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Determining between direct evaluation or vector theorems

  1. Jul 26, 2017 #1
    So the main thing i'm wondering is given a question how do we determine whether to use one of the fundamentals theorems of vector calculus or just directly evaluate the integral, and if usage of one of the theorems is required how do we determine which one to use in the situation? Examples are the questions below

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

    (a) For a vector field ##\vec F = x^3 \vec {a_x} + x^2y \vec {a_y} +x^2z \vec {a_z}## determine the total flux
    ##\oint \vec {F} \cdot \vec {ds} ##passing through the surface of a cylinder of radius 4 and
    bounded by planes at z = 0 and z = 2.

    (b) If the flux density ##\vec D## in a region is given as ##\vec D = (2+16\rho^2)\hat k## , determine
    the total flux ##\int \vec D \cdot \vec {ds}## passing through a circular surface of radius ρ = 2
    in the xy plane.


    2. Relevant equations
    Greens, Stokes and Gauss theorems.
    3. The attempt at a solution

    For (a) I can't figure out whether I should be evaluating it with stokes theorem like such:
    $$\oint \vec {F} \cdot \vec {ds} = \iint_S ( \vec {\nabla} \times {\vec F}) \cdot |r_r \times r_\theta|dS$$
    or using Gauss divergence theorem
    $$\oint_S \vec {F} \cdot \vec {ds} = \iiint_V (\vec {\nabla} \cdot \vec F)dV$$

    where as with B the wording of it has me confused so i'm relatively unsure whether i'm supposed to just directly evaluate the surface integral as is or whether I should be using Stokes/Greens theorem to find the flux:
    $$\iint_S \vec D \cdot \vec {ds} = \iint_D \vec D \cdot \hat n \vec {dA}$$
    - direct eval in polar coordinates
    or Stokes:

    $$\int_S \vec D \cdot \vec {ds} = \oint_S (\vec {\nabla} \times \vec D) \cdot |r_\rho \times r_\phi|dS$$

    or maybe using greens theorem since its bounded in the x/y plane? - unsure how to set up the greens integral though.
     
  2. jcsd
  3. Jul 26, 2017 #2

    haruspex

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    For a), I would say both the integrand and the region to be integrated over are simpler with Gauss.
    For b), you could just do the plane integral with polar coordinates in less time than it would take to compare approaches.
     
  4. Jul 27, 2017 #3
    Thank you, in general whats the best way to determine what to do, to use a theorem or direct evaluation? and how do you decide when to use gauss vs stokes vs greens
     
  5. Jul 27, 2017 #4

    haruspex

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    I have no general rule. But then, it's not something I have much occasion to do.
    I would have thought that in most situations it is fairly quickly apparent whether one of those theorems will help.
    Do you have more examples?
     
  6. Jul 27, 2017 #5
    examples such as:
    (a) ##\vec F = xy^2 \hat i + (x^2y+y) \hat j##
    Determine the total flux through a surface of a circle with radius 3.

    (b) If ##\vec D = (2 + 16 r^2) \vec{a_z}##, calculate ##\int \vec D \cdot \vec {dS}## over a hemispherical surface
    bounded by r = 2 and 0 ≤ θ π/2

    (c) If $$f=xyz$$, evaluate$$\int fds$$ on the curved surface of a cylinder of
    radius 2 in the first quadrant and bounded by the planes z = 0 and
    z = 1.

    (d) Compute the outward flux of $$ \vec {F}(x,y,z) = \frac {{x \mathbf i + y \mathbf j + z \mathbf k}} {
    \sqrt {(x^2+y^2+z^2)}^3}$$ through the elipsloid ##4x^2+9y^2+6z^2=36##
     
  7. Jul 27, 2017 #6

    Ray Vickson

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    First of all, you need to be careful to distinguish between a line-integration element ##d\vec{s}## and a surface-integral element ##d\vec{S}##.

    In general, if you have a surface integral over a closed surface, you can try Gauss or direct integration. If you have a line integral around a closed loop you can try Stokes or direct integration.

    In part (a) you can try to use Gauss, since it fits the required scenario. In part (b) you can integrate directly; Stokes will not apply unless you can figure out an appropriate ##\vec{V}## that makes ##\nabla \times \vec{V} = \vec{D}##; then you could do the line integral ##\oint \vec V \cdot d\vec s##.
     
    Last edited: Jul 27, 2017
  8. Jul 27, 2017 #7

    haruspex

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    I don't understand the question. This seems to be vectors lying in a 2D plane. How is there any flux through a surface?
    I don't understand the definition of this hemisphere. Looks like this is in cylindrical coordinates, but the bounds describe a quarter of a cylindrical shell.
    This is just a scalar. Why do anything other integrate in the obvious way?
    It will be hard constructing the ##\vec{dS}## element, so I would go for the volume integral and Gauss.

    In refreshing myself on the subject, I came across an article which describes these theorems as extensions of ∫abF'.dx=F(b)-F(a) to higher dimensions. I can see that is true for Gauss, expressing a volume integral in terms of a net change at the boundary, but Stokes'/Green's is a little different. It effectively adds up little rotations around area elements to equate to one big rotation around the loop boundary of a 2D manifold.
     
    Last edited: Jul 27, 2017
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