# Plot a vector valued function in cylindrical coordinates

• Mathematica
• member 428835
PFIn summary, there is an easier way to plot a function of the form ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z## in Mathematica, using the built-in function VectorPlot3D. This function allows you to define your function as a vector field and plot it in three dimensions, without needing to decompose the components into their ##\hat x,\hat y## components and using ParametricPlot3D.
member 428835
Hi PF!

I have a function ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z##. How can I plot such a thing in Mathematica? Surely there's an easier way than decomposing ##\hat r, \hat \theta## into their ##\hat x,\hat y## components and then using ParametricPlot3D?

Hi there,

Thank you for your question! There is indeed an easier way to plot a function of the form ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z## in Mathematica. You can use the built-in function VectorPlot3D, which allows you to plot vector fields in three dimensions.

To use this function, you will need to define your function as a vector field, with the components r, t, and z as the x, y, and z components respectively. For example, you could define your function as follows:

f[s_, \[Theta]_] := {r[s, \[Theta]], t[s, \[Theta]], z[s, \[Theta]]}

Then, you can use VectorPlot3D to plot the vector field as follows:

VectorPlot3D[f[s, \[Theta]], {s, smin, smax}, {\[Theta], \[Theta]min, \[Theta]max}]

Where smin, smax, \[Theta]min, and \[Theta]max are the minimum and maximum values for s and \[Theta] that you want to plot.

I hope this helps! Let me know if you have any further questions.

## 1. What is a vector valued function in cylindrical coordinates?

A vector valued function in cylindrical coordinates is a mathematical function that maps a set of inputs in the form of cylindrical coordinates (radius, angle, height) to a set of outputs in the form of a vector. This type of function is commonly used in physics and engineering to represent physical quantities that have both magnitude and direction.

## 2. How do you plot a vector valued function in cylindrical coordinates?

To plot a vector valued function in cylindrical coordinates, you first need to define the function in terms of the cylindrical coordinates (r, θ, z). Then, you can plot the function by choosing a range of values for each coordinate and plotting the corresponding vector at each point. Alternatively, you can use software such as Mathematica or MATLAB to plot the function.

## 3. What are some applications of plotting vector valued functions in cylindrical coordinates?

Some common applications of plotting vector valued functions in cylindrical coordinates include analyzing the motion of objects in circular or cylindrical paths, studying fluid flow in pipes or channels, and modeling the magnetic field around a cylindrical wire or solenoid.

## 4. Can you convert a vector valued function from cylindrical coordinates to Cartesian coordinates?

Yes, you can convert a vector valued function from cylindrical coordinates to Cartesian coordinates using the following equations: x = rcos(θ), y = rsin(θ), and z = z. This transformation can be useful when working with vector valued functions in different coordinate systems or when plotting the function in 3D.

## 5. Are there any limitations to plotting vector valued functions in cylindrical coordinates?

One limitation of plotting vector valued functions in cylindrical coordinates is that it can only represent functions with cylindrical symmetry. This means that the function must have the same values at points with the same radius and height, regardless of the angle. Additionally, plotting in cylindrical coordinates can become more complex for higher dimensions and may require advanced mathematical techniques.

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