# NEED HELP! Very Lost!!!!! Vector Calc

• Jul 28th 2008, 03:01 PM
davidson89
NEED HELP! Very Lost!!!!! Vector Calc
I posted some of the same questions on my other post a few days back. Mr. Fantastic helped me but i want to put them on here to start afresh. Please help me understand on how to do these problems. I have to equation and the book right here but i really dont understand it. If you could help me step by step i would really appreciate it!!!!!

1) If F = (3z - sinx)i + (x^2 + e^y)j + (y^3 - cosz)k, use Stoke's theorem to evaluate integral with lower terminal C [ F.dr] where C is given by x=cost, y=sint, z=1; 0<= t <= 2pie

2) Let S be the first-octant portion of the plane x + y + z =1. Verify Stokes' Theorem for the vector field F = y^2i + z^2j + x^2k

THe part i really dont get is how to do the line integral and the conditions of each terminal. Please help
• Jul 28th 2008, 08:42 PM
mr fantastic
Quote:

Originally Posted by davidson89
[snip]
1) If F = (3z - sinx)i + (x^2 + e^y)j + (y^3 - cosz)k, use Stoke's theorem to evaluate integral with lower terminal C [ F.dr] where C is given by x=cost, y=sint, z=1; 0<= t <= 2pie

[snip]

You should know that $\displaystyle \int F \cdot dr = \int \left( F \cdot \frac{dr}{dt} \right) dt$.

On the curve:

1. $\displaystyle F = (3 - \sin ( \cos t) i + (\cos^2 t + e^{\sin t}) j + (\sin^3 t - \cos 1)$.

2. $\displaystyle \frac{dr}{dt} = (- \sin t) i + (\cos t) j + 0 k$.

Take the dot product $\displaystyle F \cdot \frac{dr}{dt}$ to get a function of t, g(t) say.

Since no surface is given in the question I don't see how Stoke's Theorem is relevant here ......

Now evaluate $\displaystyle \int_{0}^{2 \pi} g(t) \, dt$.
• Jul 28th 2008, 09:01 PM
mr fantastic
Quote:

Originally Posted by davidson89
[snip]
2) Let S be the first-octant portion of the plane x + y + z =1. Verify Stokes' Theorem for the vector field F = y^2i + z^2j + x^2k

THe part i really dont get is how to do the line integral and the conditions of each terminal. Please help

The edges of S are:

xy-plane (z = 0): x + y = 1.

xz-plane (y = 0): x + z = 1.

yz-plane (x = 0): y + z = 1.

So the closed curve is defined by joining the line segments x + y = 1, x + z = 1 and y + z = 1. Integrate F along each of these line segments.

Along the line segment x + y = 1:

1. y = 1 - x and z = 0.

2. dx + dy = 0 => dy = -dx and dz = 0.

Therefore along the line segment x + y = 1:

$\displaystyle \int F \cdot dr = \int_1^0 \left( [1-x]^2 i + 0^2 j + x^2 k \right) \cdot (dx \, i - dx \, j + 0 dz \, k) = \int_1^0 (1 - x)^2 dx = \, ......$

The integration along the other two line segments is done in a similar way.

You have to go back and extensively revise the basics.