# Green's Theorem in 3 dimensions problem

1. May 12, 2009

### duki

1. The problem statement, all variables and given/known data

Evaluate: $$\int _C{xydx - yzdy + xzdz}$$
C: $$\vec{r}(t) = t\vec{i} + t^2\vec{j} + t^4\vec{k}$$
o <= t <= 1

2. Relevant equations

3. The attempt at a solution

I understand that you cannot use Green's Theorem in 3 dimensions. How else can I go about solving this?

Last edited by a moderator: May 12, 2009
2. May 12, 2009

### HallsofIvy

Staff Emeritus
Re: Curve

Just use the standard definition of a "line integral".

On C, x= t, y= t2, and z= t4
dx= dt, dy= 2tdt, and dz= 4t3.

xydx- yzdy+ xzdz= (t)(t2)dt- (t2)(t4)(2tdt)+ (t)(t4)(4t3dt.

Integrate that from t= 0 to t= 1.

3. May 12, 2009

### duki

Re: Curve

Ohhh snap. I remember talking about the fundamental theorem of line integrals now. Thanks!

4. May 12, 2009

### gabbagabbahey

Re: Curve

$x$, $y$ and $z$ will all be functions of $t$ along the curve....You should be able to simply look at the parametric equation for your curve $\vec{r}(t) = t\hat{i} + t^2\hat{j} + t^4\hat{k}$ and read off what those functions are.....Then you can easily express $dx$, $dy$ and $dz$ in terms of $dt$ and you integral will simply be a single variable integration...

EDIT: Halls beat me to it.

5. May 12, 2009

### duki

Re: Curve

Ok so I got:

EDIT:
$$\int _0^1{4t^4 dt} = \frac{4t^5}{5} => \frac{4}{5}$$ ?

Last edited: May 12, 2009