Question in divergence and curl

In summary, we can conclude that the vector function ##\vec {F}(\vec {r}')## does not depend on the variables ##x, y, z##, as it is a function of the position vector ##\vec {r}'##. Therefore, its partial derivatives with respect to these variables are all equal to zero. Additionally, the equality given in the question, ##\vec {F}(\vec {r}')=\frac {1}{4\pi r r'}##, is impossible because the right hand side is not vectorial. However, the general line of thinking is correct.
  • #1
yungman
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Let ##\vec {F}(\vec {r}')## be a vector function of position vector ##\vec {r}'=\hat x x'+\hat y y'+\hat z z'##.

Question is why:
[tex]\nabla\cdot\vec {F}(\vec{r}')=\nabla\times\vec {F}(\vec{r}')=0[/tex]
I understand ##\nabla## work on ##x,y,z##, not ##x',y',z'##. But what if
[tex]\vec {F}(\vec {r}')=\frac {1}{4\pi r r'}\;\hbox { where }\; r=\sqrt{x^2+y^2+z^2}[/tex]

My answer is if ##\vec {F}(\vec {r}')=\frac {1}{4\pi r r'}## , then it is a vector function of both ##r,r'##...##\vec{F}(\vec {r},\vec {r}')## not just ##\vec {F}(\vec {r}')##. So ##\vec {F}(\vec {r}')## cannot have variable of ##x,y,z##. Am I correct?

Thanks
 
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  • #2
Yes. By
yungman said:
Let ##\vec {F}(\vec {r}')## be a vector function of position vector ##\vec {r}'=\hat x x'+\hat y y'+\hat z z'##.

we mean that it does not depend on x, y or z, so $$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial y} = \frac{\partial F}{\partial z} = 0.$$
 
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  • #3
yungman said:
LBut what if
[tex]\vec {F}(\vec {r}')=\frac {1}{4\pi r r'}\;\hbox { where }\; r=\sqrt{x^2+y^2+z^2}[/tex]

This equality is impossible because the right hand side is not vectorial.

But your general line of thinking is correct.
 
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  • #4
Thanks for verifying this.
 

FAQ: Question in divergence and curl

What is divergence and curl?

Divergence and curl are mathematical operations used in vector calculus to measure the rate of change of a vector field at a given point. Divergence measures the extent to which a vector field flows outwards or inwards from a point, while curl measures the tendency of a vector field to rotate around a point.

What is the physical significance of divergence and curl?

Divergence and curl have important physical interpretations. Divergence can be used to determine the flow of a fluid or the electric charge distribution in a region, while curl can be used to describe the rotation of a fluid or the magnetic field around a current-carrying wire.

How are divergence and curl related?

One of the most famous theorems in vector calculus, known as the Stokes' theorem, states that the circulation of a vector field around a closed loop is equal to the surface integral of the curl of the vector field over the enclosed area. This shows the close relationship between divergence and curl, as they are both used to describe the behavior of vector fields in different ways.

What are some real-life applications of divergence and curl?

Divergence and curl have a wide range of applications in fields such as fluid mechanics, electromagnetism, and thermal dynamics. They are used in the design of aircraft wings, the analysis of ocean currents, and the development of electronic devices like computers and smartphones.

How can I calculate divergence and curl in practice?

To calculate divergence, you would need to compute the partial derivatives of each component of the vector field with respect to its corresponding variable. To calculate curl, you would need to take the cross product of the gradient of the vector field and the vector itself. These calculations can be done using mathematical software programs or by hand if the vector field is simple enough.

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