U = {q, r, s, t, u, v, w, x, y, z}
A = {q,...
U = {q, r, s, t, u, v, w, x, y, z}
A = {q, s, u, w, y}
B = {q, s, y, z}
C = {v, w, x, y, z}.
List the elements in A - B.
Suppose u(t,x) and v(t,x ) is C^2 functions defined on R^2 that
satisfy the first-order system...
Suppose u(t,x) and v(t,x ) is C^2 functions defined on R^2 that
satisfy the first-order system of PDE Ut=Vx,
Vt=Ux,
A.) Show that both U and V are classical solutions to the wave
equations Utt= Uxx.
Which result from multivariable calculus do you need to justify
the conclusion.
B)Given a classical sol. u(t,x) to the wave equation, can you
construct a function v(t,x) such that u(t,x), v(t,x)
form of sol. to the first order system.
Let f : R → R be defined by f(x) = x^3 + 3x, for all...
Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i)
Prove that if y > 0, then there is a solution x to the equation
f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove
that the function f : R → R is strictly monotone. (iii) By
(i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why
the derivative of the inverse function,...
Given signal x(t) = sinc(t):
1. Find out the Fourier transform of x(t), find X(f), sketch...
Given signal x(t) = sinc(t):
1. Find out the Fourier transform of x(t), find X(f), sketch
them.
2. Find out the Nyquist sampling frequency of x(t).
3. Given sampling rate fs, write down the expression of the
Fourier transform of xs(t), Xs(f) in terms of X(f).
4. Let sampling frequency fs = 1Hz.
Sketch the sampled signal xs(t) = x(kTs) and the Fourier
transform of xs(t), Xs(f).
5. Let sampling frequency fs = 2Hz. Repeat 4.
6. Let sampling frequency...
Consider the function f : R 2 → R defined by f(x, y) = 4 +...
Consider the function f : R 2 → R defined by f(x, y) = 4 + x 3 +
y 3 − 3xy.
(a)Compute the directional derivative of f at the point (a, b) =
( 1 2 , 1 2 ), in the direction u = ( √ 1 2 , − √ 1 2 ). At the
point ( 1 2 , 1 2 ), is u the direction of steepest ascent,
steepest descent, or neither? Justify your...
How many distinct invariant subspaces does the linear operator
T: R^3 --> R^3 defined by T(x,y,z)...
How many distinct invariant subspaces does the linear operator
T: R^3 --> R^3 defined by T(x,y,z) = (4z-y, x+2z, 3z) have?
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