Determining between direct evaluation or vector theorems

In summary: F}_{S}##. Then, the Gauss divergence theorem would be more appropriate in the first case, whereas the Stokes theorem would be more appropriate in the second case.
  • #1
elements
29
0
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. Homework Statement

(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.

Homework Equations


Greens, Stokes and Gauss theorems.

The Attempt at a Solution


[/B]
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.
 
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  • #2
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.
 
  • #3
haruspex said:
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.
Thank you, in general what's 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
 
  • #4
elements said:
Thank you, in general what's 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
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?
 
  • #5
haruspex said:
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?

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##
 
  • #6
elements said:
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. Homework Statement

(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.

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##.
 
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  • #7
elements said:
(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.
I don't understand the question. This seems to be vectors lying in a 2D plane. How is there any flux through a surface?
elements said:
(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
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.
elements said:
(c) If
f=xyz​
This is just a scalar. Why do anything other integrate in the obvious way?
elements said:
(d) Compute the outward flux of
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.
 
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1. What is the difference between direct evaluation and vector theorems?

Direct evaluation is a method of calculation that involves plugging values into a mathematical equation or formula to obtain a numerical result. Vector theorems, on the other hand, involve using geometric and algebraic properties of vectors to solve problems.

2. When should I use direct evaluation and when should I use vector theorems?

Direct evaluation is generally used when dealing with simple calculations or when the properties of the vectors involved are not well-defined. Vector theorems are useful for solving more complex problems involving vectors, such as finding the magnitude or direction of a vector.

3. Can I use both direct evaluation and vector theorems in the same problem?

Yes, it is possible to use both direct evaluation and vector theorems in the same problem. In some cases, one method may be more efficient or appropriate than the other.

4. What are some common vector theorems used in scientific calculations?

Some common vector theorems used in scientific calculations include the Pythagorean theorem, the dot product, the cross product, and the vector projection theorem.

5. Are there any limitations to using either direct evaluation or vector theorems?

Direct evaluation can only be used for calculations involving numerical values, while vector theorems can also be applied to geometric and algebraic problems. Additionally, vector theorems may require a deeper understanding of vector properties and concepts, making them more challenging for some individuals.

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