Divergence homework problem help

In summary, the divergence of r (cap)/ |r|^2 being equal to zero was discussed in the conversation. It was explained that this can be solved by using the equation r(cap)= x(cap)+y(cap)+z(cap) and |r|^2 as x^2+y^2+z^2. However, the person trying to solve the problem ended up with a different solution and asked for help in finding where they went wrong. The expert requested for the calculation that led to the wrong answer to pinpoint the error, and provided guidance in approaching the problem correctly.
  • #1
manimaran1605
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Homework Statement



How divergence of r (cap)/ |r|^2 is equal to zero?



Homework Equations



r(cap)= x(cap)+y(cap)+z(cap)

|r|^2 as x^2+y^2+z^2


The Attempt at a Solution



I tried the problem and end up with with a different solution I took r(cap)= x(cap)+y(cap)+z(cap)
|r|^2 as x^2+y^2+z^2. Tell me where i went wrong?

where r(cap) is unit vector of r vector
|r|^2 is the square of modulus of r
 
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  • #2
Could you please provide the calculation that lead you to the wrong answer? It is hard to pinpoint where you went wrong otherwise.
 
  • #3
You didn't differentiate correctly. Show your work.
 
  • #4
Here's my attempt to the problem
 

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  • #5
Well ##\mathbf{\hat r} = \mathbf{\hat x} + \mathbf{\hat y} + \mathbf{\hat z}## seems false. You have ##\mathbf{r} = (x,y,z)##. Work from there.
 
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1. What is the definition of divergence?

Divergence is a mathematical concept that describes the spread of a vector field. It measures the rate of change of a vector field at a given point and indicates whether the vectors are moving away from or towards that point.

2. How is divergence calculated?

Divergence is calculated using the divergence operator, which is represented by the symbol ∇·. It is applied to a vector field, and the resulting value is a scalar. The formula for divergence is: ∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z, where F is the vector field and x, y, and z are the coordinates.

3. What is the physical interpretation of divergence?

The physical interpretation of divergence depends on the context in which it is used. In fluid dynamics, for example, it represents the rate of change of the density of a fluid at a given point. In electricity and magnetism, it describes the flow of electric and magnetic fields. In general, divergence indicates the presence or absence of a source or sink at a particular point in a vector field.

4. How is divergence related to other vector operations?

Divergence is related to other vector operations, such as curl and gradient, through the fundamental theorem of calculus for vector fields. This theorem states that the divergence of a vector field is equal to the scalar field obtained by integrating the dot product of the vector field with the differential volume element. In other words, divergence is the flux of a vector field over a closed surface.

5. What are some real-world applications of divergence?

Divergence has numerous applications in physics, engineering, and other fields. It is used in fluid dynamics to analyze fluid flow and in electromagnetism to study electric and magnetic fields. It is also used in computer graphics to create realistic motion in animated objects. In addition, divergence is used in economics to measure the flow of goods and services between different regions.

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