# Converting to cylindrical and then taking div and curl

• EngageEngage
In summary, the conversation discusses converting a vector function into cylindrical coordinates and finding the divergence. The formula for divergence in cylindrical coordinates is provided and the attempt at a solution is described. It is mentioned that the i,j,k unit vectors must also be converted to the corresponding basis vectors in order to use vector operations. After correctly converting the basis vectors, the correct answer is obtained. The conversation ends with the acknowledgement that the book does not provide examples for this topic.
EngageEngage

## Homework Statement

Change to cylindrical coordinates and find the divergence

F = <x, y, 0>/(x^2 + y^2)

## Homework Equations

$$\nabla$$ . F = $$\frac{1}{\rho}$$$$\frac{\partial\rho F}{\partial\rho}$$+$$\frac{1}{\rho}$$$$\frac{\partial F}{\partial\theta}$$+$$\frac{\partial F}{\partial z}$$

## The Attempt at a Solution

i changed into cylindrical by simply having x = $$\rho$$cos$$\theta$$
y =$$\rho$$sin$$\theta$$ and x^2 +y^2 = $$\rho$$^2. THen, using the above formula I get cos(theta)/rho^2. The answer is 0 however. Can someone please tell me what i could have messed up?

Wait, must I also convert the i,j,k unit vectors to e(rho), e(theta), and ez before I can use vector operations? If i do this i get the right answer, but I'm still not sure -- my book doesn't have a single example excersise so I don't have too much to work off of. If someone could help me out that would be appreciated greatly. Thank you

EngageEngage said:
Wait, must I also convert the i,j,k unit vectors to e(rho), e(theta), and ez before I can use vector operations? If i do this i get the right answer, but I'm still not sure -- my book doesn't have a single example excersise so I don't have too much to work off of. If someone could help me out that would be appreciated greatly. Thank you

Sure. You have to convert the basis vectors as well. You should get F_r=1/r and the other components zero, so div(F)=0.

that is in fact what I got. Thank you very much for the help -- unfortunately my book offers only theory and no examples.

## 1. What is the process of converting to cylindrical coordinates?

Converting to cylindrical coordinates involves replacing the rectangular coordinates (x, y, z) with the cylindrical coordinates (r, θ, z). This can be done using the formulas r = √(x^2 + y^2), θ = tan^-1(y/x), and z = z. This allows for a more convenient way to represent points and vectors in three-dimensional space.

## 2. Why is it useful to convert to cylindrical coordinates?

Converting to cylindrical coordinates can be useful in solving problems involving cylindrical symmetry, such as in electromagnetism or fluid dynamics. It also simplifies certain calculations, such as taking derivatives or integrals in certain coordinate systems.

## 3. What is the divergence in cylindrical coordinates?

The divergence in cylindrical coordinates is a mathematical operation that measures the rate of flow of a vector field outward from a given point. In cylindrical coordinates, the divergence is given by ∇ · F = (1/r) ∂(rFr)/∂r + (1/r) ∂Fθ/∂θ + ∂Fz/∂z, where F is the vector field and r, θ, and z are the cylindrical coordinates.

## 4. How is the curl calculated in cylindrical coordinates?

The curl in cylindrical coordinates is calculated using the formula ∇ x F = (1/r) ∂(rFz)/∂θ - ∂Fθ/∂z + Fz/r, where F is the vector field and r, θ, and z are the cylindrical coordinates. This operation measures the rotation or circulation of a vector field around a given point.

## 5. What are some examples of problems where converting to cylindrical coordinates and taking div and curl is useful?

Converting to cylindrical coordinates and taking div and curl can be useful in problems involving cylindrical structures, such as a pipe or a wire, where there is cylindrical symmetry. It can also be used in problems involving fluid flow in a cylindrical container or electromagnetic fields around a cylindrical conductor. Additionally, it can be used in solving certain differential equations involving cylindrical coordinates.

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