É = ( 2x3 + y°, y3 + z³, 3y²z), S is the surface of the solid bounded by the paraboloid z = 1– x2 – y² and the xy plane. (Ans: 7)
É = ( 2x3 + y°, y3 + z³, 3y²z), S is the surface of the solid bounded by the paraboloid z = 1– x2 – y² and the xy plane. (Ans: 7)
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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Verify the answer using F.ndS
NOT divergence theorem.
![The vector field is given by
\[
\vec{F} = \langle 2x^3 + y^3, y^3 + z^3, 3y^2z \rangle
\]
\( S \) denotes the surface of the solid bounded by the paraboloid
\[
z = 1 - x^2 - y^2
\]
and the \( xy \)-plane.
(Answer: \(\pi\))](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ae14ade-6f74-4154-97ef-b44e7d0350a4%2Fb2600b5f-55b2-496f-8e42-9057d204c751%2Fvtyx0u6_processed.png&w=3840&q=75)
Transcribed Image Text:The vector field is given by
\[
\vec{F} = \langle 2x^3 + y^3, y^3 + z^3, 3y^2z \rangle
\]
\( S \) denotes the surface of the solid bounded by the paraboloid
\[
z = 1 - x^2 - y^2
\]
and the \( xy \)-plane.
(Answer: \(\pi\))
![The image shows the formula for the surface integral of a vector field. The expression is:
\[
\iint_S \mathbf{F} \cdot \hat{\mathbf{n}} \, dS
\]
### Explanation:
- **\(\iint_S\):** This denotes a surface integral over a surface \(S\).
- **\(\mathbf{F}\):** This represents a vector field.
- **\(\hat{\mathbf{n}}\):** This is the unit normal vector to the surface \(S\).
- **\(\cdot\):** Represents the dot product between the vector field \(\mathbf{F}\) and the unit normal vector \(\hat{\mathbf{n}}\).
- **\(dS\):** A differential element of the surface \(S\).
### Description:
This integral computes the flux of the vector field \(\mathbf{F}\) across the surface \(S\), essentially measuring how much of the vector field "flows" through the surface. The dot product \(\mathbf{F} \cdot \hat{\mathbf{n}}\) captures how much of \(\mathbf{F}\) is in the direction of the normal vector to the surface, which affects the contribution to the overall flux.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ae14ade-6f74-4154-97ef-b44e7d0350a4%2Fb2600b5f-55b2-496f-8e42-9057d204c751%2F1ypn47g_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows the formula for the surface integral of a vector field. The expression is:
\[
\iint_S \mathbf{F} \cdot \hat{\mathbf{n}} \, dS
\]
### Explanation:
- **\(\iint_S\):** This denotes a surface integral over a surface \(S\).
- **\(\mathbf{F}\):** This represents a vector field.
- **\(\hat{\mathbf{n}}\):** This is the unit normal vector to the surface \(S\).
- **\(\cdot\):** Represents the dot product between the vector field \(\mathbf{F}\) and the unit normal vector \(\hat{\mathbf{n}}\).
- **\(dS\):** A differential element of the surface \(S\).
### Description:
This integral computes the flux of the vector field \(\mathbf{F}\) across the surface \(S\), essentially measuring how much of the vector field "flows" through the surface. The dot product \(\mathbf{F} \cdot \hat{\mathbf{n}}\) captures how much of \(\mathbf{F}\) is in the direction of the normal vector to the surface, which affects the contribution to the overall flux.
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