Vector Fields in Cartesian and Cylindrical Coordinates, The Curl

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.
  • #1
sriracha
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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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  • #2
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.
 

Related to Vector Fields in Cartesian and Cylindrical Coordinates, The Curl

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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