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(1) Review from class notes the notion of a surface. Now consider F: R³ → R, a differentiable (smooth) function. Thus, the partial derivatives of F, Fr OF OF OF F = Oxy and F. = ду მ2 exist and are continuous on all of R³. We say a point p = (P1, P2, P3) € R³ is a critical point of F if Fr(p) Fy(p) = F(p) = 0. Otherwise p is said to be a regular point of F. For any cЄR, the level set F-1(c) is a regular level set if F-¹(c) 0 and consists entirely of regular points. Regular Level Set Theorem. Let F: R³ → R be a differentiable function. Then every regular level set of F is a surface in R³. Note: The Regular Level Set Theorem does not say that non-regular level sets of F are never surfaces! It only says that regular level sets always are. In each case below decide if the set is (or is not) a surface in R³. Give a one-line explanation of your answer in each case. The Regular Level Set Theorem may be helpful in deciding that a set is a surface. It will not help you in arguing that a set is not a surface. (In the latter situation, just state where the problem points are. No further explanation is necessary.) (i.) C = {(x, y, z) R³: x² - y²+ z² = 0};

(1) Review from class notes the notion of a surface. Now consider F: R³ → R, a differentiable (smooth) function. Thus, the partial derivatives of F, Fr OF OF OF F = Oxy and F. = ду მ2 exist and are continuous on all of R³. We say a point p = (P1, P2, P3) € R³ is a critical point of F if Fr(p) Fy(p) = F(p) = 0. Otherwise p is said to be a regular point of F. For any cЄR, the level set F-1(c) is a regular level set if F-¹(c) 0 and consists entirely of regular points. Regular Level Set Theorem. Let F: R³ → R be a differentiable function. Then every regular level set of F is a surface in R³. Note: The Regular Level Set Theorem does not say that non-regular level sets of F are never surfaces! It only says that regular level sets always are. In each case below decide if the set is (or is not) a surface in R³. Give a one-line explanation of your answer in each case. The Regular Level Set Theorem may be helpful in deciding that a set is a surface. It will not help you in arguing that a set is not a surface. (In the latter situation, just state where the problem points are. No further explanation is necessary.) (i.) C = {(x, y, z) R³: x² - y²+ z² = 0};

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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(1) Review from class notes the notion of a surface. Now consider F: R³ → R, a differentiable (smooth) function. Thus, the partial derivatives of F, Fr OF OF OF F = Oxy and F. = ду მ2 exist and are continuous on all of R³. We say a point p = (P1, P2, P3) € R³ is a critical point of F if Fr(p) Fy(p) = F(p) = 0. Otherwise p is said to be a regular point of F. For any cЄR, the level set F-1(c) is a regular level set if F-¹(c) 0 and consists entirely of regular points. Regular Level Set Theorem. Let F: R³ → R be a differentiable function. Then every regular level set of F is a surface in R³. Note: The Regular Level Set Theorem does not say that non-regular level sets of F are never surfaces! It only says that regular level sets always are. In each case below decide if the set is (or is not) a surface in R³. Give a one-line explanation of your answer in each case. The Regular Level Set Theorem may be helpful in deciding that a set is a surface. It will not help you in arguing that a set is not a surface. (In the latter situation, just state where the problem points are. No further explanation is necessary.) (i.) C = {(x, y, z) R³: x² - y²+ z² = 0};
Transcribed Image Text:(1) Review from class notes the notion of a surface. Now consider F: R³ → R, a differentiable (smooth) function. Thus, the partial derivatives of F, Fr OF OF OF F = Oxy and F. = ду მ2 exist and are continuous on all of R³. We say a point p = (P1, P2, P3) € R³ is a critical point of F if Fr(p) Fy(p) = F(p) = 0. Otherwise p is said to be a regular point of F. For any cЄR, the level set F-1(c) is a regular level set if F-¹(c) 0 and consists entirely of regular points. Regular Level Set Theorem. Let F: R³ → R be a differentiable function. Then every regular level set of F is a surface in R³. Note: The Regular Level Set Theorem does not say that non-regular level sets of F are never surfaces! It only says that regular level sets always are. In each case below decide if the set is (or is not) a surface in R³. Give a one-line explanation of your answer in each case. The Regular Level Set Theorem may be helpful in deciding that a set is a surface. It will not help you in arguing that a set is not a surface. (In the latter situation, just state where the problem points are. No further explanation is necessary.) (i.) C = {(x, y, z) R³: x² - y²+ z² = 0};
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