anonymous
  • anonymous
Calculate the normal component of F to the surface. (Calc 3)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Let F = and Let S be the oriented surface parametrized by G(u, v) = (u2 − v, u, v2) for 0 ≤ u ≤ 9, −1 ≤ v ≤ 4. Calculate the normal component of F to the surface at P = (63, 8, 1) = G(8, 1). So I tried and got 10 / sqrt(21).
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anonymous
  • anonymous
Steps for tangential components: \[T_{u} = \frac{ dG }{ du } = (2u, 1, 0)\] \[T_{v} = \frac{ dG }{ dv } = (-1, 0, 2v)\] Normal vector: det \[\left[ 2u \right] \left[ 1 \right] \left[ 0 \right]\] \[\left[ -1 \right] \left[ 0 \right] \left[ 2v \right]\] \[n = <2v, -4uv, 1>\] F(P) = <1, 0, 8> n(P) = <2, -4, 1> \[e_{n}(P) = \frac{ n(P) }{ |n(P)| } = \frac{ <2, -4, 1> }{ \sqrt{21} }\] \[F(P) * e_{n}(P) = <1,0,8> * \frac{ 1 }{ \sqrt{21} } <2, -4, 1> = \frac{ 2 }{ \sqrt{21} } + \frac{ 8 }{ \sqrt{21} } = \frac{ 8 }{ \sqrt{21} }\]
anonymous
  • anonymous
Steps for tangential components: \[T_{u} = \frac{ dG }{ du } = (2u, 1, 0)\] \[T_{v} = \frac{ dG }{ dv } = (-1, 0, 2v)\] Normal vector: det \[\left[ 2u \right] \left[ 1 \right] \left[ 0 \right]\] \[\left[ -1 \right] \left[ 0 \right] \left[ 2v \right]\] \[n = <2v, -4uv, 1>\] F(P) = <1, 0, 8> n(P) = <2, -4, 1> \[e_{n}(P) = \frac{ n(P) }{ |n(P)| } = \frac{ <2, -4, 1> }{ \sqrt{21} }\] \[F(P) * e_{n}(P) = <1,0,8> * \frac{ 1 }{ \sqrt{21} } <2, -4, 1> = \frac{ 2 }{ \sqrt{21} } + \frac{ 8 }{ \sqrt{21} } = \frac{ 8 }{ \sqrt{21} }\]

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IrishBoy123
  • IrishBoy123
check algebra \(n(P) = <2, \color{red}{-32}, 1>\)
anonymous
  • anonymous
But shouldn't the end result be regardless of the y component of n(P)?
anonymous
  • anonymous
@IrishBoy123
IrishBoy123
  • IrishBoy123
what do you get for \(\hat n\) now?
IrishBoy123
  • IrishBoy123
what is "Calc 3", btw??
anonymous
  • anonymous
Calculus 3
anonymous
  • anonymous
Let me compute that..
anonymous
  • anonymous
\[7\sqrt{21}\]
anonymous
  • anonymous
that would be |n(P)|
anonymous
  • anonymous
n hat = \[\frac{ <2,-32,1> }{ 7\sqrt{21} }\]
IrishBoy123
  • IrishBoy123
well, that's correcting the glitch i found, assuming you agree. if that does not work, we need to look at something else. but the method is sound.
anonymous
  • anonymous
Can you please tell me how you computed the components of n(P)? I was just guessing from the solution.
anonymous
  • anonymous
With the correction the final answer is \[\frac{ 10 }{ 7\sqrt{21} }\] (verified correct)
IrishBoy123
  • IrishBoy123
well, n=<2v,−4uv,1> so, −4uv = -4(8)(1) = -32 is that what you are getting at?
anonymous
  • anonymous
That makes sense now. Was tripping on that haha. thank you

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