Green's Theorem in 3 dimensions problem

[duki]

Homework Statement

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

Homework Equations

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?

 

[HallsofIvy]

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.

 

[duki]

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

 

[gabbagabbahey]

$x$, $y$ 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.

 

[duki]

Ok so I got:

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