Electromagnetic Theory · Spherical Coordinate System

#04 Definition, Cartesian transformation, and differential elements in spherical coordinates

Trace the spherical metric coefficients r and r sin θ from coordinate transformations through line, surface, and volume elements.

Question

Lesson frame showing spherical r, theta, phi coordinates, transformations, and differential line, surface, and volume elements. — In spherical coordinates the metric coefficient is r along θ and r sin θ along φ.

Define the spherical variables r, θ, φ and their unit vectors; write the Cartesian transformations, position vector, and differential dl, dS, and dV elements with their metric coefficients.

Written solution and narration transcript(shows the full solution)

Below are all the lines written in the notebook together with the full narration transcript.

1. Set up the move to spherical coordinates
In spherical coordinates the metric coefficient is r along θ and r sin θ along φ.
In the previous lesson, we defined the cylindrical coordinate system with coordinates ρ, φ, and z, and derived the differential elements with the metric coefficient ρ.
Now let's move to the third and final coordinate system: spherical coordinates.
We'll define r, θ, and φ, show the transformation to Cartesian, and derive the differential elements dl, dS, and dV.
Narration transcript
In the previous lesson, we defined the cylindrical coordinate system with coordinates rho, phi, and z, and derived the differential elements with the metric coefficient rho. Now let's move to the third and final coordinate system: spherical coordinates. We'll define r, theta, and phi, show the transformation to Cartesian, and derive the differential elements dl, dS, and dV.
2. Define the r, θ, φ coordinates
In spherical coordinates the metric coefficient is r along θ and r sin θ along φ.
The spherical coordinate system uses three variables: r, θ, and φ.
r is the distance from the origin to the point, always non-negative.
Theta is the polar angle, measured from the positive z axis downward, ranging from zero to π.
Phi is the azimuthal angle, measured from the positive x axis in the x y plane, ranging from zero to two π.
At every point, we define three unit vectors: e_r pointing radially outward from the origin, e_θ pointing in the direction of increasing θ, and e_φ pointing in the direction of increasing φ.
These three unit vectors are mutually orthogonal and follow the right-hand rule.
Narration transcript
The spherical coordinate system uses three variables: r, theta, and phi. r is the distance from the origin to the point, always non-negative. Theta is the polar angle, measured from the positive z axis downward, ranging from zero to pi. Phi is the azimuthal angle, measured from the positive x axis in the x y plane, ranging from zero to two pi. At every point, we define three unit vectors: e r pointing radially outward from the origin, e theta pointing in the direction of increasing theta, and e phi pointing in the direction of increasing phi. These three unit vectors are mutually orthogonal and follow the right-hand rule.
3. Write the Cartesian transformations
In spherical coordinates the metric coefficient is r along θ and r sin θ along φ.
To convert between spherical and Cartesian, we use the geometry of the coordinate system.
The x coordinate is x = r sin θ cos φ.
The y coordinate is y = r sin θ sin φ.
The z coordinate is z = r cos θ.
Going the other way, r = √(x² + y² + z²).
$\displaystyle \theta = \arccos \left(\frac{z}{r}\right).$
And φ = arctan(y/x).
The position vector in spherical coordinates is simply r⃗ = r e_r.
This is because e_r already encodes both angular directions.
Narration transcript
To convert between spherical and Cartesian, we use the geometry of the coordinate system. The x coordinate equals r sine theta cosine phi. The y coordinate equals r sine theta sine phi. The z coordinate equals r cosine theta. Going the other way, r equals the square root of x squared plus y squared plus z squared. Theta equals the arccosine of z over r. And phi equals the arctangent of y over x. The position vector in spherical coordinates is simply r equals r e r. This is because e r already encodes both angular directions.
4. Derive the spherical line element
In spherical coordinates the metric coefficient is r along θ and r sin θ along φ.
Now let's derive the differential line element dl.
In spherical coordinates, a small displacement has three components.
In the r direction, a change dr gives dl_r = dr e_r.
In the θ direction, an angular change dθ corresponds to an arc length r dθ, so dl_θ = r dθ e_θ.
The metric coefficient for θ is r.
In the φ direction, the arc length is r sin θ dφ, because the circle of constant θ has radius r sin θ.
So dl_φ = r sin θ dφ e_φ.
The total line element is dl = dr e_r + r dθ e_θ + r sin θ dφ e_φ.
Narration transcript
Now let's derive the differential line element dl. In spherical coordinates, a small displacement has three components. In the r direction, a change dr gives dl r equals dr e r. In the theta direction, an angular change d theta corresponds to an arc length of r d theta, so dl theta equals r d theta e theta. The metric coefficient for theta is r. In the phi direction, the arc length is r sine theta d phi, because the circle of constant theta has radius r sine theta. So dl phi equals r sine theta d phi e phi. The total line element is: dl equals dr e r plus r d theta e theta plus r sine theta d phi e phi.
5. Build the spherical surface elements
In spherical coordinates the metric coefficient is r along θ and r sin θ along φ.
The differential surface element dS combines two differential lengths.
On a surface of constant r, which is a sphere, dS = r² sin θ dθ dφ e_r.
Both angular metric coefficients appear here: r for θ and r sin θ for φ, giving the r² sin θ factor.
On a surface of constant θ, which is a cone, dS = r sin θ dr dφ e_θ.
On a surface of constant φ, which is a half-plane, dS = r dr dθ e_φ.
Notice that r² sin θ appears whenever both angles are involved.
Narration transcript
The differential surface element dS combines two differential lengths. On a surface of constant r, which is a sphere, dS equals r squared sine theta d theta d phi e r. Both angular metric coefficients appear here: r for theta and r sine theta for phi, giving the r squared sine theta factor. On a surface of constant theta, which is a cone, dS equals r sine theta dr d phi e theta. On a surface of constant phi, which is a half-plane, dS equals r dr d theta e phi. Notice that r squared sine theta appears whenever both angles are involved.
6. Derive the spherical volume element
In spherical coordinates the metric coefficient is r along θ and r sin θ along φ.
The differential volume element combines all three differential lengths.
dV = dr × r dθ × r sin θ dφ = r² sin θ dr dθ dφ.
Compared to Cartesian dV = dx dy dz, the spherical volume element has an extra factor r² sin θ.
This makes sense geometrically: elements farther from the origin and closer to the equator sweep out larger volumes.
Narration transcript
The differential volume element combines all three differential lengths. dV equals dr times r d theta times r sine theta d phi, which simplifies to r squared sine theta dr d theta d phi. Compared to the Cartesian dV equals dx dy dz, the spherical volume element has an extra factor of r squared sine theta. This makes sense geometrically: elements farther from the origin and closer to the equator sweep out larger volumes.
7. Summarize the metric coefficients
In spherical coordinates the metric coefficient is r along θ and r sin θ along φ.
Let's review.
In spherical coordinates, the three variables are r, θ, and φ.
The line element is dl = dr e_r + r dθ e_θ + r sin θ dφ e_φ.
The key difference from Cartesian is the two metric coefficients: r for the θ direction and r sin θ for the φ direction.
The volume element is dV = r² sin θ dr dθ dφ.
With Cartesian, cylindrical, and spherical coordinates all defined, we now have the three coordinate systems needed for electromagnetic theory.
Narration transcript
Let's review. In spherical coordinates, the three variables are r, theta, and phi. The line element is dl equals dr e r plus r d theta e theta plus r sine theta d phi e phi. The key difference from Cartesian is the two metric coefficients: r for the theta direction and r sine theta for the phi direction. The volume element is dV equals r squared sine theta dr d theta d phi. With Cartesian, cylindrical, and spherical coordinates all defined, we now have the three coordinate systems needed for electromagnetic theory.