Verify that the divergence theorem is true for the vector field F on the region E. F(x, y, z) = (z, y, x); E is the solid ball x2 + y2 + z² s 25 First compute the divergence of F: div F = rexre = and F(r(p, 8)) = and JS.F. a Ә -(z) + ax ay J.F. F. (rxr) da - 1.³t [²² -(y) + S is a sphere of radius 5 centered at the origin which can be parametrized by r(,0) = (5 5 sin(p) cos(0), 5 sin(p) sin(0), 5 cos(p)), 0 spsx, 0 s0s 2x (similar to this example). Then -(so 5 cos(p) cos(8), 5 cos(p) sin(8), -5 sin(p)) x (-5 sin(p) sin(e), 5 sin(p) cos(6), - (25 sin³²(p) cos(6), 25 sin²(p) sin(0), [ 1) = (5 cos(p), 5 sin(p) sin(0), 5 sin(p) cos(0)). Thus, F. (rxr) 125 cos(p) sin2 (p) cos(0) + 125 sin³ (p) sin²(0)+ = 250 cos(p) sin²(p) cos(0) + dS= a (x): (250 cos(p) sin(p) cos(0) + (2x [250 in ³(m) cos(A) + ( , so II div sin³(p) sin² (6) div F dv= cos(p) sin(p) cos(6) MAC sin³(p) sin2 (0)) de de (¹ cos³(p) - cos(p)) sin²(e) de dV = V(E). This calculates to the exact value:
Verify that the divergence theorem is true for the vector field F on the region E. F(x, y, z) = (z, y, x); E is the solid ball x2 + y2 + z² s 25 First compute the divergence of F: div F = rexre = and F(r(p, 8)) = and JS.F. a Ә -(z) + ax ay J.F. F. (rxr) da - 1.³t [²² -(y) + S is a sphere of radius 5 centered at the origin which can be parametrized by r(,0) = (5 5 sin(p) cos(0), 5 sin(p) sin(0), 5 cos(p)), 0 spsx, 0 s0s 2x (similar to this example). Then -(so 5 cos(p) cos(8), 5 cos(p) sin(8), -5 sin(p)) x (-5 sin(p) sin(e), 5 sin(p) cos(6), - (25 sin³²(p) cos(6), 25 sin²(p) sin(0), [ 1) = (5 cos(p), 5 sin(p) sin(0), 5 sin(p) cos(0)). Thus, F. (rxr) 125 cos(p) sin2 (p) cos(0) + 125 sin³ (p) sin²(0)+ = 250 cos(p) sin²(p) cos(0) + dS= a (x): (250 cos(p) sin(p) cos(0) + (2x [250 in ³(m) cos(A) + ( , so II div sin³(p) sin² (6) div F dv= cos(p) sin(p) cos(6) MAC sin³(p) sin2 (0)) de de (¹ cos³(p) - cos(p)) sin²(e) de dV = V(E). This calculates to the exact value:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
 - cos(4)
sin³ (p) sin²(0)) de de
div F dv=
- C
cos(p) sin²(p) cos(6)
(p)) sin² (6)]* de
])ov.
dV= V(E). This calculates to the exact value:
calcPad](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb29ff817-7b8d-4056-886e-42f43ca6da22%2Fba50bd25-9541-47d1-b871-bce651aba735%2Fdhy74tt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Verify that the divergence theorem
true for the vector field F on the region E.
F(x, y, z) = (z, y, x); E is the solid ball x² + y² + z² s 25
@_(z) + 2/(r) +
Əx
First compute the divergence of F: div F =
and
1₁ F. ds = [[F. (₂x) da
- 1.² 1.²
- 12²-1250
2 500
-(x) + ²(x)
(250 cos(p) sin(p) cos(0) +
250 sin³ (p) cos(6) +
=
sin³(4):
sin³ (p) sin²(0)
S is a sphere of radius 5 centered at the origin which can be parametrized by r(9, 0) = (5 sin(p) cos(8), 5 sin(p) sin(0), 5 cos(p)), 0 s p ≤n, 0 ≤ 0 ≤ 27 (similar to this example). Then
roxre =
= (5₁
5 cos(p) cos(6), 5 cos(p) sin(0), -5 sin(p)) x (-5 sin(p) sin(0), 5 sin(p) cos(8),
(25 sin2 (p) cos(6), 25 sin² (p) sin(0),
and F(r(p, 8)) = (5 cos(p), 5 sin(p) sin(e), 5 sin(p) cos(e)). Thus,
F. (rxre) = 125 cos(p) sin2 (p) cos(0) + 125 sin³ (p) sin²(e) +
= 250 cos(p) sin² (p) cos(0) +
](/co
, so
III.
-cos³ (p) - cos(4)
sin³ (p) sin²(0)) de de
div F dv=
- C
cos(p) sin²(p) cos(6)
(p)) sin² (6)]* de
])ov.
dV= V(E). This calculates to the exact value:
calcPad
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