Balls And Spherical Shells As Coordinate Regions

From Distance to Coordinate Bounds

A ball centered at the origin contains points whose distance from the origin is no greater than a fixed radius. A spherical shell contains points whose distances lie between two radii. Spherical coordinates describe both regions directly: the coordinate \(\rho\) is the distance from the origin, while \(\theta\) and \(\phi\) specify direction.

The central task is therefore to translate a distance condition into a bound on \(\rho\), and then decide whether the region includes every direction or only selected directions. That distinction lets us describe a complete ball, a hemisphere, an octant, or an angular portion of a ball or shell.

Coordinate convention: Use \(\rho\ge 0\), azimuthal angle \(0\le\theta\le 2\pi\), and polar angle \(0\le\phi\le\pi\), with \(x=\rho\sin\phi\cos\theta\), \(y=\rho\sin\phi\sin\theta\), and \(z=\rho\cos\phi\). The angle \(\theta\) turns around the \(z\)-axis; \(\phi\) is measured down from the positive \(z\)-axis.

At \(\rho=0\), every angle represents the origin. At the poles, where \(\phi=0\) or \(\phi=\pi\), changing \(\theta\) does not change the point. These coordinate repetitions do not prevent us from describing regions; they only mean that a point on an axis or on the angular seam may have more than one coordinate representation.

The Radial Bound for a Ball

For a point \((x,y,z)\), its distance from the origin is \(\sqrt{x^2+y^2+z^2}\). Substituting the spherical-coordinate formulas gives a direct relation between that distance and \(\rho\).

Square and add the coordinates.
Use the formulas for \(x\), \(y\), and \(z\) to calculate \(x^2+y^2+z^2\).

Simplify the angular terms.
Use \(\cos^2\theta+\sin^2\theta=1\) and \(\sin^2\phi+\cos^2\phi=1\).

Translate the distance condition.
Since \(\rho\ge0\), a distance bound becomes a radial bound on \(\rho\).

\begin{aligned} x^2+y^2+z^2 &=\rho^2\sin^2\phi\cos^2\theta +\rho^2\sin^2\phi\sin^2\theta +\rho^2\cos^2\phi\\ &=\rho^2\sin^2\phi(\cos^2\theta+\sin^2\theta) +\rho^2\cos^2\phi\\ &=\rho^2(\sin^2\phi+\cos^2\phi)\\ &=\rho^2. \end{aligned}

This calculation proves the key distance identity: \(x^2+y^2+z^2=\rho^2\). Because \(\rho\) is nonnegative, the distance from the origin is exactly \(\rho\), not \(-\rho\). Thus being within distance \(R\) is equivalent to \(0\le\rho\le R\).

Definition: The closed ball of radius \(R>0\) centered at the origin is the set of points whose distance from the origin is at most \(R\). In spherical coordinates it is described by \(0\le\rho\le R\), \(0\le\theta\le2\pi\), and \(0\le\phi\le\pi\). Its boundary sphere is given by \(\rho=R\) with the full angular ranges.

For the open ball, the radial condition is \(0\le\rho<R\). For the spherical surface alone, \(\rho=R\); this equation does not describe the solid inside the sphere. Keeping the distinction between a surface and a solid is essential when writing coordinate bounds.

Worked Example: A Ball of Radius Four

Describe the closed ball centered at the origin with radius \(4\) in spherical coordinates, and check two points against the description.

The distance condition is \(\sqrt{x^2+y^2+z^2}\le4\). By the distance identity, this is \(0\le\rho\le4\). Every direction is included, so the angular ranges are the full ranges:

0\le\rho\le4,\qquad 0\le\theta\le2\pi,\qquad 0\le\phi\le\pi.

For \((1,2,2)\), the distance is \(\sqrt{1^2+2^2+2^2}=\sqrt9=3\), so the point is inside. Its spherical radius is \(\rho=3\), which satisfies \(0\le\rho\le4\).

For \((4,0,1)\), the distance is \(\sqrt{4^2+0^2+1^2}=\sqrt{17}\). Since \(17>16\), \(\sqrt{17}>4\), so the point is outside. Its radius \(\rho=\sqrt{17}\) fails the bound \(\rho\le4\). Both checks agree with the stated region.

Spherical Shells: An Inner and an Outer Radius

A spherical shell centered at the origin is the set of points between two concentric spheres. If the inner radius is \(a\) and the outer radius is \(b\), with \(0\le a\le b\), the distance from the origin must be at least \(a\) and at most \(b\). Therefore the radial coordinate lies between those same values.

Definition: The closed spherical shell with inner radius \(a\) and outer radius \(b\), centered at the origin, is described by \(a\le\rho\le b\), with full angular ranges when the entire shell is included. Its Cartesian description is \(a^2\le x^2+y^2+z^2\le b^2\).

If \(a=0\), this is a solid ball, not a hollow shell. If \(a=b\), the radial bounds reduce to \(\rho=a\), the spherical surface alone. To exclude either bounding sphere, use a strict inequality at that boundary; for example, \(a<\rho<b\) describes the points strictly between the two surfaces.

Worked Example: The Region Between Radii Two and Five

Describe the closed spherical shell between radii \(2\) and \(5\), then verify whether two given points lie in it.

The radial bounds are \(2\le\rho\le5\), and the full angular ranges include every direction:

2\le\rho\le5,\qquad 0\le\theta\le2\pi,\qquad 0\le\phi\le\pi.

Equivalently, squaring the nonnegative distance bounds gives \(4\le x^2+y^2+z^2\le25\).

For \((1,2,2)\), \(x^2+y^2+z^2=1+4+4=9\), so \(4\le9\le25\). Its radius is \(3\), and \(2\le3\le5\); the point is in the shell.

For \((1,1,1)\), \(x^2+y^2+z^2=1+1+1=3\), which is less than \(4\). Its radius is \(\sqrt3<2\), so the point lies inside the inner sphere and is not in the shell. The Cartesian and spherical checks give the same conclusion.

Restricting the Angles

The radial bound controls how far points extend from the origin. Angular bounds select which directions are allowed. A hemisphere is obtained by restricting the polar angle, while an octant requires restrictions on both angles. The full ball or shell still supplies the radial bounds unless a different radial interval is specified.

For the upper half-space \(z\ge0\), the spherical formula \(z=\rho\cos\phi\) shows that \(\cos\phi\ge0\), which corresponds to \(0\le\phi\le\pi/2\). Thus the upper hemisphere of a ball of radius \(R\) has full azimuth and \(0\le\phi\le\pi/2\). The lower hemisphere instead has \(\pi/2\le\phi\le\pi\).

To describe the first octant, require \(x\ge0\), \(y\ge0\), and \(z\ge0\). In the standard angle ranges, \(x\) and \(y\) are both nonnegative when \(0\le\theta\le\pi/2\), and \(z\ge0\) when \(0\le\phi\le\pi/2\). These bounds describe the first-octant directions, including their boundary planes.

Worked Example: The First-Octant Portion of a Ball

Describe the part of the ball of radius \(3\) in the first octant. Then check whether \((1,1,1)\) belongs to it.

The ball gives \(0\le\rho\le3\). The first-octant conditions give \(0\le\theta\le\pi/2\) and \(0\le\phi\le\pi/2\). Together, the bounds are:

0\le\rho\le3,\qquad 0\le\theta\le\frac{\pi}{2},\qquad 0\le\phi\le\frac{\pi}{2}.

In Cartesian coordinates this is \(x\ge0\), \(y\ge0\), \(z\ge0\), and \(x^2+y^2+z^2\le9\). For \((1,1,1)\), all three coordinates are nonnegative and \(1^2+1^2+1^2=3\le9\), so the point is in the region.

Its spherical radius is \(\rho=\sqrt3\), which satisfies \(0\le\sqrt3\le3\). Its azimuth is \(\theta=\pi/4\), since its positive \(x\)- and \(y\)-coordinates are equal. Its polar angle satisfies \(\cos\phi=z/\rho=1/\sqrt3\), so \(\phi=\arccos(1/\sqrt3)\), which lies between \(0\) and \(\pi/2\). Each coordinate bound is satisfied, confirming the Cartesian check.

A region with a restricted azimuth and polar angle as well as a radial bound is an angular portion of a ball or shell, often called a spherical sector. The radial bounds determine the inner and outer spherical surfaces; the angular bounds select the directions included. For instance, restricting \(\theta\) to a quarter-turn selects a wedge of directions around the \(z\)-axis, while the full polar range allows those directions at every height above and below the \(xy\)-plane.

Worked Example: A Spherical Sector Inside a Ball

Describe the part of the ball of radius \(2\) with \(0\le\theta\le\pi/2\), using all polar angles. Check the point \((1,1,0)\).

The ball imposes \(0\le\rho\le2\). The specified azimuth selects the first-quadrant directions around the \(z\)-axis, and the full polar range includes both positive and negative \(z\)-values:

0\le\rho\le2,\qquad 0\le\theta\le\frac{\pi}{2},\qquad 0\le\phi\le\pi.

In Cartesian terms, the angular restriction gives \(x\ge0\) and \(y\ge0\), while the radial restriction gives \(x^2+y^2+z^2\le4\). For \((1,1,0)\), \(\rho=\sqrt{1+1+0}=\sqrt2\le2\), and \(\theta=\pi/4\) is within the specified interval. Since \(z=0\), its polar angle is \(\phi=\pi/2\), within \([0,\pi]\). The point satisfies all three bounds and belongs to the sector.

A Common Mistake: Forgetting the Inner Radius

When describing a shell, it is not enough to use the outer sphere's bound. For example, the incorrect bounds \(0\le\rho\le5\) describe the entire ball of radius \(5\), including its center and the points inside the intended inner radius. For a shell from radius \(2\) to radius \(5\), the correct bounds are \(2\le\rho\le5\).

The point \((1,1,1)\) makes the error visible. Its radius is \(\sqrt3<2\). The incorrect bounds \(0\le\rho\le5\) include it, but the shell must exclude it. The corrected lower bound \(\rho\ge2\) does exclude it. Always translate both sides of the distance interval into radial bounds.

Another useful check is to translate the proposed radial bounds back into a Cartesian distance inequality. A ball \(0\le\rho\le R\) must give \(x^2+y^2+z^2\le R^2\); a shell \(a\le\rho\le b\) must give \(a^2\le x^2+y^2+z^2\le b^2\). If the inner inequality disappears, the shell has been replaced by a ball.

Putting the Bounds Together

To describe one of these regions, first identify its radial extent, then determine whether every direction is included. Use \(0\le\theta\le2\pi\) and \(0\le\phi\le\pi\) for all directions. Restrict \(\phi\) to choose an upper or lower hemisphere, or restrict both angles to select an octant or another angular portion.

Key takeaway: An origin-centered ball or shell is controlled radially by \(\rho\). A ball uses \(0\le\rho\le R\); a shell uses \(a\le\rho\le b\). Angular bounds then include all directions or select the desired portion of the region.

Check Your Understanding

For each question, identify the radial and angular conditions needed to describe the stated region.

What spherical-coordinate bounds describe the closed ball of radius \(6\) centered at the origin?
What Cartesian inequality corresponds to the shell \(3\le\rho\le7\) with full angular ranges?
Which polar-angle interval describes the lower hemisphere of a ball, and why does it impose \(z\le0\)?
Give spherical-coordinate bounds for the first-octant portion of a shell with inner radius \(1\) and outer radius \(4\).
Why do the bounds \(0\le\rho\le5\) fail to describe the shell whose inner radius is \(2\) and outer radius is \(5\)?