F(x, y, z) = (2xz + y²) i + 2xy j + (x² + 6z²) k C: x = t²2, y = t + 2, z = 2t - 1, 0≤t≤1 (a) Find a function f such that F Vf. f(x, y, z) = (b) Use part (a) to evaluate S Vf. dr along the given curve C
F(x, y, z) = (2xz + y²) i + 2xy j + (x² + 6z²) k C: x = t²2, y = t + 2, z = 2t - 1, 0≤t≤1 (a) Find a function f such that F Vf. f(x, y, z) = (b) Use part (a) to evaluate S Vf. dr along the given curve C
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Consider** **F** and **C** below.
\[ \mathbf{F}(x, y, z) = \left(2xz + y^2\right) \mathbf{i} + 2xy \, \mathbf{j} + \left(x^2 + 6z^2\right) \mathbf{k} \]
\[ C: x = t^2, \quad y = t + 2, \quad z = 2t - 1, \quad 0 \le t \le 1 \]
(a) Find a function \( f \) such that \(\mathbf{F} = \nabla f\).
\[ f(x, y, z) = \]
(b) Use part (a) to evaluate \(\int_C \nabla f \cdot d\mathbf{r}\) along the given curve \( C \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc42a53c5-a661-4e08-8bd7-4bdfe484c1d7%2Fc00e4948-478e-48de-a6cd-e61e989e5863%2Feb6o08h_processed.png&w=3840&q=75)
Transcribed Image Text:**Consider** **F** and **C** below.
\[ \mathbf{F}(x, y, z) = \left(2xz + y^2\right) \mathbf{i} + 2xy \, \mathbf{j} + \left(x^2 + 6z^2\right) \mathbf{k} \]
\[ C: x = t^2, \quad y = t + 2, \quad z = 2t - 1, \quad 0 \le t \le 1 \]
(a) Find a function \( f \) such that \(\mathbf{F} = \nabla f\).
\[ f(x, y, z) = \]
(b) Use part (a) to evaluate \(\int_C \nabla f \cdot d\mathbf{r}\) along the given curve \( C \).
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