Consider F and C below. F(x, y, z) = (y²z + 2xz?)i + 2xyzj + (xy² + 2x²z)k, C: x = Vt, y = t + 8, z = t2, ostsi (a) Find a function f such that F = Vf. f(x, у, 2) %3 (b) Use part (a) to evaluate F. dr along the given curve C.
Consider F and C below. F(x, y, z) = (y²z + 2xz?)i + 2xyzj + (xy² + 2x²z)k, C: x = Vt, y = t + 8, z = t2, ostsi (a) Find a function f such that F = Vf. f(x, у, 2) %3 (b) Use part (a) to evaluate F. 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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![**Vector Calculus Problem:**
Consider **F** and **C** below.
\[ \mathbf{F}(x, y, z) = (y^2z + 2xz^2) \mathbf{i} + (2xyz) \mathbf{j} + (xy^2 + 2x^2z) \mathbf{k}, \]
\[ \mathbf{C}: x = \sqrt{t}, \; y = t + 8, \; z = t^2, \quad 0 \leq t \leq 1 \]
**(a) Find a function \( f \) such that \(\mathbf{F} = \nabla f\).**
\[ f(x, y, z) = \boxed{\text{ }} \]
**(b) Use part (a) to evaluate \(\int_{C} \mathbf{F} \cdot d\mathbf{r} \) along the given curve \(\mathbf{C}\).**
\[ \boxed{\text{ }} \]
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Transcribed Image Text:**Vector Calculus Problem:**
Consider **F** and **C** below.
\[ \mathbf{F}(x, y, z) = (y^2z + 2xz^2) \mathbf{i} + (2xyz) \mathbf{j} + (xy^2 + 2x^2z) \mathbf{k}, \]
\[ \mathbf{C}: x = \sqrt{t}, \; y = t + 8, \; z = t^2, \quad 0 \leq t \leq 1 \]
**(a) Find a function \( f \) such that \(\mathbf{F} = \nabla f\).**
\[ f(x, y, z) = \boxed{\text{ }} \]
**(b) Use part (a) to evaluate \(\int_{C} \mathbf{F} \cdot d\mathbf{r} \) along the given curve \(\mathbf{C}\).**
\[ \boxed{\text{ }} \]
### Need Help?
**Read It** **Talk to a Tutor**
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