Question

For
map g:(x,y) --> ( e^x cos y, e^x sin y), find the image of g.
Write a proof that this is the image of g.

Then, prove that g is not bijective. make sure all of your
proofs are complete.

Answer #1

(a) Is the vector field F = <e^(−x) cos y, e^(−x) sin y>
conservative?
(b) If so, find the associated potential function φ.
(c) Evaluate Integral C F*dr, where C is the straight line path
from (0, 0) to (2π, 2π).
(d) Write the expression for the line integral as a single
integral without using the fundamental theorem of calculus.

Refer to the graph of y = sin x or y
= cos x to find the exact values of x in the
interval [0, 4π] that satisfy the equation. (Enter your answers as
a comma-separated list.)
3 sin x = −3

Using the Taylor series for e^x, sin(x), and cos(x), prove that
e^ix = cos(x) + i sin(x) (Hint: plug in ix into the Taylor series
expansion for e^x . Then separate out the terms which have i in
them and the terms which do not.)

y = (6 +cos(x))^x
Use Logarithmic Differentiation to find dy/dx
dy/dx =
Type sin(x) for sin(x)sin(x) ,
cos(x) for cos(x)cos(x), and so on.
Use x^2 to square x, x^3 to cube
x, and so on.
Use ( sin(x) )^2 to square sin(x).
Use ln( ) for the natural logarithm.

Find x and y
cos x + i sin x = cosh(y -1) + ixy
Hint: Sketch the cosine and cosh functions

Consider the linear system x' = x cos a − y sin a
y'= x sin a + y cos a
where a is a parameter. Show that as a ranges over [0, π], the
equilibrium point at the origin passes through the sequence stable
node, stable spiral, center, unstable spiral, unstable node.

Let F ( x , y , z ) =< e^z sin( y ) + 3x , e^x cos( z ) + 4y
, cos( x y ) + 5z >, and let S1 be the sphere x^2 + y^2 + z^2 =
4 oriented outwards Find the flux integral ∬ S1 (F) * dS. You may
with to use the Divergence Theorem.

8.4: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is surjective then g is surjective.
8.5: Let f : X → Y and g : Y→ Z be bijections. Prove that if
composition g o f is bijective then f is bijective.
8.6: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is bijective then f is...

Use the Mean Value Theorem and the fact that for f(x) = cos(x),
f′(x) = −sin(x), to prove that, for x, y ∈ R,
| cos x − cos y| ≤ |x − y|.

2. Is the vector field F = < z cos(y), −xz sin(y), x
cos(y)> conservative? Why or why not? If F is conservative, then
find its potential function.

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