![subject](/tpl/images/cats/mat.png)
Mathematics, 20.08.2021 03:40 andrew412603
This problem will illustrate the divergence theorem by computing the outward flux of the vector field F(x, y,z)=2xi+5yj+3zkF(x, y,z)=2xi+5yj+3zk across the boundary of the right rectangular prism: β3β€xβ€5,β5β€yβ€7,β4β€zβ€7β3β€xβ€5,β5β€yβ€7,β 4β€zβ€7 oriented outwards using a surface integral and a triple integral over the solid bounded by rectangular prism.
Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the prism to be positive.
Part 1 - Using a Surface Integral
First we parameterize the six faces using 0β€sβ€10β€sβ€1 and 0β€tβ€10β€tβ€1:
The face with z = -4 : Ο1=(x1(s),y1(t),z1(s, t))Ο1=(x1(s),y1(t),z1(s, t))
x1(s)=x1(s)=
y1(t)=y1(t)=
z1(s, t)=β4z1(s, t)=β4
The face with z = 7 : Ο2=(x2(s),y2(t),z2(s, t))Ο2=(x2(s),y2(t),z2(s, t))
x2(s)=x2(s)=
y2(t)=y2(t)=
z2(s, t)=7z2(s, t)=7
The face with x = -3 : Ο3=x3(s, t),y3(s),z3(t))Ο3=x3(s, t),y3(s),z3(t))
x3(s, t)=β3x3(s, t)=β3
y3(s)=y3(s)=
z3(t)=z3(t)=
The face with x = 5 : Ο4=(x4(s, t),y4(s),z4(t))Ο4=(x4(s, t),y4(s),z4(t))
x4(s, t)=5x4(s, t)=5
y4(s)=y4(s)=
z4(t)=z4(t)=
The face with y = -5 : Ο5=(x5(s),y5(s, t),z5(t))Ο5=(x5(s),y5(s, t),z5(t))
x5(s)=x5(s)=
y5(s, t)=β5y5(s, t)=β5
z5(t)=z5(t)=
The face with y = 7 : Ο6=(x6(s),y6(s, t),z6(t))Ο6=(x6(s),y6(s, t),z6(t))
x6(s)=x6(s)=
y6(s, t)=7y6(s, t)=7
z6(t)=z6(t)=
Then (mind the orientation)
β«β«ΟFβ
ndSβ«β«ΟFβ
ndS =β«10β«10F(Ο1)β
(βΟ1βtΓβΟ1βs)dsdt=β«01β« 01F(Ο1)β
(βΟ1βtΓβΟ1βs)dsdt +β«10β«10F(Ο2)β
(βΟ2βsΓβΟ2βt)dsdt+β«01β« 01F(Ο2)β
(βΟ2βsΓβΟ2βt)dsdt +β«10β«10F(Ο3)β
(βΟ3βtΓβΟ3βs)dsdt+β«01β« 01F(Ο3)β
(βΟ3βtΓβΟ3βs)dsdt +β«10β«10F(Ο4)β
(βΟ4βsΓβΟ4βt)dsdt+β«01β« 01F(Ο4)β
(βΟ4βsΓβΟ4βt)dsdt +β«10β«10F(Ο5)β
(βΟ5βsΓβΟ5βt)dsdt+β«01β« 01F(Ο5)β
(βΟ5βsΓβΟ5βt)dsdt +β«10β«10F(Ο6)β
(βΟ6βtΓβΟ6βs)dsdt+β«01β« 01F(Ο6)β
(βΟ6βtΓβΟ6βs)dsdt
== + + + + +
==
![ansver](/tpl/images/cats/User.png)
Answers: 1
Another question on Mathematics
![question](/tpl/images/cats/mat.png)
Mathematics, 21.06.2019 20:00
Write the point-slope form of the line passing through (2, -12) and parallel to y=3x.
Answers: 3
![question](/tpl/images/cats/mat.png)
Mathematics, 21.06.2019 20:00
Someone answer asap for ! max recorded the heights of 500 male humans. he found that the heights were normally distributed around a mean of 177 centimeters. which statements about maxβs data must be true? a. the median of maxβs data is 250 b. more than half of the data points max recorded were 177 centimeters. c. a data point chosen at random is as likely to be above the mean as it is to be below the mean. d. every height within three standard deviations of the mean is equally likely to be chosen if a data point is selected at random.
Answers: 1
![question](/tpl/images/cats/mat.png)
Mathematics, 21.06.2019 20:30
Find the solution(s) to the system of equations. select all that apply y=x^2-1 y=2x-2
Answers: 2
![question](/tpl/images/cats/mat.png)
Mathematics, 21.06.2019 21:00
A25- foot ladder leans against a building and reaches a point of 23.5 feet above the ground .find the angle that the ladder makes with the building
Answers: 2
You know the right answer?
This problem will illustrate the divergence theorem by computing the outward flux of the vector fiel...
Questions
![question](/tpl/images/cats/biologiya.png)
Biology, 03.09.2020 23:01
![question](/tpl/images/cats/mat.png)
![question](/tpl/images/cats/mat.png)
![question](/tpl/images/cats/mat.png)
Mathematics, 03.09.2020 23:01
![question](/tpl/images/cats/mat.png)
Mathematics, 03.09.2020 23:01
![question](/tpl/images/cats/obshestvoznanie.png)
Social Studies, 03.09.2020 23:01
![question](/tpl/images/cats/mat.png)
![question](/tpl/images/cats/fizika.png)
![question](/tpl/images/cats/fizika.png)
Physics, 03.09.2020 23:01
![question](/tpl/images/cats/mat.png)
![question](/tpl/images/cats/mat.png)
![question](/tpl/images/cats/health.png)
Health, 03.09.2020 23:01
![question](/tpl/images/cats/health.png)
![question](/tpl/images/cats/mat.png)
Mathematics, 03.09.2020 23:01
![question](/tpl/images/cats/istoriya.png)
![question](/tpl/images/cats/mat.png)
![question](/tpl/images/cats/himiya.png)
![question](/tpl/images/cats/mat.png)
![question](/tpl/images/cats/biologiya.png)
![question](/tpl/images/cats/istoriya.png)
History, 03.09.2020 23:01