Calculate the surface integral over scalar field
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Let f:$R^3$ → R, f (x, y, z) = ax + by + z, where a and b are given constants.
Consider the simple surface S=Φ(D), parameterized by Φ: D = {(u, v) | $u^2$ + $v^2$ ≤ 1} ⊂ $R^2$ → $R^3$, Φ(u, v) = (u, v, $u^2 + v^2)$. S is then the piece of the paraboloid $z = x^2 + y^2$ that lies below the plane z=1.
Calculate the surface integral of f over S.
My attempt
Ithink that i should apply this:
$int_s$= $int_D f(Φ(u, v)||left(frac{partial Φ}{partial u}right) left(frac{partial Φ}{partial v}right) || dudv $
the part of $||left(frac{partial Φ}{partial u}right) left(frac{partial Φ}{partial v}right) ||$ i alredy have it, but i don´t know how to write the first part, i mean, do i have to parametrized ax + by + z in terms of u and v? and how do i find the integration limits?
Can someone help me please? I´m stuck in this problem
I would thank you a lot
calculus integration multivariable-calculus surface-integrals
asked Nov 19 at 21:06
Enrique Gr
11
add a comment |
up vote
-1
down vote
favorite
Let f:$R^3$ → R, f (x, y, z) = ax + by + z, where a and b are given constants.
Consider the simple surface S=Φ(D), parameterized by Φ: D = {(u, v) | $u^2$ + $v^2$ ≤ 1} ⊂ $R^2$ → $R^3$, Φ(u, v) = (u, v, $u^2 + v^2)$. S is then the piece of the paraboloid $z = x^2 + y^2$ that lies below the plane z=1.
Calculate the surface integral of f over S.
My attempt
Ithink that i should apply this:
$int_s$= $int_D f(Φ(u, v)||left(frac{partial Φ}{partial u}right) left(frac{partial Φ}{partial v}right) || dudv $
the part of $||left(frac{partial Φ}{partial u}right) left(frac{partial Φ}{partial v}right) ||$ i alredy have it, but i don´t know how to write the first part, i mean, do i have to parametrized ax + by + z in terms of u and v? and how do i find the integration limits?
Can someone help me please? I´m stuck in this problem
I would thank you a lot
calculus integration multivariable-calculus surface-integrals
asked Nov 19 at 21:06
Enrique Gr
11
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let f:$R^3$ → R, f (x, y, z) = ax + by + z, where a and b are given constants.
Consider the simple surface S=Φ(D), parameterized by Φ: D = {(u, v) | $u^2$ + $v^2$ ≤ 1} ⊂ $R^2$ → $R^3$, Φ(u, v) = (u, v, $u^2 + v^2)$. S is then the piece of the paraboloid $z = x^2 + y^2$ that lies below the plane z=1.
Calculate the surface integral of f over S.
My attempt
Ithink that i should apply this:
$int_s$= $int_D f(Φ(u, v)||left(frac{partial Φ}{partial u}right) left(frac{partial Φ}{partial v}right) || dudv $
the part of $||left(frac{partial Φ}{partial u}right) left(frac{partial Φ}{partial v}right) ||$ i alredy have it, but i don´t know how to write the first part, i mean, do i have to parametrized ax + by + z in terms of u and v? and how do i find the integration limits?
Can someone help me please? I´m stuck in this problem
I would thank you a lot
calculus integration multivariable-calculus surface-integrals
asked Nov 19 at 21:06
Enrique Gr
11
Let f:$R^3$ → R, f (x, y, z) = ax + by + z, where a and b are given constants.
Consider the simple surface S=Φ(D), parameterized by Φ: D = {(u, v) | $u^2$ + $v^2$ ≤ 1} ⊂ $R^2$ → $R^3$, Φ(u, v) = (u, v, $u^2 + v^2)$. S is then the piece of the paraboloid $z = x^2 + y^2$ that lies below the plane z=1.
Calculate the surface integral of f over S.
My attempt
Ithink that i should apply this:
$int_s$= $int_D f(Φ(u, v)||left(frac{partial Φ}{partial u}right) left(frac{partial Φ}{partial v}right) || dudv $
the part of $||left(frac{partial Φ}{partial u}right) left(frac{partial Φ}{partial v}right) ||$ i alredy have it, but i don´t know how to write the first part, i mean, do i have to parametrized ax + by + z in terms of u and v? and how do i find the integration limits?
Can someone help me please? I´m stuck in this problem
I would thank you a lot
calculus integration multivariable-calculus surface-integrals
calculus integration multivariable-calculus surface-integrals
asked Nov 19 at 21:06
Enrique Gr
11
asked Nov 19 at 21:06
Enrique Gr
11
asked Nov 19 at 21:06
Enrique Gr
11
asked Nov 19 at 21:06
Enrique Gr
11
asked Nov 19 at 21:06
Enrique Gr
11
11
add a comment |
add a comment |
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