Consider the orthogonal coordinate system x = uv cosθ, y = uv
sinθ, z = (u2...
Consider the orthogonal coordinate system x = uv cosθ, y = uv
sinθ, z = (u2 −v2)/2, with u,v ≥ 0 and 0 ≤ θ < 2π.
Express the volume element dV in terms of u, v, θ.
3. Suppose we have a coordinate system (x, y, z) with the origin
at one corner of a cube, and the axes parallel to the edges of the
cube. We want to perform a rotation to a coordinate system (x' , y'
, z' ), where the x 0 axis is along the diagonal of the cube, and
the y' axis remains in the original x ? y plane.
(a) (0.5 points) Using the Z-Y’-Z” Euler angle convention that
is...
Consider a two-dimensional triangular lattice described by the
two primitive vectors (in an orthogonal coordinate system
Find the two primitive lattice vectors describing the reciprocal
lattice. Find the area of the 1st
Brillouin zone and its relation with
the area of the direct lattice unit cell.
A charge of -3.00 nC is placed at the origin of an x-y
coordinate system, and a charge of 2.00 nC is placed on the y-axis
at y = 4.00 cmb.
If a third charge of 5.00 nC is now placed at the point x = 3.00
cm, y = 4.00 cm, what are the x and y components of the total force
exerted on the charge by the other two charges?c.e.
What are the magnitude and direction of the...
A charge of -3.30 nC is placed at the origin of an x
y-coordinate system, and a charge of 1.55 nC is placed on the
y axis at y = 4.30 cm .
A. If a third charge, of 5.00 nC , is now placed at the point x
= 3.30 cm , y = 4.30 cm find the x and y components of the total
force exerted on this charge by the other two charges.
B. Find the magnitude...
Consider a region R bound by the coordinate axes and y = ( 9 + x
2 ) − 1 2 on 0 ≤ x ≤ 4.
a. Find the area of R.
b. Suppose R is revolved about the x-axis to form a solid. Find
the volume of the solid.
c. Suppose R is revolved about the y-axis to form a solid. Find
the volume of the solid.
Consider the following function:
f (x , y , z ) = x 2 + y 2 + z 2 − x y − y z + x + z
(a) This function has one critical point. Find it.
(b) Compute the Hessian of f , and use it to determine whether
the critical point is a local man, local min, or neither?
(c) Is the critical point a global max, global min, or neither?
Justify your answer.
A puck is moving on an air hockey table. Relative to an
x, y coordinate system at time t = 0 s,
the x components of the puck's initial velocity and
acceleration are v0x = +2.2 m/s
anda. The y components of the puck's initial
velocity and acceleration are and . Find (a) the
magnitude v and (b) the direction
? of the puck's velocity at a time of . Specify the
direction relative to the +x axis.x =
+7.7...
The curried version of let f (x,y,z) = (x,(y,z)) is
let f (x,(y,z)) = (x,(y,z))
Just f (because f is already curried)
let f x y z = (x,(y,z))
let f x y z = x (y z)
Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )