Calculating Line Integrals with Vector Fields on a Bounded Region in 3D Space

[ElDavidas]

Again, I'm stuck on a question:

"Let C be the region in space given by $0 \leq x,y,z \leq 1$ and let $\partial C$ be the boundary of C oriented by the outward pointing unit normal. Suppose that v is the vector field given by

$$v = (y^3 -2xy, y^2+3y+2zy, z-z^2)$$.

Evaluate $$\int_{\partial C} v . dA$$

Stating clearly any result used"

Thanks in advance

 

[Benny]

Well if the integral you need to evaluate is a surface integral then just use Gauss/Divergence theorem. But in this context, your notation, more specifically the dA, is unfamiliar to me.

 

[ElDavidas]

more specifically the dA, is unfamiliar to me.

This is what I don't understand. If you have to calculate the area, why are you given x,y, and z? Unless it's the surface of the cube you have to find out. I'm not sure how to go about doing that though

 

[Benny]

If it's the surface integral over the cube then it should be

$$\int\limits_{}^{} {\int\limits_{\partial W}^{} {\mathop v\limits^ \to } } .d\mathop S\limits^ \to = \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_W^{} {\nabla \bullet \mathop v\limits^ \to } } }$$

Where the terminals of the triple integral go from -1 to 1 for each of x,y and z.

 

[ElDavidas]

$$\int\limits_{}^{} {\int\limits_{\partial W}^{} {\mathop v\limits^ \to } } .d\mathop S\limits^ \to = \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_W^{} {\nabla \bullet \mathop v\limits^ \to } } }$$

Sorry but I don't understand this notation. Are you integrating the gradient of v?

 

[Benny]

It's the divergence of v (there is a dot in between grad and v). If you are unsure about what the problem is asking then you should ask whoever set the question. If it is a textbook problem then surely there should be a related theory section with relevant formulas and explanations.