Vector Calculus - Cylindrical Co-ordinates

I'm not sure what you're trying to find the volume of with cylindrical coordinates.In summary, the conversation is about using cylindrical coordinates to find the volume of an ellipsoid with the equation ${R}^{2}+{3z}^{2}=1$. The speaker knows that the Jacobian for cylindrical coordinates is $r$ and has written out an integral, but is struggling to determine the bounds for $z$, $r$, and $\theta$ as well as what to integrate. They are asking for help understanding the method for determining the limits and the purpose of the integration.
  • #1
RigB
1
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I have a question, to use cylindrical coordinates to find the volume of the ellipsoid ${R}^{2}+{3z}^{2}=1$.

I know for cylindrical coordinates the Jacobian is $r$ so I have some integral:

$$\iiint (r)dzdrd\theta$$

However I am struggling to work out the bounds of the integral for $z,r,\theta$ and also what I am integrating. Please may someone explain the method for working out the limits in this example and what I am integrating? I should be okay to do the integral itself from then on.

I would really appreciate it. Thank you.
 
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  • #2
I think there's some information missing here. What you have so far is an ellipse in $R$ and $z$.
 

FAQ: Vector Calculus - Cylindrical Co-ordinates

What are cylindrical coordinates?

Cylindrical coordinates are a type of coordinate system used in three-dimensional space to describe the position of a point. They consist of a radial distance from the origin, an angle from a fixed reference direction, and a height or z-coordinate.

How are cylindrical coordinates related to Cartesian coordinates?

Cylindrical coordinates are related to Cartesian coordinates through a set of equations. The radial distance, angle, and height can be converted into x, y, and z coordinates using the following equations:
x = rcosθ
y = rsinθ
z = z

What is the gradient in cylindrical coordinates?

The gradient in cylindrical coordinates is a vector that points in the direction of the greatest increase of a scalar function. It is given by the following formula:
∇f = (∂f/∂r)er + (1/r)(∂f/∂θ)eθ + (∂f/∂z)ez
where er, eθ, and ez are the unit vectors in the radial, angular, and height directions, respectively.

How do you find the divergence in cylindrical coordinates?

The divergence in cylindrical coordinates is calculated using the following formula:
∇·F = (1/r)(∂(rFr)/∂r) + (1/r)(∂Fθ/∂θ) + (∂Fz/∂z)
where Fr, Fθ, and Fz are the components of the vector field F in the radial, angular, and height directions, respectively.

What is the curl in cylindrical coordinates?

The curl in cylindrical coordinates is given by the following formula:
∇ × F = (1/r)(∂Fz/∂θ)er + (∂Fr/∂z)eθ + (1/r)(∂(rFθ)/∂r)ez
where Fr, Fθ, and Fz are the components of the vector field F in the radial, angular, and height directions, respectively.

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