Question

(4) Let V be an open subset of \(\mathbb{R}^3\), and let \(f: V \to \mathbb{R}\) be a smooth function for which \(a \in \mathbb{R}\) is a regular value. Take \(S = f^{-1}(a)\) and fix a point \(p \in S\). (a) Show that \(T_pS \subset \ker df_p\). (b) Argue that \(\dim \ker df_p = 2\) (Hint: Recall the rank-nullity theorem!) HOMEWORK 6 DUE TUESDAY, OCTOBER 17 (c) Conclude from parts (a) and (b) that \(\ker df_p = T_pS\).

          (4) Let V be an open subset of \(\mathbb{R}^3\), and let \(f: V \to \mathbb{R}\) be a smooth function for which \(a \in \mathbb{R}\) is a regular value. Take \(S = f^{-1}(a)\) and fix a point \(p \in S\).
(a) Show that \(T_pS \subset \ker df_p\).
(b) Argue that \(\dim \ker df_p = 2\) (Hint: Recall the rank-nullity theorem!)
HOMEWORK 6 DUE TUESDAY, OCTOBER 17
(c) Conclude from parts (a) and (b) that \(\ker df_p = T_pS\).

(4) Let V be an open subset of ℝ^3, and let f: V →ℝ be a smooth function for which a ∈ℝ is a regular value. Take S = f^-1(a) and fix a point p ∈ S.
(a) Show that TpS ⊂ dfp.
(b) Argue that dfp = 2 (Hint: Recall the rank-nullity theorem!)
HOMEWORK 6 DUE TUESDAY, OCTOBER 17
(c) Conclude from parts (a) and (b) that dfp = TpS.

Added by Dawn H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

06/05/2023

Step 1

To do this, we need to show that for any vector v in TpS, v is also in ker(df_p). Show more…

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Texts: 4. Let V be an open subset of R^3 and let f: V -> R be a smooth function for which a ∈ R is a regular value. Take S = f^(-1)(a) and fix a point p ∈ S. (a) Show that TpS ⊆ ker(df_p). (b) Argue that dim(ker(df_p)) = 2. (Hint: Recall the rank-nullity theorem!) HOMEWORK 6 DUE TUESDAY, OCTOBER 17 (c) Conclude from parts (a) and (b) that ker(df_p) = TpS.