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: −3≤x≤5,−5≤y≤7,−4≤z≤7−3≤x≤5,−5≤y≤7,− 4≤z≤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 0≤s≤10≤s≤1 and 0≤t≤10≤t≤1:
The face with z = -4 : σ1=(x1(s),y1(t),z1(s, t))σ1=(x1(s),y1(t),z1(s, t))
x1(s)=x1(s)=
y1(t)=y1(t)=
z1(s, t)=−4z1(s, t)=−4
The face with z = 7 : σ2=(x2(s),y2(t),z2(s, t))σ2=(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 : σ3=x3(s, t),y3(s),z3(t))σ3=x3(s, t),y3(s),z3(t))
x3(s, t)=−3x3(s, t)=−3
y3(s)=y3(s)=
z3(t)=z3(t)=
The face with x = 5 : σ4=(x4(s, t),y4(s),z4(t))σ4=(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 : σ5=(x5(s),y5(s, t),z5(t))σ5=(x5(s),y5(s, t),z5(t))
x5(s)=x5(s)=
y5(s, t)=−5y5(s, t)=−5
z5(t)=z5(t)=
The face with y = 7 : σ6=(x6(s),y6(s, t),z6(t))σ6=(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)
∫∫σF⋅ndS∫∫σF⋅ndS =∫10∫10F(σ1)⋅(∂σ1∂t×∂σ1∂s)dsdt=∫01∫ 01F(σ1)⋅(∂σ1∂t×∂σ1∂s)dsdt +∫10∫10F(σ2)⋅(∂σ2∂s×∂σ2∂t)dsdt+∫01∫ 01F(σ2)⋅(∂σ2∂s×∂σ2∂t)dsdt +∫10∫10F(σ3)⋅(∂σ3∂t×∂σ3∂s)dsdt+∫01∫ 01F(σ3)⋅(∂σ3∂t×∂σ3∂s)dsdt +∫10∫10F(σ4)⋅(∂σ4∂s×∂σ4∂t)dsdt+∫01∫ 01F(σ4)⋅(∂σ4∂s×∂σ4∂t)dsdt +∫10∫10F(σ5)⋅(∂σ5∂s×∂σ5∂t)dsdt+∫01∫ 01F(σ5)⋅(∂σ5∂s×∂σ5∂t)dsdt +∫10∫10F(σ6)⋅(∂σ6∂t×∂σ6∂s)dsdt+∫01∫ 01F(σ6)⋅(∂σ6∂t×∂σ6∂s)dsdt
== + + + + +
==
Answers: 1
Mathematics, 21.06.2019 17:00, KHaire2077
In a sample of 2023 u. s. adults, 373 said franklin roosevelt was the best president since world war ii. two u. s. adults are selected at random from the population of all u. s. adults without replacement. assuming the sample is representative of all u. s. adults, complete parts (a) through (d). (a) find the probability that both adults say franklin roosevelt was the best president since world war ii. the probability that both adults say franklin roosevelt was the best president since world war ii is (round to three decimal places as needed.) (b) find the probability that neither adult says franklin roosevelt was the best president since world war ii. the probability that neither adult says franklin roosevelt was the best president since world war ii is (round to three decimal places as needed.) (c) find the probability that at least one of the two adults says franklin roosevelt was the best president since world war ii. the probability that at least one of the two adults says franklin roosevelt was the best president since world war ii is (round to three decimal places as needed.) (d) which of the events can be considered unusual? explain. select all that apply. the event in part left parenthesis a right parenthesis is unusual because its probability is less than or equal to 0.05. the event in part (b) is unusual because its probability is less than or equal to 0.05. none of these events are unusual. the event in part (c) is unusual because its probability is less than or equal to 0.05.
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Mathematics, 22.06.2019 04:00, Piercey4642
Maria has $10,000 to invest in two accounts. he decides to invest one part in an account with 5% interest and the rest in an account with 8% interest. at the end of the year he has earned $702,50 in interest.¿how much did maria invest in the 8% account?
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This problem will illustrate the divergence theorem by computing the outward flux of the vector fiel...
Mathematics, 08.09.2021 01:10