Let T be the triangular region with vertices (1,0,0), (0, 1,0), and (0,0,1) oriented with upward-pointing normal vector. (1, 0, 0), (0, 0, 1) (0, 1, 0) A fluid flows with constant velocity field v = 8i + 2j m/s. Calculate: (a) The flow rate through T (b) The flow rate through the projection of T onto the xy-plane [the triangle with vertices (0, 0, 0), (1,0,0), and (0, 1, 0)] Assume distances are in meters.
Let T be the triangular region with vertices (1,0,0), (0, 1,0), and (0,0,1) oriented with upward-pointing normal vector. (1, 0, 0), (0, 0, 1) (0, 1, 0) A fluid flows with constant velocity field v = 8i + 2j m/s. Calculate: (a) The flow rate through T (b) The flow rate through the projection of T onto the xy-plane [the triangle with vertices (0, 0, 0), (1,0,0), and (0, 1, 0)] Assume distances are in meters.
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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how do i solve the attached calculus problem?
![Let \( \mathcal{T} \) be the triangular region with vertices \( (1, 0, 0), (0, 1, 0), \) and \( (0, 0, 1) \) oriented with an upward-pointing normal vector.
A diagram shows a 3D coordinate system with the triangle \( \mathcal{T} \). The vertices of \( \mathcal{T} \) are clearly labeled on the xyz-plane, forming a triangular region. The orientation of the normal vector is indicated as pointing upwards. This triangular region is positioned within the 3D space to highlight its surface orientation.
**A fluid flows with a constant velocity field \( \mathbf{v} = 8\mathbf{i} + 2\mathbf{j} \, \text{m/s}.** Calculate:
(a) The flow rate through \( \mathcal{T} \).
(b) The flow rate through the projection of \( \mathcal{T} \) onto the \( xy \)-plane [the triangle with vertices \( (0, 0, 0), (1, 0, 0), \) and \( (0, 1, 0) \)].
Assume distances are in meters.
\[
(a) \iint_S \mathbf{v} \cdot d\mathbf{S} = \boxed{\phantom{0}}
\]
\[
(b) \iint_S \mathbf{v} \cdot d\mathbf{S} = \boxed{\phantom{0}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3cfeff4-8ba6-46a8-98d4-804b4f4f620a%2F61845a94-2a6d-4f21-961e-125488c551ed%2Fywa5dv8_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( \mathcal{T} \) be the triangular region with vertices \( (1, 0, 0), (0, 1, 0), \) and \( (0, 0, 1) \) oriented with an upward-pointing normal vector.
A diagram shows a 3D coordinate system with the triangle \( \mathcal{T} \). The vertices of \( \mathcal{T} \) are clearly labeled on the xyz-plane, forming a triangular region. The orientation of the normal vector is indicated as pointing upwards. This triangular region is positioned within the 3D space to highlight its surface orientation.
**A fluid flows with a constant velocity field \( \mathbf{v} = 8\mathbf{i} + 2\mathbf{j} \, \text{m/s}.** Calculate:
(a) The flow rate through \( \mathcal{T} \).
(b) The flow rate through the projection of \( \mathcal{T} \) onto the \( xy \)-plane [the triangle with vertices \( (0, 0, 0), (1, 0, 0), \) and \( (0, 1, 0) \)].
Assume distances are in meters.
\[
(a) \iint_S \mathbf{v} \cdot d\mathbf{S} = \boxed{\phantom{0}}
\]
\[
(b) \iint_S \mathbf{v} \cdot d\mathbf{S} = \boxed{\phantom{0}}
\]
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