With regard to multi variable calculus can some explain Divergence Theorem and applications in detail. Thank you. Provide an example problem and solve too thanks!

With regard to multi variable calculus can some explain multiple integration in detail. Thank you. Provide an example problem and solve too thanks!

The demand equation for a computer desk is p = −4x + 290,  and the supply equation is p = 3x + 80.

(b) Find the equilibrium quantity x and price p. (Round your answers to one decimal place.)

(x, p) =

(c) Find the price at which the buyer stops buying.
$

(d) Find the price at which the supplier stops supplying.
$

(e) Is there a shortage or surplus when the price is $110? How much?

(f) Is there a shortage or surplus when the price is $206? How much?

A trail crew is constructing a 500-ft electric fence. Fence posts are placed every 5 feet. The location is 8 miles from the parking lot, so they must hike in everything to the site. They want to have enough posts but not too many since they will have to carry them back out, Leave No Trace. What is the minimum number of posts they will need to hike in? What other consideration might the crew want to consider? Additionally, between each post a plastic marker will be placed warning that the fence is electrified. What is the minimum number of these they should pack in?

Solve the given differential equation by using an appropriate substitution. The DE is homogeneous.

=

If p is prime, then Z^*_{p is a group under multiplication.}  

Either give an example of sequences (sn) and (tn) satisfying the properties or explain why such sequences do not exist.

(a) (sn) converges, (tn) diverges, (sn + tn) converges.

(b) (sn) converges, (tn) diverges, (sntn) converges.

(c) (sn) diverges, (tn) diverges, (sntn) converges.

(d) (sn) is bounded, (tn) converges, (sntn) diverges.

(e) (sn) converges, (tn) converges, tn 6= 0 for all n, ( sn tn ) diverges.

Unless otherwise noted, all sets in this module are finite. Prove the following statements...

1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B| and (b) |A × B| = |A||B|.

2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b) surjective then |A| ≥ |B|.

3. For each part below, there is a function f : R→R that is (a) injective and surjective. (b) injective but not surjective. (c) surjective but not injective. (d) neither injective nor surjective.

A donut shop has 10 types of donuts including chocolate. (a) How many ways are there to choose 6 donuts? (b) How many ways are there to choose 6 donuts, where at least one of the choices should include chocolate?

Unless otherwise noted, all sets in this module are finite. Prove the following statements...

1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r when 24|(k−r). If g : S→S is defined by (a) g(m) = f(7m) then g is injective and (b) g(m) = f(15m) then g is not injective.

2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is injective.

3. Let f : A→B and g : B→C be surjective. Then g ◦ f : A→C is surjective.

4. There is a surjection f : A→B such that f −1 : B→A is not a function.

If P = (1,4) in the elliptic curve E13(1, 1) , then 4P is ????

PLEASE SHOW ALL STEPS IN DETAIL AND EXPLAIN EACH STEP...

State and prove the Weighted Mean Value Theorem for integrals.

State and prove the Gaussian Quadrature Formula. Explain (do not prove) in what sense this formula is optimal.