Clearly demonstrate how you use Stokes' Theorem to evaluate f F 1- y³, x³ + e²,0) and C is the circle a² + y² = 9, counterclockwise direction. F dr, where F n
Clearly demonstrate how you use Stokes' Theorem to evaluate f F 1- y³, x³ + e²,0) and C is the circle a² + y² = 9, counterclockwise direction. F dr, where F n
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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![### Application of Stokes' Theorem
To evaluate the line integral using Stokes' Theorem, let's consider the given problem:
Evaluate the line integral \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\), where \(\mathbf{F} = \langle 1 - y^3, x^3 + e^{y^2}, 0 \rangle\) and \(C\) is the circle \(x^2 + y^2 = 9\), oriented in the counterclockwise direction.
#### Step-by-Step Solution:
1. **Stokes' Theorem Statement:**
Stokes' Theorem relates a surface integral of a curl of a vector field to a line integral over its boundary:
\[
\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} \nabla \times \mathbf{F} \cdot d\mathbf{S}
\]
Where \(S\) is a surface with boundary \(C\).
2. **Compute the Curl of \(\mathbf{F}\):**
Given \(\mathbf{F} = (1 - y^3, x^3 + e^{y^2}, 0)\), the curl of \(\mathbf{F}\) is:
\[
\nabla \times \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
1 - y^3 & x^3 + e^{y^2} & 0
\end{vmatrix}
\]
Calculating the determinant:
\[
\nabla \times \mathbf{F} = \left(0 - 0\right) \mathbf{i} - \left(0 - 0\right) \mathbf{j} + \left(\frac{\partial}{\partial x}(x^3 + e^{y^2}) - \frac{\partial}{\partial y}(1 - y^3)\right) \mathbf{k}
\]
Simplifying:
\[
\nabla \times \mathbf{F](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d3733a5-5e9e-433b-b6d1-2cfec636672d%2Fc36eff6e-f837-4951-8e70-ffe7e7c9af84%2F2lif3vm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Application of Stokes' Theorem
To evaluate the line integral using Stokes' Theorem, let's consider the given problem:
Evaluate the line integral \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\), where \(\mathbf{F} = \langle 1 - y^3, x^3 + e^{y^2}, 0 \rangle\) and \(C\) is the circle \(x^2 + y^2 = 9\), oriented in the counterclockwise direction.
#### Step-by-Step Solution:
1. **Stokes' Theorem Statement:**
Stokes' Theorem relates a surface integral of a curl of a vector field to a line integral over its boundary:
\[
\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} \nabla \times \mathbf{F} \cdot d\mathbf{S}
\]
Where \(S\) is a surface with boundary \(C\).
2. **Compute the Curl of \(\mathbf{F}\):**
Given \(\mathbf{F} = (1 - y^3, x^3 + e^{y^2}, 0)\), the curl of \(\mathbf{F}\) is:
\[
\nabla \times \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
1 - y^3 & x^3 + e^{y^2} & 0
\end{vmatrix}
\]
Calculating the determinant:
\[
\nabla \times \mathbf{F} = \left(0 - 0\right) \mathbf{i} - \left(0 - 0\right) \mathbf{j} + \left(\frac{\partial}{\partial x}(x^3 + e^{y^2}) - \frac{\partial}{\partial y}(1 - y^3)\right) \mathbf{k}
\]
Simplifying:
\[
\nabla \times \mathbf{F
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