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Chapter 16: Problem 5

Explain why \(d z r d r d \theta\) is the volume of a small "box" in cylindrical coordinates.

Short Answer

Expert verified
Answer: \(dz\, rdr\,d\theta\) is the volume of a small "box" in cylindrical coordinates because it represents the product of the dimensions in each of the coordinate directions (radial, angular, and vertical) that uniquely define the spatial region of a small box.

Step by step solution

01

Understand the cylindrical coordinate system

In the cylindrical coordinate system, a point in space is represented by the coordinates \((r, \theta, z)\), where \(r\) is the radial distance from the \(z\)-axis, \(\theta\) is the angular position with respect to the positive \(x\)-axis, and \(z\) is the height along the \(z\)-axis. These coordinates provide a way of uniquely identifying a point in space.
02

Visualize a small "box" in cylindrical coordinates

Consider a small region in space which forms a "box" in cylindrical coordinates. It has dimensions \(dr\) in the radial direction, \(r d\theta\) in the angular direction, and \(dz\) in the vertical direction.
03

Calculate the volume of the "box"

To find the volume of the small "box" in cylindrical coordinates, we must multiply the dimensions of the "box" together: \( dV = (dr)(r d\theta)(dz) \)
04

Simplify the expression

Combining the terms from Step 3, we have: \( dV = dz\, rdr\, d\theta \) This expression represents the volume of the small "box" in cylindrical coordinates. In conclusion, \(dz\, rdr\, d\theta\) is the volume of a small "box" in cylindrical coordinates because it represents the product of the dimensions in each of the coordinate directions that uniquely define the spatial region of a small box.

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