Which of the following correctly expresses the conversion from rectangular coordinates $(x,y,z)$ to cylindrical coordinates $(r,\theta ,z)$?

• A. $r=x+y,\theta =\arccos \left(\frac{x}{y}\right),z=y$
• B. $r=\sqrt{x^2+y^2},\theta =\arctan \left(\frac{y}{x}\right),z=z$
• C. $r=\sqrt{y^2+z^2},\theta =\arctan \left(\frac{x}{z}\right),z=x$
• D. $r=\sqrt{x^2+z^2},\theta =\arctan \left(\frac{z}{x}\right),z=y$

Answer: (B)

Verified step by step guidance

1. Recall that cylindrical coordinates \((r, \theta, z)\) relate to rectangular coordinates \((x, y, z)\) by expressing the position in terms of a radius \(r\) in the \(xy\)-plane, an angle \(\theta\) from the positive \(x\)-axis, and the same height \(z\).
2. Calculate the radius \(r\) as the distance from the origin to the projection of the point onto the \(xy\)-plane using the Pythagorean theorem: \[r = \sqrt{x^{2} + y^{2}}\]
3. Determine the angle \(\theta\) as the angle between the positive \(x\)-axis and the line connecting the origin to the point's projection in the \(xy\)-plane. This is given by the arctangent of \(y/x\): \[\theta = \arctan\left(\frac{y}{x}\right)\]
4. Note that the height \(z\) remains the same in both coordinate systems, so: \[z = z\]
5. Combine these results to express the conversion from rectangular to cylindrical coordinates as: \[\left(r, \theta, z\right) = \left(\sqrt{x^{2} + y^{2}}, \arctan\left(\frac{y}{x}\right), z\right)\]