Developmental Mathematics (9th Edition)
9th Edition
ISBN: 9780321997173
Author: Marvin L. Bittinger, Judith A. Beecher
Publisher: PEARSON
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Chapter 7.2, Problem 77ES
To determine
To calculate: The order of following numbers from least to the greatest
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1. Show that the vector field
F(x, y, z)
=
(2x sin ye³)ix² cos yj + (3xe³ +5)k
satisfies the necessary conditions for a conservative vector field, and find a potential function for
F.
1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude
of the gravitational force between two objects with masses m and M is
|F|
mMG
|r|2
where r is the distance between the objects, and G is the gravitational constant. Assume that the
object with mass M is located at the origin in R³. Then, the gravitational force field acting on
the object at the point r = (x, y, z) is given by
F(x, y, z) =
mMG
r3
r.
mMG
mMG
Show that the scalar vector field f(x, y, z) =
=
is a potential function for
r
√√x² + y² .
Fi.e. show that F = Vf.
Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).
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- 1- Let A = {A1, A2, ...), in which A, A, = 0, when i j. a) Is A a π-system? If not, which element(s) should be added to A to become a π-system? b) Prove that σ(A) consists of the finite or countable unions of elements of A; i.c., A E σ(A) if and only if there exists finite or countable sequence {n} such that A = U₁An (Hint: Let F be such class; prove that F is a σ-filed containing A.) c) Let p ≥ 0 be a sequence of non-negative real numbers with Σip₁ = 1. Using p₁'s, how do you construct a probability measure on σ(A)? (Hint: use extension theorem.) 2- Construct an example for which P(lim sup A,) = 1 and P(lim inf An) = 0.arrow_forward3. Let f(z) = sin (22) + cos (T2) 2(22+1)(z+1) Compute f(z)dz over each of the contours/closed curves C1, C2, C3 and C4 shown below. Don't use any Al tool Don't send the same previous answer that was Al generated L 10 -c x show ur answer pe n and paper then take Send ur answer in pe n and paper don't rep uted ur self downarrow_forwardPLEASE SOLVE STEP BY STEP WITHOUT ARTIFICIAL INTELLIGENCE OR CHATGPT SOLVE BY HAND STEP BY STEP 4.- A beer at an unknown temperature is introduced into a refrigerator that has a constant temperature of 1°C. After 20 minutes, the temperature of the beer is 10°C, and after 40 minutes, the temperature of the beer is 6°C. a) Determine the temperature at which the beer was placed inside the refrigerator.b) How long will it take for the beer to reach 2°C?arrow_forward
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