4. Given the vector field F = ye*V i+ (xe#y + zev²) j+ (yeUz + z) k Show that the field is conservative and then find the potential for F
4. Given the vector field F = ye*V i+ (xe#y + zev²) j+ (yeUz + z) k Show that the field is conservative and then find the potential for F
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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Please help with the solution to the following two questions. This is
![**Problem 4:**
*Given the vector field*
\[ \mathbf{F} = ye^{xy} \mathbf{i} + (xe^{xy} + ze^{yz}) \mathbf{j} + (ye^{yz} + z) \mathbf{k} \]
*Show that the field is conservative and then find the potential for \( \mathbf{F} \).*
**Problem 5:**
*Evaluate the integral*
\[ \int_{C} \mathbf{G} \cdot d\mathbf{R} \]
*where*
\[ \mathbf{G} = ye^{xy} \mathbf{i} + (xe^{xy} + ze^{yz} - 3z) \mathbf{j} + (ye^{yz} - y + z) \mathbf{k} \]
*and the curve \( C \) is given by*
\[ x = 1 + 2t, \quad y = 1 - t^2, \quad z = 2t - 3, \quad 0 \leq t \leq 1 \]
*(Hint: Note that \(\mathbf{G}\) can be written as a sum of \(\mathbf{F}\) in the previous question and an additional non-conservative part.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1345a87f-3d4a-45ee-a13b-cb772771cc65%2Fe5e7e7c0-c9dc-4778-bae2-da61c179fed0%2Fd8puqhqb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4:**
*Given the vector field*
\[ \mathbf{F} = ye^{xy} \mathbf{i} + (xe^{xy} + ze^{yz}) \mathbf{j} + (ye^{yz} + z) \mathbf{k} \]
*Show that the field is conservative and then find the potential for \( \mathbf{F} \).*
**Problem 5:**
*Evaluate the integral*
\[ \int_{C} \mathbf{G} \cdot d\mathbf{R} \]
*where*
\[ \mathbf{G} = ye^{xy} \mathbf{i} + (xe^{xy} + ze^{yz} - 3z) \mathbf{j} + (ye^{yz} - y + z) \mathbf{k} \]
*and the curve \( C \) is given by*
\[ x = 1 + 2t, \quad y = 1 - t^2, \quad z = 2t - 3, \quad 0 \leq t \leq 1 \]
*(Hint: Note that \(\mathbf{G}\) can be written as a sum of \(\mathbf{F}\) in the previous question and an additional non-conservative part.)*
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