Solution Found!
Answer: Computing flux Use the Divergence Theorem to
Chapter 14, Problem 21E(choose chapter or problem)
Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S.
\(\mathbf{F}=\left\langle y-2 x, x^{3}-y, y^{2}-z\right\rangle\); S is the sphere \(\left\{(x, y, z): x^{2}+y^{2}+z^{2}=4\right\}\)
Text Transcription:
F = langle y - 2x, x^3 - y, y^2 - z rangle
{(x, y, z): x^2 + y^2 + z^2 = 4}
Questions & Answers
QUESTION:
Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S.
\(\mathbf{F}=\left\langle y-2 x, x^{3}-y, y^{2}-z\right\rangle\); S is the sphere \(\left\{(x, y, z): x^{2}+y^{2}+z^{2}=4\right\}\)
Text Transcription:
F = langle y - 2x, x^3 - y, y^2 - z rangle
{(x, y, z): x^2 + y^2 + z^2 = 4}
ANSWER:Solution 21E
Divergence Theorem represents the volume density of the outward flux of a vector field F = 〈y - 2x, - y,
- z〉 with surface region bounded by the sphere {(x, y, z):
+
+
= 4}
Divergence Theorem : (F.n)ds =
div F dv
Closed surface triple
integral integral
where D: a closed and bounded region
S: boundary surface of D oriented outward direction
n: normal to surface
Step 1:
First solve the divergence of F () =
〈y - 2x,
- y,
- z〉
Vector form of F= 〈(y - 2x) i, - y) j,
- z )k〉
F = i+
j+
k
We know +
+
)
=+
+
=( -2) +(-1)+(-1)
= -4