Explanation Given: Let D be a disk in R 2 . It show that if ∇ f ( x , y ) = 0 for all ( x , y ) in D , then f is constant. Consider points P = ( a , b ) , Q = ( c , d ) , and R = ( c , b ) . Proof: a. For the line segment P R ¯ , the y-variable will be constant that is y = b . Now, consider S is a function of a single variable say, x So, use the given relation ∇ f ( x , y ) = 0 , we can say that, ∇ S ( x , y ) = 0 To obtain d S d x = 0 That means S is a constant with respect to the variable x . Here, S is just a function of a single variable x on the line segment P R ¯ , so S has to be a constant along the line segment P R ¯ . For the line segment R Q ¯ , the x-variable will be constant that is x = c . Now, consider S is a function of a single variable say, y So, use the given relation ∇ f ( x , y ) = 0 , we can say that, ∇ S ( x , y ) = 0 To obtain d S d y = 0 That means S is a constant with respect to the variable y ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.1, Problem 57E

To determine

To Prove:

a. $f$ is a constant along the segments $\bar{P R}$ and $\bar{R Q}$ by using single-variable calculus.

b. $f (P) = f (Q)$ for any two points $P, Q \in D$ .

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Chapter 17.1, Problem 56E

Chapter 17.2, Problem 1PQ

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Chapter 17 Solutions

CALCULUS (CLOTH)

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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To Prove: a. f is a constant along the segments P R ¯ and R Q ¯ by using single-variable calculus. b. f ( P ) = f ( Q ) for any two points P , Q ∈ D .

Solution Summary: The author proves that f is a constant along the segments P R and R Q

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To Prove:

a. $f$ is a constant along the segments $\bar{P R}$ and $\bar{R Q}$ by using single-variable calculus.

b. $f (P) = f (Q)$ for any two points $P, Q \in D$ .

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Chapter 17 Solutions

To Prove: a. f is a constant along the segments P R ¯ and R Q ¯ by using single-variable calculus. b. f ( P ) = f ( Q ) for any two points P , Q ∈ D .

Solution Summary: The author proves that f is a constant along the segments P R and R Q

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To Prove: a. f is a constant along the segments PR¯ and RQ¯ by using single-variable calculus. b. f(P)=f(Q) for any two points P,Q∈D.

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Chapter 17 Solutions

To Prove:

a. $f$ is a constant along the segments $\bar{P R}$ and $\bar{R Q}$ by using single-variable calculus.

b. $f (P) = f (Q)$ for any two points $P, Q \in D$ .