# Vector Fields in Cartesian and Cylindrical Coordinates, The Curl

• sriracha
In summary, the necessary information is attached except for the answer in Cartesian coordinates, which is -ix-jy+2kz. The poster used WolframAlpha to convert from cylindrical to Cartesian coordinates and is unsure if their method is correct. To convert the vector, a conversion matrix must be used and the vector must be multiplied by it. The resulting vector in Cartesian coordinates is -ix-jy+2kz.
sriracha
All necessary information is attached except the answer in Cartesian coordinates, which is -ix-jy+2kz and my work converting back from cylindrical to Cartesian, which I used WolframAlpha for, as the trig is a mess (that is, if the way I am doing this is correct).

http://www.wolframalpha.com/input/?i=%28icos%28tan^-1%28y%2Fx%29%29%2Bjsin%28tan^-1%28y%2Fx%29%29%29%28-sqrt%28x^2%2By^2%29%28cos^2%28tan^-1%28y%2Fx%29%29-sin^2%28tan^-1%28y%2Fx%29%29%29%29%2B%28-isin%28tan^-1%28y%2Fx%29%29%2Bjcos%28tan^-1%28y%2Fx%29%29%29%282z%28cos^2%28tan^-1%28y%2Fx%29%29-sin^2%28tan^-1%28y%2Fx%29%29%29%29&a=i_Variable

WolframAlpha, however, just gave me this mess back. Would appreciate if someone would let me know if the way I'm going about this is correct. I have a strong feeling the matrix I'm using for reconversion is off.

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The answer in Cartesian coordinates is -ix-jy+2kz. To convert the vector given in cylindrical coordinates to Cartesian coordinates, you need to use the following conversion matrix:\begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1\end{bmatrix}where $\theta$ is the angle between the vector and the x-axis. Then, multiply the vector in cylindrical coordinates by the matrix to get the vector in Cartesian coordinates. In this case, the vector in cylindrical coordinates is $(r\cos(\theta), r\sin(\theta), z)$. Multiplying this vector by the matrix above gives us the vector in Cartesian coordinates:\begin{bmatrix} -r\sin(\theta) \\ r\cos(\theta) \\ 2z\end{bmatrix}which can be written as -ix-jy+2kz.

## What are vector fields?

A vector field is a mathematical concept that assigns a vector to every point in a given space. It is represented by arrows that indicate the direction and magnitude of the vector at each point.

## What is the difference between Cartesian and cylindrical coordinates?

Cartesian coordinates use a system of x, y, and z axes to locate points in three-dimensional space, while cylindrical coordinates use a combination of a radius, angle, and height to locate points.

## What is the curl of a vector field?

The curl of a vector field measures the tendency of the vectors in the field to rotate around a given point. It is a vector quantity that is perpendicular to the plane formed by the vectors at that point.

## How is the curl of a vector field calculated in Cartesian coordinates?

In Cartesian coordinates, the curl of a vector field is calculated using the determinant of a 3x3 matrix made up of partial derivatives of the vector components with respect to x, y, and z.

## What is the significance of the curl in physics and engineering?

The curl of a vector field is an important concept in physics and engineering as it helps to describe the rotational behavior of a field. It is used in various applications, such as fluid dynamics, electromagnetism, and heat transfer.

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