eff curl(F). ds. F(x, y, z) = x² sin(z)i + y²j + xyk, S is the part of the paraboloid z = 9 - x² - y² that lies above the xy-plane, oriented upward Use Stokes' theorem to evaluate
eff curl(F). ds. F(x, y, z) = x² sin(z)i + y²j + xyk, S is the part of the paraboloid z = 9 - x² - y² that lies above the xy-plane, oriented upward Use Stokes' theorem to evaluate
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Using Stokes' Theorem to Evaluate the Surface Integral
Stokes' theorem provides a way to transform a difficult surface integral into a potentially simpler line integral. Here is an example problem demonstrating the application of Stokes' theorem:
#### Problem Statement
Use Stokes' theorem to evaluate the surface integral:
\[ \iint_S \text{curl}(\mathbf{F}) \cdot d\mathbf{S} \]
where the vector field \(\mathbf{F}\) and the surface \(S\) are defined as follows:
\[ \mathbf{F}(x, y, z) = x^2 \sin(z) \mathbf{i} + y^2 \mathbf{j} + xy \mathbf{k}, \]
\[ S \text{ is the part of the paraboloid } z = 9 - x^2 - y^2 \text{ that lies above the xy-plane, oriented upward}. \]
#### Explanation of Variables
- \(\mathbf{F}(x, y, z)\): Given vector field,
- \(S\): Surface of the paraboloid \(z = 9 - x^2 - y^2\) above the xy-plane,
- \(d\mathbf{S}\): An infinitesimal surface element vector.
### Key Concept: Stokes' Theorem
Stokes' theorem states that for a smooth vector field \(\mathbf{F}\) on a surface \(S\) with boundary curve \(C\),
\[ \iint_S \text{curl}(\mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}, \]
where \(C\) is the positively oriented (counter-clockwise looking from above) boundary curve of \(S\).
### Surface Description and Orientation
The paraboloid surface is given by the equation \(z = 9 - x^2 - y^2\). The region of interest is the part of this surface that lies above the \(xy\)-plane (\(z \geq 0\)). The surface is oriented upward, meaning the normal vectors point in the positive \(z\)-direction.
### Next Steps
1. **Compute the Boundary Curve \(C\)**: Find \(C\) by setting \(z = 0\) in the surface equation \(z = 9 - x^2 - y^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46029127-96a5-4dc2-997a-3c090d1aab2d%2F910db3b8-4894-4520-8832-7dc6e59b34e5%2F6tv0hkn_processed.png&w=3840&q=75)
Transcribed Image Text:### Using Stokes' Theorem to Evaluate the Surface Integral
Stokes' theorem provides a way to transform a difficult surface integral into a potentially simpler line integral. Here is an example problem demonstrating the application of Stokes' theorem:
#### Problem Statement
Use Stokes' theorem to evaluate the surface integral:
\[ \iint_S \text{curl}(\mathbf{F}) \cdot d\mathbf{S} \]
where the vector field \(\mathbf{F}\) and the surface \(S\) are defined as follows:
\[ \mathbf{F}(x, y, z) = x^2 \sin(z) \mathbf{i} + y^2 \mathbf{j} + xy \mathbf{k}, \]
\[ S \text{ is the part of the paraboloid } z = 9 - x^2 - y^2 \text{ that lies above the xy-plane, oriented upward}. \]
#### Explanation of Variables
- \(\mathbf{F}(x, y, z)\): Given vector field,
- \(S\): Surface of the paraboloid \(z = 9 - x^2 - y^2\) above the xy-plane,
- \(d\mathbf{S}\): An infinitesimal surface element vector.
### Key Concept: Stokes' Theorem
Stokes' theorem states that for a smooth vector field \(\mathbf{F}\) on a surface \(S\) with boundary curve \(C\),
\[ \iint_S \text{curl}(\mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}, \]
where \(C\) is the positively oriented (counter-clockwise looking from above) boundary curve of \(S\).
### Surface Description and Orientation
The paraboloid surface is given by the equation \(z = 9 - x^2 - y^2\). The region of interest is the part of this surface that lies above the \(xy\)-plane (\(z \geq 0\)). The surface is oriented upward, meaning the normal vectors point in the positive \(z\)-direction.
### Next Steps
1. **Compute the Boundary Curve \(C\)**: Find \(C\) by setting \(z = 0\) in the surface equation \(z = 9 - x^2 - y^
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