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

• Jun 26th 2008, 09:50 AM
woojae2001
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.
• Jun 26th 2008, 10:09 AM
Mathstud28
Quote:

Originally Posted by woojae2001
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

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

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

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

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