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

In summary, the problem asks for the area of a region bounded by a outward pointing unit normal and a surface. The notation used is unfamiliar and the problem does not seem to make sense.
  • #1
ElDavidas
80
0
Again, I'm stuck on a question:

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

[tex] v = (y^3 -2xy, y^2+3y+2zy, z-z^2) [/tex].

Evaluate [tex]\int_{\partial C} v . dA[/tex]

Stating clearly any result used"

Thanks in advance
 
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  • #2
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.
 
  • #3
Benny said:
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
 
  • #4
If it's the surface integral over the cube then it should be

[tex]
\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 } } }
[/tex]

Where the terminals of the triple integral go from -1 to 1 for each of x,y and z.
 
  • #5
Benny said:
[tex]
\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 } } }
[/tex]

Sorry but I don't understand this notation. Are you integrating the gradient of v?
 
  • #6
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.
 

What is Integral Q with vector field?

Integral Q with vector field is a mathematical concept that combines integral calculus with vector field analysis. It involves calculating the line integral of a vector field along a given curve or surface.

How is Integral Q with vector field used in science?

Integral Q with vector field is used in many fields of science, including physics, engineering, and biology. It can help determine the work done by a force, the flow of a fluid, or the circulation of a magnetic field.

What is the difference between integral Q and ordinary integrals?

The main difference between integral Q and ordinary integrals is that integral Q involves the integration of a vector field, while ordinary integrals involve the integration of a scalar function. Integral Q also requires the use of a parametrized curve or surface.

What is the significance of the vector field in integral Q?

The vector field in integral Q represents a physical quantity, such as force or velocity, that varies at different points in space. By integrating along a curve or surface, we can calculate the overall effect of the vector field over that region.

What are some real-life applications of integral Q with vector field?

Integral Q with vector field has many practical applications, such as calculating the work done by a varying force, analyzing fluid flow patterns, and determining the magnetic field around a current-carrying wire. It is also used in computer graphics to create realistic visual effects.

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