whose spherical
coordinates are given. (a) (1, 0, 0) (x, y, z) = (b) (14, π/3, π/4)
(x, y, z) =

Write all spherical harmonics up to l = 2 (there are
nine of them) in Cartesian form,
i.e. give expressions in terms of x, y, z, and r. You can either
use the Rodrigues formula
for the Legendre polynomials or start with the given expressions
for Y
m
l
in terms of θ
and φ. In any event you must show your work.

- write N-S equations for incompressible flows in cartesian
coordinates in long form (in x,y,z coordinates) (1 point)
- also write continuity equation and N-S equations for
incompressible flows in polar coordinates in long form (in r,θ,z)
(since one of your friends ask you can find it easily in online
resources or in books) (1 point)
- Write down N-S equations in x direction for Planar Couette-
Pouseille flow and derive an equation for velocity variation using
boundary layer conditions...

Answers for Z-values should be 2
decimals, probabilities should be 3 decimals, X-values 1 decimal.
If σX = the standard deviation (SD) of X and
µX = the mean of X, then:
4.
a) Find Z1 such that
prob(.35 < Z < Z1) = .3231
b) Find prob(-1.2 < Z < 2.7)
c) Find prob(Z = .83)
d) Find any Z0 and Z1 such that
prob(Z0 < Z < Z1) = .4444

Use the Chain Rule to find ∂z/∂s and ∂z/∂t.
3. z=x/y, x=se^t, y=1+se^-t
4. z=e^rcos θ, r=st, θ=√s^2+t^2

Bordered Hessian element
The Lagrangian is L=ln(x+y^2) -z^3/(3*y) -x*y
+λ*(x*z +3*x^2*y -r), where r is a parameter (a known real number).
Here, ln denotes the natural logarithm, ^ power, * multiplication,
/ division, + addition, - subtraction. The border is at the top and
left of the Hessian. The variables are ordered λ,x,y,z.
Find the last element in the second row of the
bordered Hessian at the point (λ,x,y,z) =(0.11, 0, 2440, 0.01167).
This point need not be stationary and...

Given the following five pairs of (x, y)
values,
x
5
3
9
6
12
y
10
6
4
3
1
Determine the least squares regression line (give the values of
b0 and b1 (in that order),
separated by a comma
(numbers correct to 4 decimals))

The following observations are obtained from a random sample of
10 individuals: Individual x y 1 9.08 5.25 2 4.23 3.58 3 6.88 4.75
4 10.3 5.38 5 8.09 4.27 6 10.6 5.79 7 4.50 3.41 8 8.32 5.75 9 7.17
4.74 10 9.45 5.43 Run a t-linear regression test on this data.
(HINT: make sure you copy the numbers correctly!) What are the
appropriate null and alternative hypotheses? H0:r=0H1:r≠0
H0:ρ=0H1:ρ<0 H0:ρ=0H1:ρ≠0 H0:r=0H1:r>0 H0:r=0H1:r<0
H0:ρ=0H1:ρ>0 What is the correlation coefficient?...

Let X and Y be independent positive random variables. Let Z=X/Y.
In what follows, all occurrences of x, y, z are assumed to be
positive numbers.
1. Suppose that X and Y are discrete, with known PMFs, pX and
pY. Then, pZ|Y(z|y)=pX(?). What is the argument in the place of the
question mark?
2. Suppose that X and Y are continuous, with known PDFs, fX and
fY. Provide a formula, analogous to the one in part (a), for
fZ|Y(z|y) in...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 8 minutes ago

asked 9 minutes ago

asked 15 minutes ago

asked 17 minutes ago

asked 25 minutes ago

asked 25 minutes ago

asked 25 minutes ago

asked 44 minutes ago

asked 53 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 1 hour ago