Finding the Curl of a Vector Field

In summary, the question is asking to find the circulation of a vector field around a curve using the formula \operatorname{circ}_C(\vec G) = \iint_A \operatorname{curl}(\vec G) \, \mathrm d\vec a. The student attempted to find the curl, but found y-3 which is not a constant. The correct approach is to integrate over the area of the disk from y = -2 to y = 2, resulting in a curl of -4.
  • #1
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Homework Statement



http://img5.imageshack.us/img5/8295/capturewmw.th.jpg

Homework Equations





The Attempt at a Solution



I tried to find the curl first and what i got is y - 3 and then I multiply that by the area of the circle which is 4pi.. am I doing something wrong?
 
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  • #2
The circulation of a vector field [itex]\vec G[/itex] around a curve C is given by
[tex]\operatorname{circ}_C(\vec G) = \iint_A \operatorname{curl}(\vec G) \, \mathrm d\vec a[/tex]
Since the curl is not a constant on the disk, the integral is not as trivial as integrand * surface area.

You could have easily seen that your answer is wrong because 4pi(y - 2) still depends on y, while it should be a number.
 
  • #3
well..how do I get around to solve this? I know the curl is y-3...
 
  • #4
If you have to solve this question I assume you have learned how to integrate a function over some area.

I suggest integrating from y = -2 to y = 2 so that the x integral will be trivial (you only need to worry about the integration boundaries):

[tex]\operatorname{circ}_{C}(\vec G) \propto \int_{-2}^{2} \int_{\cdots}^{\cdots} (y - 2) \, dx \, dy[/tex]
up to some proportionality factors... see the image below.

I hope that I have provided you with enough clues to solve the question now...
 

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  • #5
so the curl is -4? I don't get it why it's -2 to 2
 

Related to Finding the Curl of a Vector Field

1. What is a vector field?

A vector field is a mathematical concept used to describe the behavior of vector quantities, such as velocity or force, at every point in space. It consists of a set of vectors, each associated with a specific point in space, that represent the magnitude and direction of the quantity at that point.

2. Why is it important to find the curl of a vector field?

The curl of a vector field provides information about the circulation or rotation of the vector field at each point in space. This is useful in many fields of science and engineering, such as fluid dynamics, electromagnetism, and mechanics, where understanding the behavior of a vector field is crucial.

3. How is the curl of a vector field calculated?

The curl of a vector field is calculated using a mathematical operation known as the cross product. This involves taking the partial derivatives of the components of the vector field with respect to the three spatial dimensions and combining them in a specific way to obtain a new vector field representing the curl.

4. What are the steps for finding the curl of a vector field?

The steps for finding the curl of a vector field are as follows:

  1. Take the partial derivatives of the components of the vector field with respect to the three spatial dimensions.
  2. Rearrange the partial derivatives using the cross product formula.
  3. Simplify the resulting expression to obtain the components of the curl vector field.

5. What are some applications of finding the curl of a vector field?

Finding the curl of a vector field has many practical applications, such as:

  • In fluid dynamics, the curl can be used to calculate the vorticity of a fluid flow, which can help predict the formation of eddies and other complex flow patterns.
  • In electromagnetism, the curl of the electric and magnetic fields can be used to determine the strength and direction of electromagnetic forces.
  • In mechanics, the curl of a force field can be used to calculate the torque exerted on an object, which is important in understanding rotational motion.

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