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Math Help - Multi-Variable Calculus (Green's Thm, Stoke's Thm)

  1. #1
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    Multi-Variable Calculus (Green's Thm, Stoke's Thm)

    Hey everyone, I just had a few questions I ran into studying and need help!

    Here they are:

    1: Using spherical coordinates, find parametric equations for the part of the surface x^2 + y^2 + z^2 = 4 between the planes z = 0 and z = 1

    2: Compute the surface area of the surface T described by r[vector] (u,v) = <u^2, u*sin(v), u*cos(v)> and 1<=u<=2 and 0<=v<=pi

    3: Let F(x,y,z) = <y, xz, sin(xyz)> and let T be the cone z = sqrt(x^2 + y^2) between the planes z = 0 and z = 3. Evaluate ∫_T curl F *(dot) dS

    4: Evaluate F*(dot) dr where F(x,y,z) = <xy, yz, zx> and T is the surface z = xe^y, 0<=x<=1, 0<=y<=1 with downward orientation.

    5: Let F(x,y,z) = <z, 5x, e^z>, and let C be the curve given by the parametric equation r(t) = <cos(t), sin(t), 3 - cos(t) + 2sin(t)>, 0 <=t<= 2pi. Use Stokes' Thm to evaluate _c F*(dot) dr. (Hint. The curve C lies in the plane z = 3 - x + 2y)

    Thanks and by no means do all of these need answering...as much would be nice...the most important thing for me is that I understand them so a step by step explanation would be so much more helpful as opposed to a "it is (answer) by using (blah formula)" answer.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by woojae2001 View Post
    Hey everyone, I just had a few questions I ran into studying and need help!

    Here they are:

    1: Using spherical coordinates, find parametric equations for the part of the surface x^2 + y^2 + z^2 = 4 between the planes z = 0 and z = 1

    2: Compute the surface area of the surface T described by r[vector] (u,v) = <u^2, u*sin(v), u*cos(v)> and 1<=u<=2 and 0<=v<=pi

    3: Let F(x,y,z) = <y, xz, sin(xyz)> and let T be the cone z = sqrt(x^2 + y^2) between the planes z = 0 and z = 3. Evaluate ∫_T curl F *(dot) dS

    4: Evaluate F*(dot) dr where F(x,y,z) = <xy, yz, zx> and T is the surface z = xe^y, 0<=x<=1, 0<=y<=1 with downward orientation.

    5: Let F(x,y,z) = <z, 5x, e^z>, and let C be the curve given by the parametric equation r(t) = <cos(t), sin(t), 3 - cos(t) + 2sin(t)>, 0 <=t<= 2pi. Use Stokes' Thm to evaluate _c F*(dot) dr. (Hint. The curve C lies in the plane z = 3 - x + 2y)

    Thanks and by no means do all of these need answering...as much would be nice...the most important thing for me is that I understand them so a step by step explanation would be so much more helpful as opposed to a "it is (answer) by using (blah formula)" answer.
    For the first one, what have you tried, do you see that we can parametrize this as

    x^2+y^2+z^2=2^2\Rightarrow{r=2}

    Is that all you were asking? Or were you asking for a parametrization by two independent variables?

    Such as

    x=2\cos(v)\sin(u)

    y=2\sin(v)\sin(u)

    z=2\cos(u)

    ?

    So that

    x^2+y^2+z^2=(2\cos(v)\sin(u))^2+(2\sin(v)\sin(u))^  2+(2\cos(u))^2

    ............. =4\cos^2(v)\sin^2(u)+4\sin^2(v)\sin^2(u)+4\cos^2(u  )

    ............. =4\sin^2(u)(\cos^2(v)+\sin^2(v))+4\cos^2(u)

    ............. =4\sin^2(u)+4\cos^2(u)=4?
    Last edited by CaptainBlack; June 27th 2008 at 12:27 AM.
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