Let f ( x , y ) = x^ 2 + y ^3 + sin ( x ^2 + y ^3 ). Determine
the line integral of f ( x , y ) with respect to arc length over
the unit circle centered at the origin (0, 0).
f(x,y)=sin(2x)sin(y)
intervals for x and y:
-π/2 ≤ x ≤ π/2 and -π ≤ y ≤ π
find extrema and saddle points
In the solution, I mainly interested how to
findcritical points in case of the system of trigonometric
equations (fx=0 and fy=0).
,
2 Let F be a field and let R = F[x, y] be the ring of
polynomials in two variables with coefficients in F.
(a) Prove that
ev(0,0) : F[x, y] → F
p(x, y) → p(0, 0)
is a surjective ring homomorphism.
(b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x,
y) + ys(x, y) | r,s ∈ F[x, y]}
(c) Use the first isomorphism theorem to prove that (x, y) ⊆
F[x, y]...
Let F be a field.
(a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) =
2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple.
That is, a^2 + b^2 = c^2. Use this fact to explain how to generate
right triangles with integer side lengths.
(b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy
− y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...
3. Let F : X → Y and G: Y → Z be functions.
i. If G ◦ F is injective, then F is injective.
ii. If G ◦ F is surjective, then G is surjective.
iii. If G ◦ F is constant, then F is constant or G is
constant.
iv. If F is constant or G is constant, then G ◦ F is
constant.
4. (Oblique Trajectory Problem) Let F(x, y) = x 2 + xy + y 2 .
Find a formula for G(x, y) such that every curve in the
one-parameter family defined by F(x, y) = c intersects every curve
in the one-parameter family defined by G(x, y) = c at a sixty
degree angle
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)