Coordinate Conventions for Spherical Integration

Why Coordinate Conventions Matter

Spherical coordinates describe a point using its distance from the origin and two angles. They are useful when a region or an integrand has a natural relationship to distance from the origin or to direction. But the symbols and angle conventions are not universal: different books may use the same Greek letters for different angles.

For a point, the numerical values of its spherical coordinates are meaningful only after the convention has been specified. In an iterated integral, the convention also determines which angular interval describes the region. A correct calculation with the wrong convention can therefore describe a different point and produce incorrect bounds.

The Convention Used Here

Write \(\rho\) for the distance from the origin, \(\phi\) for the polar angle measured down from the positive \(z\)-axis, and \(\theta\) for the azimuthal angle measured in the \(xy\)-plane from the positive \(x\)-axis toward the positive \(y\)-axis.

The radius \(\rho\) is nonnegative. The polar angle \(\phi\) runs from the positive \(z\)-axis through the equatorial plane to the negative \(z\)-axis. The azimuthal angle \(\theta\) makes one full turn around the \(z\)-axis. The endpoint \(2\pi\) is excluded from the standard azimuthal interval because it gives the same direction as \(0\).

From Spherical Coordinates to Cartesian Coordinates

The formulas follow by considering the projection of a point onto the \(xy\)-plane. Let \(r_{\!xy}\) be the distance of that projection from the origin. In the right triangle formed by the point, its projection, and the origin, \(\rho\) is the hypotenuse and \(\phi\) is measured from the vertical side. Therefore \(z=\rho\cos\phi\) and \(r_{\!xy}=\rho\sin\phi\). In the \(xy\)-plane, the azimuthal angle gives the usual relations \(x=r_{\!xy}\cos\theta\) and \(y=r_{\!xy}\sin\theta\). Substitution yields the defining formulas.

From Cartesian Coordinates to Spherical Coordinates

For a Cartesian point \((x,y,z)\), its radius is its distance from the origin. If \(\rho>0\), the polar angle is determined by the vertical component, and the azimuth is determined by the direction of the \(xy\)-projection:

For the azimuth, use the angle in the \(xy\)-plane whose cosine and sine have the signs of \(x\) and \(y\), respectively. Equivalently, \(\theta=\operatorname{atan2}(y,x)\), adjusted to lie in \(0\leq\theta<2\pi\). This quadrant-aware choice matters: \(\arctan(y/x)\) alone does not distinguish all four quadrants.

Coordinate Expressions

The defining formulas also give useful identities. Squaring and adding the expressions for \(x\) and \(y\) removes the azimuthal angle, while including \(z\) gives the squared distance from the origin:

Also, \(z=\rho\cos\phi\) directly. Ratios such as \(x/y=\cot\theta\) require \(y\neq 0\), so using the defining equations is safer when an expression might be undefined on part of a region.

Angle Ranges and Bounds

An angular bound describes a range of directions, not a range of distances from the origin. The polar interval controls how far a direction tilts away from the positive \(z\)-axis; the azimuthal interval controls the direction around that axis. A radial interval controls the distance from the origin. Keeping these roles separate helps prevent bounds from being assigned to the wrong variable.

For example, in the first octant, \(x,y,z\geq0\). The condition \(z\geq0\) corresponds to \(0\leq\phi\leq\pi/2\). The conditions \(x,y\geq0\) place the azimuth in the first quadrant, \(0\leq\theta\leq\pi/2\). These angular ranges do not by themselves set a radial extent: the region might include all positive radii, or it might have a separate radial restriction.

Why the Convention Changes Integral Bounds

In this convention, \(\phi\) measures inclination from the \(z\)-axis and \(\theta\) measures rotation around that axis. Thus a condition about whether a point lies above or below the \(xy\)-plane affects the range of \(\phi\), while a condition restricting directions in the \(xy\)-plane affects the range of \(\theta\). If a reference uses the opposite assignment of symbols, the numerical limits may belong to the other symbol there.

For instance, a direction in the first octant has both angles between \(0\) and \(\pi/2\), so merely seeing those endpoints does not reveal which convention is being used. The defining equations do. Before setting angular bounds, identify the angle definitions, then translate the region's directional conditions using those definitions. Do not transfer a symbol's bounds from a different convention without translating them.

Common Mistake: Swapping the Angles

A frequent error is to use formulas from a convention where \(\theta\) is the polar angle and \(\phi\) is the azimuth, while keeping the present course's meanings and bounds. For the spherical point \((2,\pi/3,\pi/6)\), the correct formulas give \((x,y,z)=(3/2,\sqrt{3}/2,1)\), as checked above.

If the angles are mistakenly swapped in the coordinate formulas, one might calculate \(x=2\sin(\pi/6)\cos(\pi/3)=1/2\), \(y=2\sin(\pi/6)\sin(\pi/3)=\sqrt{3}/2\), and \(z=2\cos(\pi/6)=\sqrt{3}\). That is the different point \((1/2,\sqrt{3}/2,\sqrt{3})\), not the intended point. Its squared distance is \(1/4+3/4+3=4\), so checking only the radius would fail to catch the mistake. The correction is to keep the course's roles fixed: \(\phi\) is polar, \(\theta\) is azimuthal, and the defining equations must be used consistently.

Coordinate Checks Before Using Bounds

Before using spherical coordinates, it is useful to check three things:

• Does \(\rho\) have a nonnegative range and represent distance from the origin?
• Do the formulas use \(\phi\) as the polar angle and \(\theta\) as the azimuth?
• Do the angular intervals match the directions allowed by the region?

These checks are especially important when an interval begins or ends at an axis, when a region crosses an azimuthal wraparound, or when the formulas have been copied from a source using different symbols. Spherical coordinates are not unique on the axis or at the origin, so those exceptional points should not be treated as having uniquely determined angles.