Let S be the surfaced parameterized by 7 (u, v) = (3u², v, u) for 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1. A vector normal to S is T₁ × 7₂ = ( −1 ✓or). OT " 0 " 6u Suppose we want to find the flux of the vector field F(x, y, z) = ( − z, x, y) across S.
Let S be the surfaced parameterized by 7 (u, v) = (3u², v, u) for 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1. A vector normal to S is T₁ × 7₂ = ( −1 ✓or). OT " 0 " 6u Suppose we want to find the flux of the vector field F(x, y, z) = ( − z, x, y) across S.
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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![Let \( S \) be the surface parameterized by \( \vec{r}(u, v) = \langle 3u^2, v, u \rangle \) for \( 0 \leq u \leq 1 \) and \( 0 \leq v \leq 1 \).
A vector normal to \( S \) is \( \vec{r}_u \times \vec{r}_v = \langle -1, 0, 6u \rangle \).
Suppose we want to find the flux of the vector field \( \vec{F}(x, y, z) = \langle -z, x, y \rangle \) across \( S \).
In order to do this, we need to compute a double integral. The integrand of that double integral will be
\[
\begin{aligned}
&\bigcirc \ \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \\
&\bigcirc \ \vec{F}(\vec{r}(u, v)) \cdot \vec{r}_v \\
&\bigcirc \ \vec{F}(\vec{r}(u, v)) \cdot (\vec{r}_u \times \vec{r}_v) \quad \text{(selected)} \\
&\bigcirc \ \vec{F}(\vec{r}(u, v)) \cdot \vec{r}_u \\
&\bigcirc \ \vec{F}(\vec{r}(t)) \cdot \vec{n}(t)
\end{aligned}
\]
Thus, the flux of the vector field \( \vec{F} \) across the surface \( S \) is
\[
\int_0^1 \int_0^1 \quad dudv
\]
\[
=
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09d105f3-6c69-4cbc-8997-6988f1733e6f%2F1d5f20a2-c279-40a2-be37-f28947ea10bc%2Fouwlpjq_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( S \) be the surface parameterized by \( \vec{r}(u, v) = \langle 3u^2, v, u \rangle \) for \( 0 \leq u \leq 1 \) and \( 0 \leq v \leq 1 \).
A vector normal to \( S \) is \( \vec{r}_u \times \vec{r}_v = \langle -1, 0, 6u \rangle \).
Suppose we want to find the flux of the vector field \( \vec{F}(x, y, z) = \langle -z, x, y \rangle \) across \( S \).
In order to do this, we need to compute a double integral. The integrand of that double integral will be
\[
\begin{aligned}
&\bigcirc \ \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \\
&\bigcirc \ \vec{F}(\vec{r}(u, v)) \cdot \vec{r}_v \\
&\bigcirc \ \vec{F}(\vec{r}(u, v)) \cdot (\vec{r}_u \times \vec{r}_v) \quad \text{(selected)} \\
&\bigcirc \ \vec{F}(\vec{r}(u, v)) \cdot \vec{r}_u \\
&\bigcirc \ \vec{F}(\vec{r}(t)) \cdot \vec{n}(t)
\end{aligned}
\]
Thus, the flux of the vector field \( \vec{F} \) across the surface \( S \) is
\[
\int_0^1 \int_0^1 \quad dudv
\]
\[
=
\]
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