Let⇀F and⇀G be vector fields defined on R3 whose component
functions have continuous partial derivatives. Furthermore,...
Let⇀F and⇀G be vector fields defined on R3 whose component
functions have continuous partial derivatives. Furthermore, assume
that ⇀∇×⇀F=⇀∇×⇀G.Show that there is a scalar function f such that
⇀G=⇀F+⇀∇f.
Let f and g be two functions whose first and second order
derivative functions are continuous, all defined on R. What
assumptions on f and g guarantee that the composite function f ◦g
is concave?
Let F and G~be two vector fields in R2 . Prove that
if F~ and G~ are both conservative, then F~ +G~ is also
conservative. Note: Give a mathematical proof, not just an
example.
Let f : N → N and g : N → N be the functions defined as ∀k ∈ N
f(k) = 2k and g(k) = (k/2 if k is even, (k + 1) /2 if k is
odd).
(1) Are the functions f and g injective? surjective? bijective?
Justify your answers.
(2) Give the expressions of the functions g ◦ f and f ◦ g?
(3) Are the functions g ◦ f and f ◦ g injective? surjective?
bijective?...
Let C(R) be the vector space of continuous functions from R to R
with the usual addition and scalar multiplication. Determine if W
is a subspace of C(R). Show algebraically and explain your answers
thoroughly.
a. W = C^n(R) = { f ∈ C(R) | f has a continuous nth
derivative}
b. W = {f ∈ C^2(R) | f''(x) + f(x) = 0}
c. W = {f ∈ C(R) | f(-x) = f(x)}.
Let V be the vector space of all functions f : R → R. Consider
the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The
function T : W → W given by taking the derivative is a linear
transformation
a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the
matrix for T relative to B.
b)Find all the eigenvalues of the matrix you found in the
previous part and describe their eigenvectors. (One...
Let p, q, g : R → R be continuous functions.
Let L[y] := y'' + py' + qy.
(i) Explain what it means for a pair of functions y1 and y2 to
be a fundamental solution set for the equation L[y] = 0.
(ii) State a theorem detailing the general solution of the
differential equation L[y] = g(t) in terms of solutions to this,
and a related, equation.
A polymodal brain region is an association are whose neurons
have
receptive fields defined by MORE THAN ONE sensory modality (e.g.,
vision,
somatosensations or vision/hearing, etc.) Discuss the adaptive
significance of the
expansion of the polymodal regions PG and STS in humans compared to
other
primates.
In this question we study the recursively defined functions f, g
and h given by the following defining equations
f(0) = −1 base case 0,
f(1) = 0 base case 1, and
f(n) = n · f(n − 1) + f(n − 2)^2 recursive case for n ≥ 2.
and
g(0, m, r, k) = m base case 0, and
g(n, m, r, k) = g(n − 1, r,(k + 2)r + m^2 , k + 1) recursive
case for...