Without using the formula (Gauss), calculate E∫∇.F(x)dx where F(x) = x32k, E :x12+x22≤ x3, 0≤x3≤1 (that is, E is the solid bounded by the surfaces x12 + x22=x3 e x3 = 1)
Without using the formula (Gauss), calculate E∫∇.F(x)dx where F(x) = x32k, E :x12+x22≤ x3, 0≤x3≤1 (that is, E is the solid bounded by the surfaces x12 + x22=x3 e x3 = 1)
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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Without using the formula (Gauss), calculate E∫∇.F(x)dx where F(x) = x32k, E :x12+x22≤ x3, 0≤x3≤1 (that is, E is the solid bounded by the surfaces x12 + x22=x3 e x3 = 1)
___________________________________
Given:
In the formulas below E, S and l always denote a solid, a surface, and a line, respectively. While n(x) denotes the normal unitary exterior of S in x, and T(x) denotes the unitary tangent of l in x. (given image with formulas)
![1. S: g(x) = 0, n(x) = ±T E, l: y(t), t e [a, b], T(x) =FOT
Vg(z)
= 7(t).
%3!
2. |n(x).k|ds = dr,dr2, |n(x).j|ds = dxıdr3, |n(x).i|ds = drzdr3, i = (1,0,0), j = (0, 1, 0), k =
(0,0, 1), .
3. T(x).idl = dx1, T(x).jdl = dx2, T(x).kdl = dr3, donde dl = ||r'(t)||dt, .
4. (Gauss)ſ V.F(x)dx = S F(x).n(x)ds, onde dr = dx,dx,dr3, V = i+ +k
E
S=DE
5. (Stokes) (V x F(x)).n(x)ds = S F(x).T(x)dl, F(x) = F;(x)i+ F2(x)j + F3(x)k.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0944c0f9-7a1b-44ef-bcc7-13fa55cc7eb0%2F9f27fce9-df17-4b48-81d6-d3a81c2ce860%2Flg36s1d_processed.png&w=3840&q=75)
Transcribed Image Text:1. S: g(x) = 0, n(x) = ±T E, l: y(t), t e [a, b], T(x) =FOT
Vg(z)
= 7(t).
%3!
2. |n(x).k|ds = dr,dr2, |n(x).j|ds = dxıdr3, |n(x).i|ds = drzdr3, i = (1,0,0), j = (0, 1, 0), k =
(0,0, 1), .
3. T(x).idl = dx1, T(x).jdl = dx2, T(x).kdl = dr3, donde dl = ||r'(t)||dt, .
4. (Gauss)ſ V.F(x)dx = S F(x).n(x)ds, onde dr = dx,dx,dr3, V = i+ +k
E
S=DE
5. (Stokes) (V x F(x)).n(x)ds = S F(x).T(x)dl, F(x) = F;(x)i+ F2(x)j + F3(x)k.
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