Discrete Mathematics With Applications
5th Edition
ISBN: 9781337694193
Author: EPP, Susanna S.
Publisher: Cengage Learning,
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Chapter 6.3, Problem 54ES
To determine
To derive:
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(7) (12 points) Let F(x, y, z) = (y, x+z cos yz, y cos yz).
Ꮖ
(a) (4 points) Show that V x F = 0.
(b) (4 points) Find a potential f for the vector field F.
(c) (4 points) Let S be a surface in R3 for which the Stokes' Theorem is valid. Use
Stokes' Theorem to calculate the line integral
Jos
F.ds;
as denotes the boundary of S. Explain your answer.
(3) (16 points) Consider
z = uv,
u = x+y,
v=x-y.
(a) (4 points) Express z in the form z = fog where g: R² R² and f: R² →
R.
(b) (4 points) Use the chain rule to calculate Vz = (2, 2). Show all intermediate
steps otherwise no credit.
(c) (4 points) Let S be the surface parametrized by
T(x, y) = (x, y, ƒ (g(x, y))
(x, y) = R².
Give a parametric description of the tangent plane to S at the point p = T(x, y).
(d) (4 points) Calculate the second Taylor polynomial Q(x, y) (i.e. the quadratic
approximation) of F = (fog) at a point (a, b). Verify that
Q(x,y) F(a+x,b+y).
=
Chapter 6 Solutions
Discrete Mathematics With Applications
Ch. 6.1 - The notation is read”______” and means that___Ch. 6.1 - To use an element argument for proving that a set...Ch. 6.1 - Prob. 3TYCh. 6.1 - An element x is in AB if , and only if,_______Ch. 6.1 - An element x in AB if, and only if,______Ch. 6.1 - An element x is in B-A if, and only if,______Ch. 6.1 - An elements x is in Acif, and only if.______Ch. 6.1 - The empty set is a set with ______Ch. 6.1 - The power set of a set A is _____Ch. 6.1 - Prob. 10TY
Ch. 6.1 - A collection of nonempty set is a partition of a...Ch. 6.1 - Prob. 1ESCh. 6.1 - Complete the proof from Example 6.1.3: Prove that...Ch. 6.1 - Let sets R, S, and T be defined as follows:...Ch. 6.1 - Let A={nZn=5rforsomeintegerr} and...Ch. 6.1 - Prob. 5ESCh. 6.1 - Let...Ch. 6.1 - ...Ch. 6.1 - Prob. 8ESCh. 6.1 - Complete the following sentences without using the...Ch. 6.1 - ...Ch. 6.1 - Let the universal set be R, the set of all real...Ch. 6.1 - Let the universal set be R, the set of all real...Ch. 6.1 - Let S be the set of all strings of 0’s and 1’s of...Ch. 6.1 - Prob. 14ESCh. 6.1 - Prob. 15ESCh. 6.1 - Prob. 16ESCh. 6.1 - Prob. 17ESCh. 6.1 - a. Is the number 0 in ? Why? b. Is ={} ? Why ? c....Ch. 6.1 - Prob. 19ESCh. 6.1 - Let Bi={xR0xi} for each integer i=1,2,3,4. a....Ch. 6.1 - Let Ci={i,i} for each nonnegative integer i.Ch. 6.1 - Let Di={xR-ixi}=[i,i] for each nonnegative integer...Ch. 6.1 - Let Vi={xR1ix1i}=[1i,1i] for each positive integer...Ch. 6.1 - Let Wi={xRxi}=(i,) for each nonnegative integer i....Ch. 6.1 - Let Ri={xR1x1+1i}=[1,1+1i]foreachpositiveintegeri....Ch. 6.1 - Let Si={xR1x1+1i}=(1,1+1i) for each positive...Ch. 6.1 - Prob. 27ESCh. 6.1 - Let E be the set of all even integers and O the...Ch. 6.1 - Let R be the set of all real number. Is a...Ch. 6.1 - Let Z be the set of all integers and let...Ch. 6.1 - Prob. 31ESCh. 6.1 - Suppose A={1} and B={u,v} . Find P(AB) . Suppose...Ch. 6.1 - Find P() FindP(p()). Find p(p(p())) .Ch. 6.1 - Prob. 34ESCh. 6.1 - Prob. 35ESCh. 6.1 - Prob. 36ESCh. 6.1 - Prob. 37ESCh. 6.1 - Write an algorithm to determine whether a given...Ch. 6.2 - Prob. 1TYCh. 6.2 - Prob. 2TYCh. 6.2 - Prob. 3TYCh. 6.2 - Prob. 4TYCh. 6.2 - Prob. 5TYCh. 6.2 - Prob. 6TYCh. 6.2 - To say that an element is in A(BC) means that it...Ch. 6.2 - The following are two proofs that for all sets A...Ch. 6.2 - In 3 and 4, supply explanations of the steps in...Ch. 6.2 - Prob. 4ESCh. 6.2 - Prob. 5ESCh. 6.2 - Let and stand for the words “intersection” and...Ch. 6.2 - Prob. 7ESCh. 6.2 - Prob. 8ESCh. 6.2 - Prob. 9ESCh. 6.2 - Prob. 10ESCh. 6.2 - Prob. 11ESCh. 6.2 - Prob. 12ESCh. 6.2 - Prob. 13ESCh. 6.2 - Prob. 14ESCh. 6.2 - Prob. 15ESCh. 6.2 - Prob. 16ESCh. 6.2 - Prob. 17ESCh. 6.2 - Prob. 18ESCh. 6.2 - Prob. 19ESCh. 6.2 - Prob. 20ESCh. 6.2 - Prob. 21ESCh. 6.2 - Prob. 22ESCh. 6.2 - Prob. 23ESCh. 6.2 - Prob. 24ESCh. 6.2 - Prob. 25ESCh. 6.2 - Prob. 26ESCh. 6.2 - Fill in the blanks in the following proof that for...Ch. 6.2 - Prob. 28ESCh. 6.2 - Prob. 29ESCh. 6.2 - Prob. 30ESCh. 6.2 - Prob. 31ESCh. 6.2 - Prob. 32ESCh. 6.2 - Prob. 33ESCh. 6.2 - Prob. 34ESCh. 6.2 - Prob. 35ESCh. 6.2 - Prob. 36ESCh. 6.2 - Prob. 37ESCh. 6.2 - Prob. 38ESCh. 6.2 - Prove each statement is 39-44. For all sets A and...Ch. 6.2 - Prove each statement in 39-44. For every positive...Ch. 6.2 - Prob. 41ESCh. 6.2 - Prove each statement in 39-44. For every positive...Ch. 6.2 - Prob. 43ESCh. 6.2 - Prob. 44ESCh. 6.3 - Given a proposed set identity set identity...Ch. 6.3 - When using algebraic method for proving a set...Ch. 6.3 - Prob. 3TYCh. 6.3 - Prob. 1ESCh. 6.3 - Prob. 2ESCh. 6.3 - Prob. 3ESCh. 6.3 - Prob. 4ESCh. 6.3 - Prob. 5ESCh. 6.3 - Prob. 6ESCh. 6.3 - Prob. 7ESCh. 6.3 - Prob. 8ESCh. 6.3 - Prob. 9ESCh. 6.3 - Prob. 10ESCh. 6.3 - Prob. 11ESCh. 6.3 - Prob. 12ESCh. 6.3 - Prob. 13ESCh. 6.3 - Prob. 14ESCh. 6.3 - Prob. 15ESCh. 6.3 - Prob. 16ESCh. 6.3 - Prob. 17ESCh. 6.3 - Prob. 18ESCh. 6.3 - Prob. 19ESCh. 6.3 - Prob. 20ESCh. 6.3 - Prob. 21ESCh. 6.3 - Write a negation for each of the following...Ch. 6.3 - Let S={a,b,c} and for each integer i = 0, 1, 2, 3,...Ch. 6.3 - Let A={t,u,v,w} , and let S1 be the set of all...Ch. 6.3 - Prob. 25ESCh. 6.3 - Prob. 26ESCh. 6.3 - Prob. 27ESCh. 6.3 - Prob. 28ESCh. 6.3 - Some steps are missing from the following proof...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - Prob. 31ESCh. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - Prob. 33ESCh. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30—40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - Prob. 41ESCh. 6.3 - Prob. 42ESCh. 6.3 - Prob. 43ESCh. 6.3 - Prob. 44ESCh. 6.3 - Consider the following set property: For all sets...Ch. 6.3 - Prob. 46ESCh. 6.3 - Prob. 47ESCh. 6.3 - Prob. 48ESCh. 6.3 - Prob. 49ESCh. 6.3 - Prob. 50ESCh. 6.3 - Prob. 51ESCh. 6.3 - Prob. 52ESCh. 6.3 - Prob. 53ESCh. 6.3 - Prob. 54ESCh. 6.4 - In the comparison between the structure of the set...Ch. 6.4 - Prob. 2TYCh. 6.4 - Prob. 3TYCh. 6.4 - Prob. 1ESCh. 6.4 - Prob. 2ESCh. 6.4 - In 1-3 assume that B is a Boolean algebra with...Ch. 6.4 - Prob. 4ESCh. 6.4 - Prob. 5ESCh. 6.4 - Prob. 6ESCh. 6.4 - Prob. 7ESCh. 6.4 - Prob. 8ESCh. 6.4 - Prob. 9ESCh. 6.4 - In 4—10 assume that B is a Boolean algebra with...Ch. 6.4 - Prob. 11ESCh. 6.4 - Exercises 12-15 provide an outline for a proof...Ch. 6.4 - Prob. 13ESCh. 6.4 - Exercises 12-15 provide an outline for a proof...Ch. 6.4 - Prob. 15ESCh. 6.4 - Prob. 16ESCh. 6.4 - Prob. 17ESCh. 6.4 - In 16-21 determine where each sentence is a...Ch. 6.4 - In 16-21 determin whether each sentence is a...Ch. 6.4 - In 16-21 determine wherether each sentence is a...Ch. 6.4 - In 16-21 determine wherether each sentence is a...Ch. 6.4 - Prob. 22ESCh. 6.4 - Prob. 23ESCh. 6.4 - Can there exist a cimputer program that has as...Ch. 6.4 - Can there exist a book that refers to all those...Ch. 6.4 - Some English adjectives are descriptive of...Ch. 6.4 - As strange as it may seem, it is possible to give...Ch. 6.4 - Is there an alogroithm whichm for a fixed quantity...Ch. 6.4 - Prob. 29ES
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