(d) Find the flux of F across S, the surface of C, given that F(x, y, z) = (2x+cosh(5yz)] i+ [zxe¯* − 6z]j + [1 + xy²]k (e) What is the flux of F across the surface of a spherical body if F(x, y, z) = A k, where A is a real and negative constant?
(d) Find the flux of F across S, the surface of C, given that F(x, y, z) = (2x+cosh(5yz)] i+ [zxe¯* − 6z]j + [1 + xy²]k (e) What is the flux of F across the surface of a spherical body if F(x, y, z) = A k, where A is a real and negative constant?
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
100%
![Components
equal mass.
B and C have different geometries and density but must be designed to have
Component B occupies the region outside the sphere r= 2 cos(p) and inside
the sphere r= 2 with $ € [0, π/2]. Assume B has a uniform density of PB.
• Component C is a curved wedge that lies inside the region enclosed by the
cylinder (x-2)²+y² =4 and the planes z=0 and z=-y. Assume C has a uniform
density of pc.
(b) Sketch B and C.
(c) Find the volume of B and C, and the ratio PB/PC to achieve equal mass.
(d) Find the flux of F across S, the surface of C, given that
F(x, y, z) = [2x+cosh(5yz)] i+ [zxe-*- 6z] j + [1 + xy²] k
(e) What is the flux of F across the surface of a spherical body if F(x, y, z) = A k, where A
is a real and negative constant?
Solutions of (b) and (c) are given as references, please just solve for question (d)
and (e)
(b):
-2.0
0.3 0.0
as
x
(C)VR= 4T,
20
25
2.0
VC=
2.5
16
3
.
-1.5
-2.0
0.0
-0.5 Y
-1.0
PB
PC
1.0
0.5
2.0
1.5
1.5
1.0
as
+0.0
+0.5
2.0
-1.0
1.5
+-2.0
Z
16
12T
4
3π](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feaff9b52-1faa-46f9-b131-4c9ee1e61991%2F1fb30707-5207-4e2f-bb1b-902e1a7f0d28%2F6wkpyik_processed.png&w=3840&q=75)
Transcribed Image Text:Components
equal mass.
B and C have different geometries and density but must be designed to have
Component B occupies the region outside the sphere r= 2 cos(p) and inside
the sphere r= 2 with $ € [0, π/2]. Assume B has a uniform density of PB.
• Component C is a curved wedge that lies inside the region enclosed by the
cylinder (x-2)²+y² =4 and the planes z=0 and z=-y. Assume C has a uniform
density of pc.
(b) Sketch B and C.
(c) Find the volume of B and C, and the ratio PB/PC to achieve equal mass.
(d) Find the flux of F across S, the surface of C, given that
F(x, y, z) = [2x+cosh(5yz)] i+ [zxe-*- 6z] j + [1 + xy²] k
(e) What is the flux of F across the surface of a spherical body if F(x, y, z) = A k, where A
is a real and negative constant?
Solutions of (b) and (c) are given as references, please just solve for question (d)
and (e)
(b):
-2.0
0.3 0.0
as
x
(C)VR= 4T,
20
25
2.0
VC=
2.5
16
3
.
-1.5
-2.0
0.0
-0.5 Y
-1.0
PB
PC
1.0
0.5
2.0
1.5
1.5
1.0
as
+0.0
+0.5
2.0
-1.0
1.5
+-2.0
Z
16
12T
4
3π
Transcribed Image Text:1. Change of Variable of Integration in 2D
[ f(x,y) drdy = f(z(u, v),y(u, 0)). («, ») ducho
2. Transformation to Polar Coordinates
The useful formulas
3. Change of Variable of Integration in 3D
[, (2, 2) dadydz = [[F(w, x, w)|J(u, v, w)| dududw
4. Transformation to Cylindrical Coordinates
z=rcos, y=rsin, Jr.)=r
6. Line Integrals
5. Transformation to Spherical Coordinates
x=rcos 0, y = rsin, ===, J(r,0,2)=r
x=rcos@sind, y=rsin@sind, 2=rcoso, Jr.,0,0)=²sin
7. Work Integrals
[1(x, y, z) ds = [ f(x(t), y(t),= (t)) √√x²(1)² + y′(t)² + 2′(t)²³ dt
8. Surface Integrals
[F(x, y, z) - dx = [° R² + R$/ + d
dr
dt
[ 92,9,2) ds = [[ 9(2.9.1(2.9)) √ 12 + 12 + 1 dady
9. Flux Integrals For a surface with upward unit normal,
J.P.
11. Stokes' Theorem
= [[₁-Fife - Faly + Pa dyda
F-nds=
10. Gauss' (Divergence) Theorem
JIL V. FdV =
dv = [[₁, F
[[
F. ÂdS
(V x F). ÂdS=
s = [ F..
F.dr
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