Verify the Divergence Theorem for the vector field F(x, y, z) =
< y, x , z^2 > on the region E bounded by the planes y + z =
2, z = 0 and the cylinder x^2 + y^2 = 1.
By Surface Integral:
By Triple Integral:
Compute the line integral of the vector field F(x, y, z) = ⟨−y, x,
z⟩ along the curve which is given by the intersection of the
cylinder x 2 + y 2 = 4 and the plane x + y + z = 2 starting from
the point (2, 0, 0) and ending at the point (0, 2, 0) with the
counterclockwise orientation.
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)
Compute the derivative of the given vector field F. Evaluate the line integral of
F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )over the path C consisting of line segments joining (1,1,1) to (1,1,2), (1, 1, 2) to (1, 3, 2), and (1, 3, 2) to (4, 3, 2) in 3 different ways, along the given path, along the line from (1,1,1) to (4,3,2), and finally by finding an anti-derivative, f, for F.
Consider the scalar functions
f(x,y,z)g(x,y,z)=x^2+y^2+z^2,
g(x,y,z)=xy+xz+yz,
and=h(x,y,z)=√xyz
Which of the three vector fields ∇f∇f, ∇g∇g and ∇h∇h are
conservative?
5) Let the function f : ℝ3 → ℝ3 be given
by f(x, y, z) = (2x + 2y, 2y + 2z, z + x).
a) Prove that f is one to one and onto
b) Find the inverse of f, i.e., f−1.
3. a) Consider the vector field F(x, y, z) = (2xy2 z, 2x 2 yz,
x2 y 2 ) and the curve r(t) = (sin t,sin t cost, cost) on the
interval [ π 4 , 3π 4 ]. Calculate R C F · dr using the definition
of the line integral. [5] b) Find a function f : R 3 → R so that F
= ∇f. [5] c) Verify your answer from (a) using (b) and the
Fundamental...
TOPOLOGY
Let f : X → Y be a function.
Prove that f is one-to-one and onto if and only if f[A^c] =
(f[A])^c for every subset A of X. (prove both directions)
Let f: X→Y and g: Y→Z be both onto. Prove that g◦f is an onto
function
Let f: X→Y and g: Y→Z be both onto. Prove that f◦g is an onto
function
Let f: X→Y and g: Y→Z be both one to one. Prove that g◦f is an
one to one function
Let f: X→Y and g: Y→Z be both one to one. Prove that f◦g is an
one to one function