8 practice problem Tutorial Exercise Use the Divergence Theorem to calculate the surface integral F(x, y, z) = 3xy²i + xe²j + z³k, S is the surface of the solid bounded by the cylinder y2 + z2 = 4. and the planes x = 2 and x = 4 div F = Step 1 If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that 1 [ F · ds = [ ] [ div F dv. For F(x, y, z) = 3xy²1 + xe²j + z³k we have 3(y² +2²) '[/² 3y² +3:2 [[ P-as - [ ] [ div Pav F. F Step 2 Since S bounds the cylinder y2 + z² = 4 between the planes x = -2 and x = 4, we will use cylindrical coordinates, with polar coordinates in the yz-plane. Therefore, y = r cos(0), z = r sin(0), and x = x. We, therefore, have the following. = F. ds; that is, calculate the flux of F across S. -6²61² 3²³ 37-3 [²*²* (³3r² cos²(0) + 3 (sin 6)² X dx dr de )r dx dr d
8 practice problem Tutorial Exercise Use the Divergence Theorem to calculate the surface integral F(x, y, z) = 3xy²i + xe²j + z³k, S is the surface of the solid bounded by the cylinder y2 + z2 = 4. and the planes x = 2 and x = 4 div F = Step 1 If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that 1 [ F · ds = [ ] [ div F dv. For F(x, y, z) = 3xy²1 + xe²j + z³k we have 3(y² +2²) '[/² 3y² +3:2 [[ P-as - [ ] [ div Pav F. F Step 2 Since S bounds the cylinder y2 + z² = 4 between the planes x = -2 and x = 4, we will use cylindrical coordinates, with polar coordinates in the yz-plane. Therefore, y = r cos(0), z = r sin(0), and x = x. We, therefore, have the following. = F. ds; that is, calculate the flux of F across S. -6²61² 3²³ 37-3 [²*²* (³3r² cos²(0) + 3 (sin 6)² X dx dr de )r dx dr d
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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Transcribed Image Text:8 practice problem
Tutorial Exercise
Use the Divergence Theorem to calculate the surface integral
F(x, y, z) = 3xy²i + xe²j + z³k,
S is the surface of the solid bounded by the cylinder y² + z² = 4. and the planes x = 2 and x = 4
div F =
Step 1
If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that
/ [F-as = // [ div F dv.
For F(x, y, z) = 3xy²i + xe²j + z³k we have
3y² +3:²
+
SS₁ F
div F dv
F. ds; that is, calculate the flux of F across S.
Step 2
Since S bounds the cylinder y² + z² = 4 between the planes x = -2 and x = 4, we will use cylindrical coordinates, with polar coordinates in the yz-plane.
Therefore, y = r cos(0), z = r sin(0), and x = x. We, therefore, have the following.
[ [₁ F · ds = [[[
- [ ²² 6²2 [² (3² cos²(0)+ 3 (sin 0)²
- [² TL²
37.³3
X
dx dr de
)r dx dra
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