Let F = (5x4 + 16z, 2x³ + 3y? + 16z, x + y+ ev*+1) and C be the curve defined by 7 (t) = (2 cos(t), 2 sin(t), 16 cos? (t) sin? (t)) for 0
Let F = (5x4 + 16z, 2x³ + 3y? + 16z, x + y+ ev*+1) and C be the curve defined by 7 (t) = (2 cos(t), 2 sin(t), 16 cos? (t) sin? (t)) for 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Calc 4 stoke's theorem
![### Application of Stokes' Theorem in Vector Calculus
Consider the vector field \( \mathbf{F} \) defined as:
\[ \mathbf{F} = \left( 5x^4 + 16z, 2x^3 + 3y^2 + 16z, x + y + e^{\sqrt{z^3 + 1}} \right) \]
where \( x, y, z \) are spatial coordinates.
We also define the curve \( C \) using the parameterization \( \mathbf{r}(t) \):
\[ \mathbf{r}(t) = \left( 2 \cos(t), 2 \sin(t), 16 \cos^2(t) \sin^2(t) \right) \]
for the range \( 0 \le t \le \pi \).
The objective is to evaluate the line integral:
\[ \int_C \mathbf{F} \cdot d\mathbf{r} \]
using Stokes' Theorem.
### Stokes' Theorem
Stokes' Theorem relates a surface integral of a curl of a vector field over a surface \( S \) to a line integral of the vector field over the boundary curve \( C \):
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} \nabla \times \mathbf{F} \cdot d\mathbf{S} \]
where:
- \( \nabla \times \mathbf{F} \) is the curl of the vector field \( \mathbf{F} \)
- \( d\mathbf{S} \) is the vector area element of the surface \( S \)
**Steps to Apply Stokes' Theorem:**
1. **Determine the Boundary Curve \( C \):** We have been given \( \mathbf{r}(t) \) that describes the curve C.
2. **Find a Surface \( S \) with Boundary \( C \):** Choose a convenient surface whose boundary is \( C \).
3. **Evaluate \( \nabla \times \mathbf{F} \):** Calculate the curl of the vector field \( \mathbf{F} \).
4. **Parameterize the Surface \( S \):** Express the chosen surface \( S \) in terms of parameters.
5. **Compute the Surface Integral:** Evaluate](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5f558a7-14fc-4024-84d6-4debb1adc6f6%2Feeef0413-f74d-43db-ab95-0443ee878a20%2Fij0efv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Application of Stokes' Theorem in Vector Calculus
Consider the vector field \( \mathbf{F} \) defined as:
\[ \mathbf{F} = \left( 5x^4 + 16z, 2x^3 + 3y^2 + 16z, x + y + e^{\sqrt{z^3 + 1}} \right) \]
where \( x, y, z \) are spatial coordinates.
We also define the curve \( C \) using the parameterization \( \mathbf{r}(t) \):
\[ \mathbf{r}(t) = \left( 2 \cos(t), 2 \sin(t), 16 \cos^2(t) \sin^2(t) \right) \]
for the range \( 0 \le t \le \pi \).
The objective is to evaluate the line integral:
\[ \int_C \mathbf{F} \cdot d\mathbf{r} \]
using Stokes' Theorem.
### Stokes' Theorem
Stokes' Theorem relates a surface integral of a curl of a vector field over a surface \( S \) to a line integral of the vector field over the boundary curve \( C \):
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} \nabla \times \mathbf{F} \cdot d\mathbf{S} \]
where:
- \( \nabla \times \mathbf{F} \) is the curl of the vector field \( \mathbf{F} \)
- \( d\mathbf{S} \) is the vector area element of the surface \( S \)
**Steps to Apply Stokes' Theorem:**
1. **Determine the Boundary Curve \( C \):** We have been given \( \mathbf{r}(t) \) that describes the curve C.
2. **Find a Surface \( S \) with Boundary \( C \):** Choose a convenient surface whose boundary is \( C \).
3. **Evaluate \( \nabla \times \mathbf{F} \):** Calculate the curl of the vector field \( \mathbf{F} \).
4. **Parameterize the Surface \( S \):** Express the chosen surface \( S \) in terms of parameters.
5. **Compute the Surface Integral:** Evaluate
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