Consider F and C below. F(x, y, z) = yzexzi + ex²j + xyexzk, C: r(t) = (t² + 3)i + (t² − 1)j + (t² − 4t)k, - (a) Find a function f such that F = Vf. f(x, y, z) = (b) Use part (a) to evaluate Ic 0 ≤t≤ 4 F. dr along the given curve C
Consider F and C below. F(x, y, z) = yzexzi + ex²j + xyexzk, C: r(t) = (t² + 3)i + (t² − 1)j + (t² − 4t)k, - (a) Find a function f such that F = Vf. f(x, y, z) = (b) Use part (a) to evaluate Ic 0 ≤t≤ 4 F. dr along the given curve C
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## Problem Statement:
Consider **F** and **C** below.
\[
\mathbf{F}(x, y, z) = yz e^{xz} \mathbf{i} + e^{xz} \mathbf{j} + xy e^{xz} \mathbf{k},
\]
\[
C: \mathbf{r}(t) = (t^2 + 3)\mathbf{i} + (t^2 - 1)\mathbf{j} + (t^2 - 4t)\mathbf{k}, \quad 0 \leq t \leq 4
\]
### Tasks:
(a) Find a function \( f \) such that \( \mathbf{F} = \nabla f \).
\[ f(x, y, z) = \underline{\quad\quad\quad\quad\quad\quad\quad} \]
(b) Use part (a) to evaluate \(\int_{C} \mathbf{F} \cdot d\mathbf{r}\) along the given curve \( C \).
\[ \underline{\quad\quad\quad\quad\quad\quad\quad} \]
### Additional Information:
There are no graphs or diagrams accompanying this problem. The task primarily involves vector calculus operations, specifically dealing with gradient fields and line integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc42a53c5-a661-4e08-8bd7-4bdfe484c1d7%2F6409d418-c215-4f0c-be27-207fd85bb539%2Fspzcdi8_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement:
Consider **F** and **C** below.
\[
\mathbf{F}(x, y, z) = yz e^{xz} \mathbf{i} + e^{xz} \mathbf{j} + xy e^{xz} \mathbf{k},
\]
\[
C: \mathbf{r}(t) = (t^2 + 3)\mathbf{i} + (t^2 - 1)\mathbf{j} + (t^2 - 4t)\mathbf{k}, \quad 0 \leq t \leq 4
\]
### Tasks:
(a) Find a function \( f \) such that \( \mathbf{F} = \nabla f \).
\[ f(x, y, z) = \underline{\quad\quad\quad\quad\quad\quad\quad} \]
(b) Use part (a) to evaluate \(\int_{C} \mathbf{F} \cdot d\mathbf{r}\) along the given curve \( C \).
\[ \underline{\quad\quad\quad\quad\quad\quad\quad} \]
### Additional Information:
There are no graphs or diagrams accompanying this problem. The task primarily involves vector calculus operations, specifically dealing with gradient fields and line integrals.
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